Introduction
Lie triple systems describe the infinitesimal structure of symmetric spaces. Their algebraic structure was developed by Jacobson and Lister [6 , 9 ] . In the finite-dimensional real setting, the correspondence between Lie triple systems and germs of pointed symmetric spaces identifies a triple morphism with the tangent map of a local symmetric-space morphism [11 ] . Thus a deformation of a triple morphism describes a variation of the infinitesimal data of a map preserving the local symmetric-space structure. This raises geometric questions about whether nearby maps are equivalent under inner symmetries, whether infinitesimal variations extend to actual families, and whether a map persists when the structures on its source and target change.
For fixed triple products π \pi π μ \mu μ g \mathfrak{g} g h \mathfrak{h} h Hom ( g , h ) \operatorname{Hom}(\mathfrak{g},\mathfrak{h}) Hom ( g , h ) Mor ( g , h ) \operatorname{Mor}(\mathfrak{g},\mathfrak{h}) Mor ( g , h )
S π , μ ( q ) ( x , y , z ) = μ ( q x , q y , q z ) − q π ( x , y , z ) . S_{\pi,\mu}(q)(x,y,z)=\mu(qx,qy,qz)-q\pi(x,y,z). S π , μ ( q ) ( x , y , z ) = μ ( q x , q y , q z ) − q π ( x , y , z ) . Its local geometry at a morphism f f f f f f π \pi π μ \mu μ
Cohomology gives the infinitesimal description of this geometry. The derivative of the defect at f f f δ f 1 \delta_f^1 δ f 1 Z f 1 = ker δ f 1 Z_f^1=\ker\delta_f^1 Z f 1 = ker δ f 1 B f 1 B_f^1 B f 1 H f 1 = Z f 1 / B f 1 H_f^1=Z_f^1/B_f^1 H f 1 = Z f 1 / B f 1 H f 2 H_f^2 H f 2
2020 Mathematics Subject Classification. 17A40, 17B56.
Key words and phrases. Lie triple system, Lie algebra homomorphism, universal enveloping Lie algebra, cohomology, rigidity, stability.
Yamaguti introduced a cohomology theory intrinsic to Lie triple systems [14 ] . Harris studied a related theory using Lie algebras with involution [5 ] . The deformation theory of triple products was developed by Kubo and Taniguchi [8 ] , and Zhang explained Yamaguti cohomology through Leibniz cohomology [15 ] . Xia, Sheng and Tang constructed a graded Lie algebra that controls Lie triple systems and recovers the Yamaguti differential [13 ] . We use this graded Lie algebra with its last-three-variable symmetry conditions throughout the paper.
The derived bracket method of Voronov [12 ] , together with twisting of L ∞ L_\infty L ∞ [3 ] , gives a convenient description of equations for linear maps. Frégier and Zambon applied this method to simultaneous deformations of algebras and morphisms [2 ] . In the setting of Lie triple systems, Hajjaji constructed the controlling algebra and cohomology of crossed homomorphisms [4 ] . For the zero action, crossed homomorphisms are ordinary morphisms. Thus the Maurer–Cartan description used below is a specialization of that construction. Cohomology for simultaneous deformations of Lie–Yamaguti algebra morphisms was studied by Mondal and Saha [10 ] . The positive-degree complex considered here keeps the two Lie triple systems fixed; varying their products is treated separately as a parameter-dependent stability problem.
Our geometric approach follows the rigidity and stability framework for Lie algebras described by Crainic, Schätz and Struchiner [1 ] and the corresponding study of 3 3 3 [7 ] . Two features of Lie triple systems enter the argument. First, the triple product is skew in its first two arguments and satisfies a cyclic identity. We use the corresponding cochain spaces and coefficient representation. Second, the degree-zero term records inner equivalence. This term gives the first boundary space and enters the comparison with the normal complex of a graph subsystem.
With these conventions, our main results give cohomological criteria for the local structure of the morphism space. If H f 1 = 0 H_f^1=0 H f 1 = 0 f f f f f f f f f H f 2 = 0 H_f^2=0 H f 2 = 0 f f f
T f Mor ( g , h ) = Z f 1 , dim f Mor ( g , h ) = dim Z f 1 . T_f\operatorname{Mor}(\mathfrak{g},\mathfrak{h})=Z_f^1,\qquad\dim_f\operatorname{Mor}(\mathfrak{g},\mathfrak{h})=\dim Z_f^1. T f Mor ( g , h ) = Z f 1 , dim f Mor ( g , h ) = dim Z f 1 . Every first cocycle is then realized as the derivative of a smooth family of morphisms. These assertions identify the local dimension and distinguish orbit directions from the infinitesimal directions measured by H f 1 H_f^1 H f 1 H f 2 H_f^2 H f 2
The stability result describes how this geometry depends on the two triple products. Under the same condition H f 2 = 0 H_f^2=0 H f 2 = 0 π \pi π μ \mu μ f f f f f f f f f
Triple subsystems give a second geometric realization of these results. The k k k E E E Gr k ( E ) \operatorname{Gr}_k(E) Gr k ( E ) s \mathfrak{s} s Hom ( s , E / s ) \operatorname{Hom}(\mathfrak{s},E/\mathfrak{s}) Hom ( s , E / s ) G f = { ( x , f x ) : x ∈ g } G_f=\{(x,f x):x\in\mathfrak{g}\} G f = {( x , f x ) : x ∈ g } f f f G f G_f G f Gr dim g ( g ⊕ h ) \operatorname{Gr}_{\dim\mathfrak{g}}(\mathfrak{g}\oplus\mathfrak{h}) Gr d i m g ( g ⊕ h )
A further purpose of this paper is to relate this local geometry to that of Lie algebra homomorphisms. The universal enveloping Lie algebra of a Lie triple system is an ordinary Lie algebra with a Z 2 \mathbb{Z}_2 Z 2 [6 , 11 ] . We apply this established correspondence to first cohomology and formal deformations. In finite dimension, polynomial extension and linear restriction identify the local morphism spaces and preserve smooth deformations and inner orbits. Local smoothness and rigidity therefore have equivalent interpretations in these two settings. The recognition criterion for universal envelopes also gives stability under small graded Lie bracket perturbations. The Lie equivalences in this comparison are generated by even inner derivations.
There is another comparison when the source and target triple products are induced by prescribed binary Lie brackets through [ x , y , z ] = [ [ x , y ] , z ] [x,y,z]=[[x,y],z] [ x , y , z ] = [[ x , y ] , z ]
The paper is organized as follows. In Sect. 2, we recall the controlling graded Lie algebra and give the Maurer–Cartan characterization, the universal Lie correspondence and the binary defect identity. In Sect. 3, we study augmented cohomology and formal obstructions, and compare them with the Chevalley–Eilenberg theory of Lie homomorphisms. In Sect. 4, we prove local rigidity and stability, together with their graded Lie and binary Lie consequences. In Sect. 5, we study triple subsystems, their normal complexes and their relationship with Lie subalgebra graphs.
Unless otherwise stated, vector spaces are over a field K \mathbb{K} K K \mathbb{K} K R \mathbb{R} R
Derived brackets and the morphism equation
In this section, we recall the graded Lie algebra for Lie triple systems and apply higher derived brackets to the morphism equation. Subsequently, we establish its relationship with graded Lie homomorphisms through universal envelopes. For triple products induced by binary Lie brackets, we compare the two morphism defects and give an injectivity criterion.
Definition 2.1. A Lie triple system is a vector space L L L
[ x , x , y ] = 0 , (1) [x,x,y]=0, \tag*{(1)} [ x , x , y ] = 0 , (1) [ x , y , z ] + [ y , z , x ] + [ z , x , y ] = 0 , (2) [x,y,z]+[y,z,x]+[z,x,y]=0, \tag*{(2)} [ x , y , z ] + [ y , z , x ] + [ z , x , y ] = 0 , (2) [ x , y , [ u , v , w ] ] = [ [ x , y , u ] , v , w ] + [ u , [ x , y , v ] , w ] + [ u , v , [ x , y , w ] ] . (3) [x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]]. \tag*{(3)} [ x , y , [ u , v , w ]] = [[ x , y , u ] , v , w ] + [ u , [ x , y , v ] , w ] + [ u , v , [ x , y , w ]] . (3) A morphism f : ( g , π ) ⟶ ( h , μ ) f:(\mathfrak{g},\pi)\longrightarrow(\mathfrak{h},\mu) f : ( g , π ) ⟶ ( h , μ )
μ ( f x , f y , f z ) = f π ( x , y , z ) ( x , y , z ∈ g ) . (4) \mu(fx,fy,fz)=f\pi(x,y,z)\qquad(x,y,z\in\mathfrak{g}). \tag*{(4)} μ ( f x , f y , f z ) = f π ( x , y , z ) ( x , y , z ∈ g ) . (4) For x , y ∈ L x,y \in L x , y ∈ L ad x ∧ y z = [ x , y , z ] \operatorname{ad}_{x\wedge y}z=[x,y,z] ad x ∧ y z = [ x , y , z ] (3) . These maps span a Lie algebra InnDer ( L ) \operatorname{InnDer}(L) InnDer ( L )
[ ad x ∧ y , ad u ∧ v ] = ad [ x , y , u ] ∧ v + ad u ∧ [ x , y , v ] . (5) [\operatorname{ad}_{x\wedge y},\operatorname{ad}_{u\wedge v}]=\operatorname{ad}_{[x,y,u]\wedge v}+\operatorname{ad}_{u\wedge[x,y,v]}. \tag*{(5)} [ ad x ∧ y , ad u ∧ v ] = ad [ x , y , u ] ∧ v + ad u ∧ [ x , y , v ] . (5) The notation ad X \operatorname{ad}_{X} ad X X ∈ ∧ 2 L X \in\wedge^{2}L X ∈ ∧ 2 L ∧ 2 L → InnDer ( L ) \wedge^{2}L \to\operatorname{InnDer}(L) ∧ 2 L → InnDer ( L )
For vector spaces L , V L,V L , V C 0 ( L , V ) = Hom ( L , V ) C^{0}(L,V)=\operatorname{Hom}(L,V) C 0 ( L , V ) = Hom ( L , V ) p ≥ 1 p \ge1 p ≥ 1 C p ( L , V ) C^{p}(L,V) C p ( L , V )
P : ( L ⊗ L ) ⊗ p ⊗ L ⟶ V P:(L\otimes L)^{\otimes p}\otimes L\longrightarrow V P : ( L ⊗ L ) ⊗ p ⊗ L ⟶ V that satisfy
P ( X 1 , … , X p − 1 , x , x , y ) = 0 , (6) P(X_{1},\ldots,X_{p-1},x,x,y)=0, \tag*{(6)} P ( X 1 , … , X p − 1 , x , x , y ) = 0 , (6) P ( X 1 , … , X p − 1 , x , y , z ) + P ( X 1 , … , X p − 1 , y , z , x ) + P ( X 1 , … , X p − 1 , z , x , y ) = 0. (7) P(X_{1},\ldots,X_{p-1},x,y,z)+P(X_{1},\ldots,X_{p-1},y,z,x)+P(X_{1},\ldots,X_{p-1},z,x,y)=0. \tag*{(7)} P ( X 1 , … , X p − 1 , x , y , z ) + P ( X 1 , … , X p − 1 , y , z , x ) + P ( X 1 , … , X p − 1 , z , x , y ) = 0. (7) Here X i ∈ L ⊗ L X_i \in L\otimes L X i ∈ L ⊗ L [13 ] . The cochain symmetries are exactly (6) –(7) , imposed on the last three entries; the preceding arguments remain tensor pairs.
We use the following insertion convention. If P ∈ C p ( L , L ) P \in C^{p}(L,L) P ∈ C p ( L , L ) Q ∈ C q ( L , L ) Q \in C^{q}(L,L) Q ∈ C q ( L , L ) X i = x i ⊗ y i X_i=x_i\otimes y_i X i = x i ⊗ y i 1 ≤ k ≤ p 1\le k\le p 1 ≤ k ≤ p
( P ∘ k Q ) ( X 1 , … , X p + q , z ) = ∑ σ ∈ Sh ( k − 1 , q ) sgn ( σ ) P ( X σ ( 1 ) , … , X σ ( k − 1 ) , Q ( X σ ( k ) , … , X σ ( k + q − 1 ) , x k + q ) ⊗ y k + q , X k + q + 1 , … , X p + q , z ) + ∑ σ ∈ Sh ( k − 1 , q ) sgn ( σ ) P ( X σ ( 1 ) , … , X σ ( k − 1 ) , x k + q ⊗ Q ( X σ ( k ) , … , X σ ( k + q − 1 ) , y k + q ) , X k + q + 1 , … , X p + q , z ) . (8) \begin{aligned}
(P\circ_k Q)(X_1,\ldots,X_{p+q},z)
={}&\sum_{\sigma\in\operatorname{Sh}(k-1,q)}\operatorname{sgn}(\sigma)P(X_{\sigma(1)},\ldots,X_{\sigma(k-1)},\\
&Q(X_{\sigma(k)},\ldots,X_{\sigma(k+q-1)},x_{k+q})\otimes y_{k+q},X_{k+q+1},\ldots,X_{p+q},z)\\
&+\sum_{\sigma\in\operatorname{Sh}(k-1,q)}\operatorname{sgn}(\sigma)P(X_{\sigma(1)},\ldots,X_{\sigma(k-1)},\\
&x_{k+q}\otimes Q(X_{\sigma(k)},\ldots,X_{\sigma(k+q-1)},y_{k+q}),X_{k+q+1},\ldots,X_{p+q},z). \tag*{(8)}
\end{aligned} ( P ∘ k Q ) ( X 1 , … , X p + q , z ) = σ ∈ Sh ( k − 1 , q ) ∑ sgn ( σ ) P ( X σ ( 1 ) , … , X σ ( k − 1 ) , Q ( X σ ( k ) , … , X σ ( k + q − 1 ) , x k + q ) ⊗ y k + q , X k + q + 1 , … , X p + q , z ) + σ ∈ Sh ( k − 1 , q ) ∑ sgn ( σ ) P ( X σ ( 1 ) , … , X σ ( k − 1 ) , x k + q ⊗ Q ( X σ ( k ) , … , X σ ( k + q − 1 ) , y k + q ) , X k + q + 1 , … , X p + q , z ) . (8) In this formula an unshuffle permutes the first k + q − 1 k+q-1 k + q − 1
( P ∘ p + 1 Q ) ( X 1 , … , X p + q , z ) = ∑ σ ∈ Sh ( p , q ) sgn ( σ ) P ( X σ ( 1 ) , … , X σ ( p ) , Q ( X σ ( p + 1 ) , … , X σ ( p + q ) , z ) ) . (9) (P\circ_{p+1}Q)(X_1,\ldots,X_{p+q},z)
=\sum_{\sigma\in\operatorname{Sh}(p,q)}\operatorname{sgn}(\sigma)P(X_{\sigma(1)},\ldots,X_{\sigma(p)},Q(X_{\sigma(p+1)},\ldots,X_{\sigma(p+q)},z)). \tag*{(9)} ( P ∘ p + 1 Q ) ( X 1 , … , X p + q , z ) = σ ∈ Sh ( p , q ) ∑ sgn ( σ ) P ( X σ ( 1 ) , … , X σ ( p ) , Q ( X σ ( p + 1 ) , … , X σ ( p + q ) , z )) . (9) Empty strings of arguments are omitted. Put
P ∘ Q = ∑ k = 1 p + 1 ( − 1 ) ( k − 1 ) q P ∘ k Q , [ P , Q ] B = P ∘ Q − ( − 1 ) p q Q ∘ P . (10) P\circ Q=\sum_{k=1}^{p+1}(-1)^{(k-1)q}P\circ_k Q,\qquad[P,Q]_{\mathrm B}=P\circ Q-(-1)^{pq}Q\circ P. \tag*{(10)} P ∘ Q = k = 1 ∑ p + 1 ( − 1 ) ( k − 1 ) q P ∘ k Q , [ P , Q ] B = P ∘ Q − ( − 1 ) pq Q ∘ P . (10) Proposition 2.2 ([13 ] ). The bracket (10) makes C ∙ ( L , L ) C^{\bullet}(L,L) C ∙ ( L , L ) a graded Lie algebra. A map π ∈ C 1 ( L , L ) \pi\in C^{1}(L,L) π ∈ C 1 ( L , L ) is a Lie triple system product if and only if [ π , π ] B = 0 [\pi,\pi]_{\mathrm B}=0 [ π , π ] B = 0
For a degree-one cochain satisfying (6) –(7) , the Maurer–Cartan equation is precisely (3) .
We use the symmetric convention for L ∞ [ 1 ] L_{\infty}[1] L ∞ [ 1 ] ℓ k \ell_k ℓ k 1 1 1
∑ i = 1 n ∑ σ ∈ Sh ( i , n − i ) ϵ ( σ ) ℓ n − i + 1 ( ℓ i ( a σ ( 1 ) , … , a σ ( i ) ) , a σ ( i + 1 ) , … , a σ ( n ) ) = 0. \sum_{i=1}^{n}\sum_{\sigma\in\operatorname{Sh}(i,n-i)}\epsilon(\sigma)\ell_{n-i+1}(\ell_i(a_{\sigma(1)},\ldots,a_{\sigma(i)}),a_{\sigma(i+1)},\ldots,a_{\sigma(n)})=0. i = 1 ∑ n σ ∈ Sh ( i , n − i ) ∑ ϵ ( σ ) ℓ n − i + 1 ( ℓ i ( a σ ( 1 ) , … , a σ ( i ) ) , a σ ( i + 1 ) , … , a σ ( n ) ) = 0. Here ϵ ( σ ) \epsilon(\sigma) ϵ ( σ ) a a a ∑ k ≥ 1 ℓ k ( a , … , a ) / k ! = 0 \sum_{k\ge1}\ell_k(a,\ldots,a)/k!=0 ∑ k ≥ 1 ℓ k ( a , … , a ) / k ! = 0
Let G = g ⊕ h G=\mathfrak{g}\oplus\mathfrak{h} G = g ⊕ h L = C ∙ ( G , G ) \mathcal{L}=C^{\bullet}(G,G) L = C ∙ ( G , G ) a ∈ C p ( g , h ) a\in C^{p}(\mathfrak{g},\mathfrak{h}) a ∈ C p ( g , h ) G G G g \mathfrak{g} g h \mathfrak{h} h a ^ \hat a a ^ h \mathfrak{h} h
A = ⨁ p ≥ 0 C p ( g , h ) \mathcal{A}=\bigoplus_{p\ge0}C^{p}(\mathfrak{g},\mathfrak{h}) A = p ≥ 0 ⨁ C p ( g , h ) with the resulting subspace of L \mathcal{L} L P : L → A P:\mathcal{L}\to\mathcal{A} P : L → A g \mathfrak{g} g h \mathfrak{h} h
Δ ( ( x , u ) , ( y , v ) , ( z , w ) ) = ( π ( x , y , z ) , μ ( u , v , w ) ) . \Delta((x,u),(y,v),(z,w))=(\pi(x,y,z),\mu(u,v,w)). Δ (( x , u ) , ( y , v ) , ( z , w )) = ( π ( x , y , z ) , μ ( u , v , w )) . Proposition 2.3. The quadruple ( L , A , P , Δ ) (\mathcal{L},\mathcal{A},P,\Delta) ( L , A , P , Δ )
ℓ 1 ( a ) = P [ π , a ^ ] B , (11) \ell_1(a)=P[\pi,\hat{a}]_{\mathrm{B}}, \tag*{(11)} ℓ 1 ( a ) = P [ π , a ^ ] B , (11) ℓ 3 ( a , b , c ) = P [ [ [ μ , a ^ ] B , b ^ ] B , c ^ ] B , (12) \ell_3(a,b,c)=P[[[\mu,\hat{a}]_{\mathrm{B}},\hat{b}]_{\mathrm{B}},\hat{c}]_{\mathrm{B}}, \tag*{(12)} ℓ 3 ( a , b , c ) = P [[[ μ , a ^ ] B , b ^ ] B , c ^ ] B , (12) ℓ 2 = 0 , ℓ k = 0 ( k ≥ 4 ) . (13) \ell_2=0,\qquad\ell_k=0\quad(k\geq4). \tag*{(13)} ℓ 2 = 0 , ℓ k = 0 ( k ≥ 4 ) . (13) They define an L ∞ [ 1 ] L_\infty[1] L ∞ [ 1 ] A \mathcal{A} A
Proof. The insertion of an element of A \mathcal{A} A A \mathcal{A} A h \mathfrak{h} h h \mathfrak{h} h A \mathcal{A} A ker P \ker P ker P g \mathfrak{g} g g \mathfrak{g} g ker P \ker P ker P
[ π + μ , π + μ ] B = [ π , π ] B + 2 [ π , μ ] B + [ μ , μ ] B = 0 , P ( π + μ ) = 0. [\pi+\mu,\pi+\mu]_{\mathrm{B}}=[\pi,\pi]_{\mathrm{B}}+2[\pi,\mu]_{\mathrm{B}}+[\mu,\mu]_{\mathrm{B}}=0,\qquad P(\pi+\mu)=0. [ π + μ , π + μ ] B = [ π , π ] B + 2 [ π , μ ] B + [ μ , μ ] B = 0 , P ( π + μ ) = 0. Voronov’s theorem [12 ] therefore gives
ℓ k ( a 1 , … , a k ) = P [ ⋯ [ [ π + μ , a ^ 1 ] B , a ^ 2 ] B , … , a ^ k ] B . \ell_k(a_1,\ldots,a_k)=P[\cdots[[\pi+\mu,\hat{a}_1]_{\mathrm{B}},\hat{a}_2]_{\mathrm{B}},\ldots,\hat{a}_k]_{\mathrm{B}}. ℓ k ( a 1 , … , a k ) = P [ ⋯ [[ π + μ , a ^ 1 ] B , a ^ 2 ] B , … , a ^ k ] B . For the source product, [ π , a ^ 1 ] B [\pi,\hat{a}_1]_{\mathrm{B}} [ π , a ^ 1 ] B g \mathfrak{g} g h \mathfrak{h} h μ \mu μ A \mathcal{A} A μ \mu μ
Theorem 2.4. A linear map f : g → h f:\mathfrak{g}\to\mathfrak{h} f : g → h ( A , ℓ 1 , ℓ 3 ) (\mathcal{A},\ell_1,\ell_3) ( A , ℓ 1 , ℓ 3 )
Proof. For x , y , z ∈ g x,y,z\in\mathfrak{g} x , y , z ∈ g
ℓ 1 ( f ) ( x , y , z ) = − f π ( x , y , z ) , ℓ 3 ( f , f , f ) ( x , y , z ) = 6 μ ( f x , f y , f z ) . \ell_1(f)(x,y,z)=-f\pi(x,y,z),\qquad\ell_3(f,f,f)(x,y,z)=6\mu(fx,fy,fz). ℓ 1 ( f ) ( x , y , z ) = − f π ( x , y , z ) , ℓ 3 ( f , f , f ) ( x , y , z ) = 6 μ ( f x , f y , f z ) . Consequently, the Maurer–Cartan equation evaluated at x , y , z x,y,z x , y , z
− f π ( x , y , z ) + 1 6 μ ( f x , f y , f z ) = 0. -f\pi(x,y,z)+\frac{1}{6}\mu(fx,fy,fz)=0. − f π ( x , y , z ) + 6 1 μ ( f x , f y , f z ) = 0. This is precisely the morphism identity (4) . □
If f f f [3 ] are
ℓ 1 f ( a ) = ℓ 1 ( a ) + 1 2 ℓ 3 ( f , f , a ) , ℓ 2 f ( a , b ) = ℓ 3 ( f , a , b ) , ℓ 3 f = ℓ 3 . (14) \ell_1^f(a)=\ell_1(a)+\frac{1}{2}\ell_3(f,f,a),\qquad\ell_2^f(a,b)=\ell_3(f,a,b),\qquad\ell_3^f=\ell_3. \tag*{(14)} ℓ 1 f ( a ) = ℓ 1 ( a ) + 2 1 ℓ 3 ( f , f , a ) , ℓ 2 f ( a , b ) = ℓ 3 ( f , a , b ) , ℓ 3 f = ℓ 3 . (14) Consequently, f + a f+a f + a
ℓ 1 f ( a ) + 1 2 ℓ 2 f ( a , a ) + 1 6 ℓ 3 f ( a , a , a ) = 0. (15) \ell_1^f(a)+\frac{1}{2}\ell_2^f(a,a)+\frac{1}{6}\ell_3^f(a,a,a)=0. \tag*{(15)} ℓ 1 f ( a ) + 2 1 ℓ 2 f ( a , a ) + 6 1 ℓ 3 f ( a , a , a ) = 0. (15) For linear maps a , b a,b a , b
ℓ 2 f ( a , b ) ( x , y , z ) = μ ( a x , b y , f z ) + μ ( b x , a y , f z ) + μ ( a x , f y , b z ) + μ ( b x , f y , a z ) + μ ( f x , a y , b z ) + μ ( f x , b y , a z ) . \ell_2^f(a,b)(x,y,z)=\mu(ax,by,fz)+\mu(bx,ay,fz)+\mu(ax,fy,bz)+\mu(bx,fy,az)
+\mu(fx,ay,bz)+\mu(fx,by,az). ℓ 2 f ( a , b ) ( x , y , z ) = μ ( a x , b y , f z ) + μ ( b x , a y , f z ) + μ ( a x , f y , b z ) + μ ( b x , f y , a z ) + μ ( f x , a y , b z ) + μ ( f x , b y , a z ) . For three linear maps a 1 , a 2 , a 3 a_1,a_2,a_3 a 1 , a 2 , a 3
ℓ 3 ( a 1 , a 2 , a 3 ) ( x , y , z ) = ∑ σ ∈ S 3 μ ( a σ ( 1 ) x , a σ ( 2 ) y , a σ ( 3 ) z ) . \ell_3(a_1,a_2,a_3)(x,y,z)=\sum_{\sigma\in S_3}\mu(a_{\sigma(1)}x,a_{\sigma(2)}y,a_{\sigma(3)}z). ℓ 3 ( a 1 , a 2 , a 3 ) ( x , y , z ) = σ ∈ S 3 ∑ μ ( a σ ( 1 ) x , a σ ( 2 ) y , a σ ( 3 ) z ) . For any linear map q q q
S π , μ ( q ) ( x , y , z ) = μ ( q x , q y , q z ) − q π ( x , y , z ) , (16) S_{\pi,\mu}(q)(x,y,z)=\mu(qx,qy,qz)-q\pi(x,y,z), \tag*{(16)} S π , μ ( q ) ( x , y , z ) = μ ( q x , q y , q z ) − q π ( x , y , z ) , (16) B π , μ , q ( b ) = ℓ 1 ( b ) + 1 2 ℓ 3 ( q , q , b ) , b ∈ C 1 ( g , h ) . (17) B_{\pi,\mu,q}(b)=\ell_1(b)+\frac{1}{2}\ell_3(q,q,b),\qquad b\in C^1(\mathfrak{g},\mathfrak{h}). \tag*{(17)} B π , μ , q ( b ) = ℓ 1 ( b ) + 2 1 ℓ 3 ( q , q , b ) , b ∈ C 1 ( g , h ) . (17) The value of B π , μ , q B_{\pi,\mu,q} B π , μ , q C 2 ( g , h ) C^2(\mathfrak{g},\mathfrak{h}) C 2 ( g , h )
Lemma 2.5. For every q ∈ Hom ( g , h ) q\in\operatorname{Hom}(\mathfrak{g},\mathfrak{h}) q ∈ Hom ( g , h )
B π , μ , q ( S π , μ ( q ) ) = 0. (18) B_{\pi,\mu,q}\left(S_{\pi,\mu}(q)\right)=0. \tag*{(18)} B π , μ , q ( S π , μ ( q ) ) = 0. (18) Proof. The extension of q q q h \mathfrak{h} h q ^ 2 = 0 \widehat{q}^{\,2}=0 q 2 = 0 ( id + q ^ ) − 1 = id − q ^ (\operatorname{id}+\widehat{q})^{-1}=\operatorname{id}-\widehat{q} ( id + q ) − 1 = id − q id + q ^ \operatorname{id}+\widehat{q} id + q
( id − q ^ ) △ ( id + q ^ , id + q ^ , id + q ^ ) . (\operatorname{id}-\widehat{q})\mathbin{\triangle}(\operatorname{id}+\widehat{q},\operatorname{id}+\widehat{q},\operatorname{id}+\widehat{q}). ( id − q ) △ ( id + q , id + q , id + q ) . Its value on ( x , u ) , ( y , v ) , ( z , w ) (x,u),(y,v),(z,w) ( x , u ) , ( y , v ) , ( z , w )
( π ( x , y , z ) , μ ( u + q x , v + q y , w + q z ) − q π ( x , y , z ) ) . \left(\pi(x,y,z),\mu(u+qx,v+qy,w+qz)-q\pi(x,y,z)\right). ( π ( x , y , z ) , μ ( u + q x , v + q y , w + q z ) − q π ( x , y , z ) ) . Conjugation preserves the Lie triple identities. Since ker P \ker P ker P A \mathcal{A} A
0 = 2 P [ ( id − q ^ ) △ ( id + q ^ , id + q ^ , id + q ^ ) , μ ( q ⋅ , q ⋅ , q ⋅ ) − q π ( ⋅ , ⋅ , ⋅ ) ‾ ] B . 0=2P\left[\left(\operatorname{id}-\widehat{q}\right)\mathbin{\triangle}\left(\operatorname{id}+\widehat{q},\operatorname{id}+\widehat{q},\operatorname{id}+\widehat{q}\right),\overline{\mu(q\cdot,q\cdot,q\cdot)-q\pi(\cdot,\cdot,\cdot)}\right]_{\mathrm{B}}. 0 = 2 P [ ( id − q ) △ ( id + q , id + q , id + q ) , μ ( q ⋅ , q ⋅ , q ⋅ ) − q π ( ⋅ , ⋅ , ⋅ ) ] B . For every b ∈ C 1 ( g , h ) b\in C^1(\mathfrak{g},\mathfrak{h}) b ∈ C 1 ( g , h )
P [ ( id − q ^ ) △ ( id + q ^ , id + q ^ , id + q ^ ) , b ^ ] B = P [ π , b ^ ] B + 1 2 P [ [ [ μ , q ^ ] B , q ^ ] B , b ^ ] B . P\left[\left(\operatorname{id}-\widehat{q}\right)\mathbin{\triangle}\left(\operatorname{id}+\widehat{q},\operatorname{id}+\widehat{q},\operatorname{id}+\widehat{q}\right),\widehat{b}\right]_{\mathrm{B}}
=P[\pi,\widehat{b}]_{\mathrm{B}}+\frac{1}{2}P[[[\mu,\widehat{q}]_{\mathrm{B}},\widehat{q}]_{\mathrm{B}},\widehat{b}]_{\mathrm{B}}. P [ ( id − q ) △ ( id + q , id + q , id + q ) , b ] B = P [ π , b ] B + 2 1 P [[[ μ , q ] B , q ] B , b ] B . Substituting the full trilinear expression μ ( q ⋅ , q ⋅ , q ⋅ ) − q π ( ⋅ , ⋅ , ⋅ ) \mu(q\cdot,q\cdot,q\cdot)-q\pi(\cdot,\cdot,\cdot) μ ( q ⋅ , q ⋅ , q ⋅ ) − q π ( ⋅ , ⋅ , ⋅ )
We now establish the relationship with Lie algebra homomorphisms. Two constructions will be used. The universal enveloping Lie algebra applies to every Lie triple system, whereas an iterated binary bracket gives a comparison with a prescribed pair of Lie algebras.
For a Lie triple system L L L F ( L ) \mathcal{F}(L) F ( L )
U ( L ) = F ( L ) / ⟨ [ [ x , y ] , z ] − [ x , y , z ] : x , y , z ∈ L ⟩ . (19) \mathcal{U}(L)=\mathcal{F}(L)\big/\langle[\![x,y],z]-[x,y,z]:x,y,z\in L\rangle. \tag*{(19)} U ( L ) = F ( L ) / ⟨[ [ x , y ] , z ] − [ x , y , z ] : x , y , z ∈ L ⟩ . (19) Here [ x , y , z ] [x,y,z] [ x , y , z ] L L L Z 2 \mathbb{Z}_2 Z 2 Z 2 \mathbb{Z}_2 Z 2 [6 , 11 ] .
Proposition 2.6. The generating map L → U ( L ) L\to\mathcal{U}(L) L → U ( L )
U ( L ) 1 = L , U ( L ) 0 = [ L , L ] . (20) \mathcal{U}(L)_1=L,\qquad\mathcal{U}(L)_0=[L,L]. \tag*{(20)} U ( L ) 1 = L , U ( L ) 0 = [ L , L ] . (20) Every morphism f : g → h f:\mathfrak{g}\to\mathfrak{h} f : g → h
U ( f ) : U ( g ) ⟶ U ( h ) , U ( f ) x = f x , U ( f ) [ x , y ] = [ f x , f y ] . (21) \mathcal{U}(f):\mathcal{U}(\mathfrak{g})\longrightarrow\mathcal{U}(\mathfrak{h}),\qquad\mathcal{U}(f)x=fx,\qquad\mathcal{U}(f)[x,y]=[fx,fy]. \tag*{(21)} U ( f ) : U ( g ) ⟶ U ( h ) , U ( f ) x = f x , U ( f ) [ x , y ] = [ f x , f y ] . (21) Conversely, restriction to the odd component gives a Lie triple system morphism. Thus these two operations give a bijection between the two morphism spaces.
Proof. The injectivity and the decomposition in (2.20) are the universal embedding theorem. For x , y , z ∈ g x,y,z\in\mathfrak{g} x , y , z ∈ g f f f
[ [ f x , f y ] , f z ] − f [ x , y , z ] . [[fx,fy],fz]-f[x,y,z]. [[ f x , f y ] , f z ] − f [ x , y , z ] . In U ( h ) \mathcal{U}(\mathfrak{h}) U ( h )
[ f x , f y , f z ] − f [ x , y , z ] , [fx,fy,fz]-f[x,y,z], [ f x , f y , f z ] − f [ x , y , z ] , which vanishes exactly when f f f
Conversely, for a grading-preserving Lie homomorphism from U ( g ) \mathcal{U}(\mathfrak{g}) U ( g ) U ( h ) \mathcal{U}(\mathfrak{h}) U ( h ) [ [ x , y ] , z ] [[x,y],z] [[ x , y ] , z ] [ [ x , y ] , z ] = [ x , y , z ] [[x,y],z]=[x,y,z] [[ x , y ] , z ] = [ x , y , z ]
Proposition 2.6 extends every Lie triple morphism through the universal envelopes. For the standard envelope InnDer ( L ) ⊕ L \operatorname{InnDer}(L)\oplus L InnDer ( L ) ⊕ L ad x ∧ y ↦ ad f x ∧ f y \operatorname{ad}_{x\wedge y}\mapsto\operatorname{ad}_{f x\wedge f y} ad x ∧ y ↦ ad f x ∧ f y
Next, let a \mathfrak{a} a b \mathfrak{b} b a t r \mathfrak{a}^{\mathrm{tr}} a tr b t r \mathfrak{b}^{\mathrm{tr}} b tr
π ( x , y , z ) = [ [ x , y ] a , z ] a , μ ( u , v , w ) = [ [ u , v ] b , w ] b . \pi(x,y,z)=[[x,y]_{\mathfrak{a}},z]_{\mathfrak{a}},\qquad\mu(u,v,w)=[[u,v]_{\mathfrak{b}},w]_{\mathfrak{b}}. π ( x , y , z ) = [[ x , y ] a , z ] a , μ ( u , v , w ) = [[ u , v ] b , w ] b . For a linear map q : a → b q:\mathfrak{a}\to\mathfrak{b} q : a → b
F ( q ) ( x , y ) = [ q x , q y ] b − q [ x , y ] a . (22) F(q)(x,y)=[qx,qy]_{\mathfrak{b}}-q[x,y]_{\mathfrak{a}}. \tag*{(22)} F ( q ) ( x , y ) = [ q x , q y ] b − q [ x , y ] a . (22) ( T q b ) ( x , y , z ) = [ b ( x , y ) , q z ] b + b ( [ x , y ] a , z ) , (23) (T_qb)(x,y,z)=[b(x,y),qz]_{\mathfrak{b}}+b([x,y]_{\mathfrak{a}},z), \tag*{(23)} ( T q b ) ( x , y , z ) = [ b ( x , y ) , q z ] b + b ([ x , y ] a , z ) , (23) where b : ⋀ 2 a → b b:\bigwedge^{2}\mathfrak{a}\to\mathfrak{b} b : ⋀ 2 a → b T q T_q T q Hom ( a ⊗ 3 , b ) \operatorname{Hom}(\mathfrak{a}^{\otimes3},\mathfrak{b}) Hom ( a ⊗ 3 , b )
Proposition 2.7. For every linear map q : a → b q:\mathfrak{a}\to\mathfrak{b} q : a → b we have
S π , μ ( q ) = T q F ( q ) . (24) S_{\pi,\mu}(q)=T_qF(q). \tag*{(24)} S π , μ ( q ) = T q F ( q ) . (24) Consequently, every Lie algebra homomorphism is a morphism of the associated Lie triple systems. The converse does not hold in general.
Proof. For every x , y , z ∈ a x,y,z\in\mathfrak{a} x , y , z ∈ a
[ [ q x , q y ] b , q z ] b − q [ [ x , y ] a , z ] a = [ [ q x , q y ] b − q [ x , y ] a , q z ] b + [ q [ x , y ] a , q z ] b − q [ [ x , y ] a , z ] a . [[qx,qy]_{\mathfrak{b}},qz]_{\mathfrak{b}}-q[[x,y]_{\mathfrak{a}},z]_{\mathfrak{a}}=[[qx,qy]_{\mathfrak{b}}-q[x,y]_{\mathfrak{a}},qz]_{\mathfrak{b}}+[q[x,y]_{\mathfrak{a}},qz]_{\mathfrak{b}}-q[[x,y]_{\mathfrak{a}},z]_{\mathfrak{a}}. [[ q x , q y ] b , q z ] b − q [[ x , y ] a , z ] a = [[ q x , q y ] b − q [ x , y ] a , q z ] b + [ q [ x , y ] a , q z ] b − q [[ x , y ] a , z ] a . The right-hand side is the full expression obtained from (2.23) by substituting [ q ⋅ , q ⋅ ] b − q [ ⋅ , ⋅ ] a [q\cdot,q\cdot]_{\mathfrak{b}}-q[\cdot,\cdot]_{\mathfrak{a}} [ q ⋅ , q ⋅ ] b − q [ ⋅ , ⋅ ] a
On any nonabelian Lie algebra one has
[ [ − x , − y ] , − z ] = − [ [ x , y ] , z ] , [ − x , − y ] = [ x , y ] ≠ − [ x , y ] [[-x,-y],-z]=-[[x,y],z],\qquad[-x,-y]=[x,y]\ne-[x,y] [[ − x , − y ] , − z ] = − [[ x , y ] , z ] , [ − x , − y ] = [ x , y ] = − [ x , y ] for some x , y x,y x , y − id -\operatorname{id} − id
The following condition will give a converse near a fixed homomorphism. Write
C b ( f ( a ) ) = { v ∈ b : [ v , f x ] b = 0 for all x ∈ a } . C_{\mathfrak{b}}(f(\mathfrak{a}))=\{v\in\mathfrak{b}:[v,fx]_{\mathfrak{b}}=0\text{ for all }x\in\mathfrak{a}\}. C b ( f ( a )) = { v ∈ b : [ v , f x ] b = 0 for all x ∈ a } . Lemma 2.8. Let f : a → b f:\mathfrak{a}\to\mathfrak{b} f : a → b be a Lie algebra homomorphism. If
[ a , a ] = a , C b ( f ( a ) ) = 0 , (25) [\mathfrak{a},\mathfrak{a}]=\mathfrak{a},\qquad C_{\mathfrak{b}}(f(\mathfrak{a}))=0, \tag*{(25)} [ a , a ] = a , C b ( f ( a )) = 0 , (25) then T f T_f T f is injective on Hom ( ⋀ 2 a , b ) \operatorname{Hom}(\bigwedge^{2}\mathfrak{a},\mathfrak{b}) Hom ( ⋀ 2 a , b )
Proof. Suppose b ∈ Hom ( ⋀ 2 a , b ) b\in\operatorname{Hom}(\bigwedge^{2}\mathfrak{a},\mathfrak{b}) b ∈ Hom ( ⋀ 2 a , b )
[ b ( x , y ) , f z ] b + b ( [ x , y ] a , z ) = 0 ( x , y , z ∈ a ) . [b(x,y),fz]_{\mathfrak{b}}+b([x,y]_{\mathfrak{a}},z)=0\qquad(x,y,z\in\mathfrak{a}). [ b ( x , y ) , f z ] b + b ([ x , y ] a , z ) = 0 ( x , y , z ∈ a ) . If ∑ i [ x i , y i ] a = 0 \sum_i[x_i,y_i]_{\mathfrak{a}}=0 ∑ i [ x i , y i ] a = 0
[ ∑ i b ( x i , y i ) , f z ] b = 0 ( z ∈ a ) . \left[\sum_i b(x_i,y_i),fz\right]_{\mathfrak{b}}=0\qquad(z\in\mathfrak{a}). [ i ∑ b ( x i , y i ) , f z ] b = 0 ( z ∈ a ) . The zero-centralizer assumption implies ∑ i b ( x i , y i ) = 0 \sum_i b(x_i,y_i)=0 ∑ i b ( x i , y i ) = 0 b b b
Applying this observation to the Jacobi identity gives
b ( [ u , v ] a , z ) = b ( u , [ v , z ] a ) − b ( v , [ u , z ] a ) = − [ f u , b ( v , z ) ] b + [ f v , b ( u , z ) ] b . b([u,v]_{\mathfrak{a}},z)=b(u,[v,z]_{\mathfrak{a}})-b(v,[u,z]_{\mathfrak{a}})=-[fu,b(v,z)]_{\mathfrak{b}}+[fv,b(u,z)]_{\mathfrak{b}}. b ([ u , v ] a , z ) = b ( u , [ v , z ] a ) − b ( v , [ u , z ] a ) = − [ f u , b ( v , z ) ] b + [ f v , b ( u , z ) ] b . Here the second equality follows from the displayed kernel equation and skewness of b b b a \mathfrak{a} a z ∈ a z \in\mathfrak{a} z ∈ a z = ∑ i [ x i , y i ] a z = \sum_i [x_i,y_i]_{\mathfrak{a}} z = ∑ i [ x i , y i ] a
b ( u , z ) = − [ f u , ∑ i b ( x i , y i ) ] b b(u,z)=-\left[f u,\sum_i b(x_i,y_i)\right]_{\mathfrak{b}} b ( u , z ) = − [ f u , i ∑ b ( x i , y i ) ] b Substituting this expression twice in the preceding Jacobi identity yields
b ( [ u , v ] a , z ) = [ f u , [ f v , ∑ i b ( x i , y i ) ] b ] b − [ f v , [ f u , ∑ i b ( x i , y i ) ] b ] b = [ f [ u , v ] a , ∑ i b ( x i , y i ) ] b . b([u,v]_{\mathfrak{a}},z)=\left[f u,\left[f v,\sum_i b(x_i,y_i)\right]_{\mathfrak{b}}\right]_{\mathfrak{b}}-\left[f v,\left[f u,\sum_i b(x_i,y_i)\right]_{\mathfrak{b}}\right]_{\mathfrak{b}}=\left[f[u,v]_{\mathfrak{a}},\sum_i b(x_i,y_i)\right]_{\mathfrak{b}}. b ([ u , v ] a , z ) = [ f u , [ f v , i ∑ b ( x i , y i ) ] b ] b − [ f v , [ f u , i ∑ b ( x i , y i ) ] b ] b = [ f [ u , v ] a , i ∑ b ( x i , y i ) ] b . On the other hand, the formula for b ( u , z ) b(u,z) b ( u , z ) u u u [ u , v ] a [u,v]_{\mathfrak{a}} [ u , v ] a
b ( [ u , v ] a , z ) = − [ f [ u , v ] a , ∑ i b ( x i , y i ) ] b . b([u,v]_{\mathfrak{a}},z)=-\left[f[u,v]_{\mathfrak{a}},\sum_i b(x_i,y_i)\right]_{\mathfrak{b}}. b ([ u , v ] a , z ) = − [ f [ u , v ] a , i ∑ b ( x i , y i ) ] b . Since the characteristic is zero, these two equalities imply
[ f [ u , v ] a , ∑ i b ( x i , y i ) ] b = 0. \left[f[u,v]_{\mathfrak{a}},\sum_i b(x_i,y_i)\right]_{\mathfrak{b}}=0. [ f [ u , v ] a , i ∑ b ( x i , y i ) ] b = 0. Perfectness now shows that ∑ i b ( x i , y i ) \sum_i b(x_i,y_i) ∑ i b ( x i , y i ) f ( a ) f(\mathfrak{a}) f ( a ) b ( u , z ) b(u,z) b ( u , z ) u , z u,z u , z b = 0 b=0 b = 0
Cohomology and formal deformations
In this section, we define the augmented cohomology of a Lie triple system morphism and study formal deformations. We then compare first cohomology and formal equivalence with those of the universal graded Lie homomorphism. Finally, for morphisms induced by binary Lie brackets, we compare the Chevalley–Eilenberg differentials and obstruction classes.
Definition 3.1. [14 , 13 ] A representation of a Lie triple system L L L V V V θ : L ⊗ L → End ( V ) \theta:L\otimes L\to\operatorname{End}(V) θ : L ⊗ L → End ( V ) D ( x , y ) = θ ( y , x ) − θ ( x , y ) D(x,y)=\theta(y,x)-\theta(x,y) D ( x , y ) = θ ( y , x ) − θ ( x , y )
[ D ( x , y ) , θ ( u , v ) ] = θ ( [ x , y , u ] , v ) + θ ( u , [ x , y , v ] ) . (26) [D(x,y),\theta(u,v)]=\theta([x,y,u],v)+\theta(u,[x,y,v]). \tag*{(26)} [ D ( x , y ) , θ ( u , v )] = θ ([ x , y , u ] , v ) + θ ( u , [ x , y , v ]) . (26) θ ( x , [ u , v , w ] ) = θ ( v , w ) θ ( x , u ) − θ ( u , w ) θ ( x , v ) + D ( u , v ) θ ( x , w ) . (27) \theta(x,[u,v,w])=\theta(v,w)\theta(x,u)-\theta(u,w)\theta(x,v)+D(u,v)\theta(x,w). \tag*{(27)} θ ( x , [ u , v , w ]) = θ ( v , w ) θ ( x , u ) − θ ( u , w ) θ ( x , v ) + D ( u , v ) θ ( x , w ) . (27) Equivalently, the following product makes L ⊕ V L\oplus V L ⊕ V
[ ( x , a ) , ( y , b ) , ( z , c ) ] = [ x , y , z ] , θ ( y , z ) a − θ ( x , z ) b + D ( x , y ) c . (28) [(x,a),(y,b),(z,c)]=[x,y,z],\theta(y,z)a-\theta(x,z)b+D(x,y)c. \tag*{(28)} [( x , a ) , ( y , b ) , ( z , c )] = [ x , y , z ] , θ ( y , z ) a − θ ( x , z ) b + D ( x , y ) c . (28) Proposition 3.2. A morphism f : ( g , π ) → ( h , μ ) f:(\mathfrak{g},\pi)\to(\mathfrak{h},\mu) f : ( g , π ) → ( h , μ ) h \mathfrak{h} h
θ f ( x , y ) u = μ ( u , f x , f y ) , D f ( x , y ) u = μ ( f x , f y , u ) . (29) \theta_f(x,y)u=\mu(u,fx,fy),\qquad D_f(x,y)u=\mu(fx,fy,u). \tag*{(29)} θ f ( x , y ) u = μ ( u , f x , f y ) , D f ( x , y ) u = μ ( f x , f y , u ) . (29) Proof. The adjoint representation of h \mathfrak{h} h θ a d ( u , v ) w = μ ( w , u , v ) \theta_{\mathrm{ad}}(u,v)w=\mu(w,u,v) θ ad ( u , v ) w = μ ( w , u , v ) [14 , 13 ] . For x , y , u , v ∈ g x,y,u,v\in\mathfrak{g} x , y , u , v ∈ g w ∈ h w\in\mathfrak{h} w ∈ h
μ ( f x , f y , μ ( w , f u , f v ) ) − μ ( μ ( f x , f y , w ) , f u , f v ) = μ ( w , μ ( f x , f y , f u ) , f v ) + μ ( w , f u , μ ( f x , f y , f v ) ) = μ ( w , f π ( x , y , u ) , f v ) + μ ( w , f u , f π ( x , y , v ) ) . \begin{aligned}
\mu(fx,fy,\mu(w,fu,fv))-\mu(\mu(fx,fy,w),fu,fv)
&=\mu(w,\mu(fx,fy,fu),fv)+\mu(w,fu,\mu(fx,fy,fv))\\
&=\mu(w,f\pi(x,y,u),fv)+\mu(w,fu,f\pi(x,y,v)).
\end{aligned} μ ( f x , f y , μ ( w , f u , f v )) − μ ( μ ( f x , f y , w ) , f u , f v ) = μ ( w , μ ( f x , f y , f u ) , f v ) + μ ( w , f u , μ ( f x , f y , f v )) = μ ( w , f π ( x , y , u ) , f v ) + μ ( w , f u , f π ( x , y , v )) . This is (3.1) for θ f \theta_f θ f x , u , v , z ∈ g x,u,v,z\in\mathfrak{g} x , u , v , z ∈ g w ∈ h w\in\mathfrak{h} w ∈ h
μ ( w , f x , f π ( u , v , z ) ) = μ ( w , f x , μ ( f u , f v , f z ) ) = μ ( μ ( w , f x , f u ) , f v , f z ) − μ ( μ ( w , f x , f v ) , f u , f z ) + μ ( f u , f v , μ ( w , f x , f z ) ) . \begin{aligned}
\mu(w,fx,f\pi(u,v,z))
&=\mu(w,fx,\mu(fu,fv,fz))\\
&=\mu(\mu(w,fx,fu),fv,fz)-\mu(\mu(w,fx,fv),fu,fz)\\
&\quad+\mu(fu,fv,\mu(w,fx,fz)).
\end{aligned} μ ( w , f x , f π ( u , v , z )) = μ ( w , f x , μ ( f u , f v , f z )) = μ ( μ ( w , f x , f u ) , f v , f z ) − μ ( μ ( w , f x , f v ) , f u , f z ) + μ ( f u , f v , μ ( w , f x , f z )) . This is (27) . Finally, skewness in the first pair and the cyclic identity imply
μ ( u , f y , f x ) − μ ( u , f x , f y ) = μ ( f x , f y , u ) , \mu(u,fy,fx)-\mu(u,fx,fy)=\mu(fx,fy,u), μ ( u , f y , f x ) − μ ( u , f x , f y ) = μ ( f x , f y , u ) , which gives the stated formula for D f ( x , y ) u D_f(x,y)u D f ( x , y ) u
Put
C f 0 = ⋀ 2 g ⊕ ⋀ 2 h , C f n = C n − 1 ( g , h ) ( n ≥ 1 ) . (30) C_f^0=\bigwedge^2\mathfrak{g}\oplus\bigwedge^2\mathfrak{h},\qquad C_f^n=C^{n-1}(\mathfrak{g},\mathfrak{h})\quad(n\geq1). \tag*{(30)} C f 0 = ⋀ 2 g ⊕ ⋀ 2 h , C f n = C n − 1 ( g , h ) ( n ≥ 1 ) . (30) Thus C f 1 = Hom ( g , h ) C_f^1=\operatorname{Hom}(\mathfrak{g},\mathfrak{h}) C f 1 = Hom ( g , h ) C f 2 C_f^2 C f 2 (30) is by cohomological degree. In the traditional odd-arity notation, C f n C_f^n C f n C Y 2 n − 1 ( g , h f ) C_Y^{2n-1}(\mathfrak{g},\mathfrak{h}_f) C Y 2 n − 1 ( g , h f ) h f \mathfrak{h}_f h f (29) .
Define
( δ f 0 ( X , U ) ) ( z ) = ad U f z − f ad X z , (31) (\delta_f^0(X,U))(z)=\operatorname{ad}_U fz-f\operatorname{ad}_X z, \tag*{(31)} ( δ f 0 ( X , U )) ( z ) = ad U f z − f ad X z , (31) δ f n a = ( − 1 ) n − 1 ℓ 1 f ( a ) ( n ≥ 1 ) . (32) \delta_f^n a=(-1)^{n-1}\ell_1^f(a)\quad(n\geq1). \tag*{(32)} δ f n a = ( − 1 ) n − 1 ℓ 1 f ( a ) ( n ≥ 1 ) . (32) To state the full formula, for X i = x i ⊗ y i X_i=x_i\otimes y_i X i = x i ⊗ y i
[ X i , X j ] π = π ( x i , y i , x j ) ⊗ y j + x j ⊗ π ( x i , y i , y j ) . [X_i,X_j]_\pi=\pi(x_i,y_i,x_j)\otimes y_j+x_j\otimes\pi(x_i,y_i,y_j). [ X i , X j ] π = π ( x i , y i , x j ) ⊗ y j + x j ⊗ π ( x i , y i , y j ) . For a ∈ C f n a\in C_f^n a ∈ C f n
( δ f n a ) ( X 1 , … , X n , z ) = ∑ 1 ≤ i < j ≤ n ( − 1 ) i a ( X 1 , … , X i ^ , … , X j − 1 , [ X i , X j ] π , X j + 1 , … , X n , z ) + ∑ i = 1 n ( − 1 ) i a ( X 1 , … , X i ^ , … , X n , π ( x i , y i , z ) ) + ∑ i = 1 n ( − 1 ) i + 1 μ ( f x i , f y i , a ( X 1 , … , X i ^ , … , X n , z ) ) + ( − 1 ) n + 1 { μ ( a ( X 1 , … , X n − 1 , x n ) , f y n , f z ) + μ ( f x n , a ( X 1 , … , X n − 1 , y n ) , f z ) } . (33) \begin{aligned}
(\delta_f^n a)(X_1,\ldots,X_n,z)
={}&\sum_{1\leq i<j\leq n}(-1)^i a(X_1,\ldots,\widehat{X_i},\ldots,X_{j-1},[X_i,X_j]_\pi,X_{j+1},\ldots,X_n,z)\\
&+\sum_{i=1}^{n}(-1)^i a(X_1,\ldots,\widehat{X_i},\ldots,X_n,\pi(x_i,y_i,z))\\
&+\sum_{i=1}^{n}(-1)^{i+1}\mu(fx_i,fy_i,a(X_1,\ldots,\widehat{X_i},\ldots,X_n,z))\\
&+(-1)^{n+1}\{\mu(a(X_1,\ldots,X_{n-1},x_n),f y_n,fz)+\mu(fx_n,a(X_1,\ldots,X_{n-1},y_n),fz)\}. \tag*{(33)}
\end{aligned} ( δ f n a ) ( X 1 , … , X n , z ) = 1 ≤ i < j ≤ n ∑ ( − 1 ) i a ( X 1 , … , X i , … , X j − 1 , [ X i , X j ] π , X j + 1 , … , X n , z ) + i = 1 ∑ n ( − 1 ) i a ( X 1 , … , X i , … , X n , π ( x i , y i , z )) + i = 1 ∑ n ( − 1 ) i + 1 μ ( f x i , f y i , a ( X 1 , … , X i , … , X n , z )) + ( − 1 ) n + 1 { μ ( a ( X 1 , … , X n − 1 , x n ) , f y n , f z ) + μ ( f x n , a ( X 1 , … , X n − 1 , y n ) , f z )} . (33) The hat indicates omission. Formula (33) is the Yamaguti differential for (29) . The second endpoint term is obtained from − θ f ( x n , z ) a ( … , y n ) -\theta_f(x_n,z)a(\ldots,y_n) − θ f ( x n , z ) a ( … , y n )
In the first two degrees, (33) reads
( δ f 1 a ) ( x , y , z ) = μ ( a x , f y , f z ) + μ ( f x , a y , f z ) + μ ( f x , f y , a z ) − a π ( x , y , z ) , (34) (\delta_f^1a)(x,y,z)=\mu(ax,fy,fz)+\mu(fx,ay,fz)+\mu(fx,fy,az)-a\pi(x,y,z), \tag*{(34)} ( δ f 1 a ) ( x , y , z ) = μ ( a x , f y , f z ) + μ ( f x , a y , f z ) + μ ( f x , f y , a z ) − aπ ( x , y , z ) , (34) ( δ f 2 b ) ( x , y , u , v , w ) = μ ( f x , f y , b ( u , v , w ) ) − μ ( f u , f v , b ( x , y , w ) ) − μ ( b ( x , y , u ) , f v , f w ) − μ ( f u , b ( x , y , v ) , f w ) + b ( x , y , π ( u , v , w ) ) − b ( u , v , π ( x , y , w ) ) − b ( π ( x , y , u ) , v , w ) − b ( u , π ( x , y , v ) , w ) . (35) \begin{aligned}
(\delta_f^2b)(x,y,u,v,w)={}&\mu(fx,fy,b(u,v,w))-\mu(fu,fv,b(x,y,w))\\
&-\mu(b(x,y,u),fv,fw)-\mu(fu,b(x,y,v),fw)\\
&+b(x,y,\pi(u,v,w))-b(u,v,\pi(x,y,w))\\
&-b(\pi(x,y,u),v,w)-b(u,\pi(x,y,v),w). \tag*{(35)}
\end{aligned} ( δ f 2 b ) ( x , y , u , v , w ) = μ ( f x , f y , b ( u , v , w )) − μ ( f u , f v , b ( x , y , w )) − μ ( b ( x , y , u ) , f v , f w ) − μ ( f u , b ( x , y , v ) , f w ) + b ( x , y , π ( u , v , w )) − b ( u , v , π ( x , y , w )) − b ( π ( x , y , u ) , v , w ) − b ( u , π ( x , y , v ) , w ) . (35) Theorem 3.3. The maps (31) –(32) define a cochain complex. In positive degrees this is the Yamaguti complex for h f \mathfrak{h}_f h f (30) .
Proof. Insertion in (32) gives (33) , which preserves the cochain symmetries by Proposition 2.2 .
The unary identity of the twisted L ∞ [ 1 ] L_\infty[1] L ∞ [ 1 ] n ≥ 1 n\geq1 n ≥ 1 a ∈ C f n a\in C_f^n a ∈ C f n
δ f n + 1 ( δ f n a ) = ( − 1 ) n ( − 1 ) n − 1 ℓ 1 f ( ℓ 1 f ( a ) ) = 0. \delta_f^{n+1}(\delta_f^n a)=(-1)^n(-1)^{n-1}\ell_1^f(\ell_1^f(a))=0. δ f n + 1 ( δ f n a ) = ( − 1 ) n ( − 1 ) n − 1 ℓ 1 f ( ℓ 1 f ( a )) = 0. For degree zero, take X ∈ ⋀ 2 g X\in\bigwedge^2\mathfrak{g} X ∈ ⋀ 2 g U ∈ ⋀ 2 h U\in\bigwedge^2\mathfrak{h} U ∈ ⋀ 2 h ad U \operatorname{ad}_U ad U
( δ f 1 ( ad U ∘ f ) ) ( x , y , z ) = μ ( ad U f x , f y , f z ) + μ ( f x , ad U f y , f z ) + μ ( f x , f y , ad U f z ) − ad U f π ( x , y , z ) = ad U μ ( f x , f y , f z ) − ad U f π ( x , y , z ) = 0. \begin{aligned}
(\delta_f^1(\operatorname{ad}_U\circ f))(x,y,z)
={}&\mu(\operatorname{ad}_U fx,fy,fz)+\mu(fx,\operatorname{ad}_U fy,fz)\\
&+\mu(fx,fy,\operatorname{ad}_U fz)-\operatorname{ad}_U f\pi(x,y,z)\\
={}&\operatorname{ad}_U\mu(fx,fy,fz)-\operatorname{ad}_U f\pi(x,y,z)=0.
\end{aligned} ( δ f 1 ( ad U ∘ f )) ( x , y , z ) = = μ ( ad U f x , f y , f z ) + μ ( f x , ad U f y , f z ) + μ ( f x , f y , ad U f z ) − ad U f π ( x , y , z ) ad U μ ( f x , f y , f z ) − ad U f π ( x , y , z ) = 0. Similarly, since ad X \operatorname{ad}_{X} ad X
( δ f 1 ( f ∘ ad X ) ) ( x , y , z ) = f ( π ( ad X x , y , z ) + π ( x , ad X y , z ) + π ( x , y , ad X z ) − ad X π ( x , y , z ) ) = 0. \left(\delta^{1}_{f}(f\circ\operatorname{ad}_{X})\right)(x,y,z)=f\left(\pi(\operatorname{ad}_{X}x,y,z)+\pi(x,\operatorname{ad}_{X}y,z)+\pi(x,y,\operatorname{ad}_{X}z)-\operatorname{ad}_{X}\pi(x,y,z)\right)=0. ( δ f 1 ( f ∘ ad X ) ) ( x , y , z ) = f ( π ( ad X x , y , z ) + π ( x , ad X y , z ) + π ( x , y , ad X z ) − ad X π ( x , y , z ) ) = 0. Subtracting these two identities proves δ f 1 ( δ f 0 ( X , U ) ) = 0 \delta^{1}_{f}(\delta^{0}_{f}(X,U))=0 δ f 1 ( δ f 0 ( X , U )) = 0
Define
Z f n = ker δ f n , B f n = im δ f n − 1 , H f n = Z f n / B f n ( n ≥ 1 ) . Z^{n}_{f}=\ker\delta^{n}_{f},\qquad B^{n}_{f}=\operatorname{im}\delta^{n-1}_{f},\qquad H^{n}_{f}=Z^{n}_{f}/B^{n}_{f}\quad(n\ge1). Z f n = ker δ f n , B f n = im δ f n − 1 , H f n = Z f n / B f n ( n ≥ 1 ) . We augment the positive-degree Yamaguti complex, which starts with Hom ( g , h ) \operatorname{Hom}(\mathfrak{g},\mathfrak{h}) Hom ( g , h ) δ f 0 \delta^{0}_{f} δ f 0
H f 1 = Z Y 1 ( g , h f ) / im δ f 0 , H f n ≅ H Y 2 n − 1 ( g , h f ) ( n ≥ 2 ) . (36) H^{1}_{f}=Z^{1}_{Y}(\mathfrak{g},\mathfrak{h}_{f})/\operatorname{im}\delta^{0}_{f},\qquad H^{n}_{f}\cong H^{2n-1}_{Y}(\mathfrak{g},\mathfrak{h}_{f})\quad(n\ge2). \tag*{(36)} H f 1 = Z Y 1 ( g , h f ) / im δ f 0 , H f n ≅ H Y 2 n − 1 ( g , h f ) ( n ≥ 2 ) . (36) In particular, the second group used below is the odd-arity third Yamaguti cohomology group, whereas the first group has an additional quotient.
Lemma 3.4. The first coboundaries can be obtained using target inner derivations alone:
B f 1 = { ad U ∘ f : U ∈ ∧ 2 h } . (37) B^{1}_{f}=\{\operatorname{ad}_{U}\circ f:U\in\wedge^{2}\mathfrak{h}\}. \tag*{(37)} B f 1 = { ad U ∘ f : U ∈ ∧ 2 h } . (37) Proof. For every X ∈ ∧ 2 g X\in\wedge^{2}\mathfrak{g} X ∈ ∧ 2 g x ∈ g x\in\mathfrak{g} x ∈ g
f ad X x = ad ( ∧ 2 f ) X f x . f\operatorname{ad}_{X}x=\operatorname{ad}_{(\wedge^{2}f)X}fx. f ad X x = ad ( ∧ 2 f ) X f x . Consequently, for U ∈ ∧ 2 h U\in\wedge^{2}\mathfrak{h} U ∈ ∧ 2 h
( δ f 0 ( X , U ) ) ( x ) = ad U f x − ad ( ∧ 2 f ) X f x = ad U − ( ∧ 2 f ) X f x . \left(\delta^{0}_{f}(X,U)\right)(x)=\operatorname{ad}_{U}fx-\operatorname{ad}_{(\wedge^{2}f)X}fx=\operatorname{ad}_{U-(\wedge^{2}f)X}fx. ( δ f 0 ( X , U ) ) ( x ) = ad U f x − ad ( ∧ 2 f ) X f x = ad U − ( ∧ 2 f ) X f x . Thus every first coboundary is obtained by postcomposition with a target inner derivation. Conversely, δ f 0 ( 0 , U ) = ad U ∘ f \delta^{0}_{f}(0,U)=\operatorname{ad}_{U}\circ f δ f 0 ( 0 , U ) = ad U ∘ f U ∈ ∧ 2 h U\in\wedge^{2}\mathfrak{h} U ∈ ∧ 2 h
There is also a natural pullback of positive cochains from the adjoint complex of h \mathfrak{h} h b ∈ C n − 1 ( h , h ) b\in C^{n-1}(\mathfrak{h},\mathfrak{h}) b ∈ C n − 1 ( h , h )
( f ∗ b ) ( x 1 , … , x 2 n − 1 ) = b ( f x 1 , … , f x 2 n − 1 ) . (f^{*}b)(x_{1},\ldots,x_{2n-1})=b(fx_{1},\ldots,fx_{2n-1}). ( f ∗ b ) ( x 1 , … , x 2 n − 1 ) = b ( f x 1 , … , f x 2 n − 1 ) . Substitution in (33) gives δ f ( f ∗ b ) = f ∗ ( δ ad b ) \delta_{f}(f^{*}b)=f^{*}(\delta_{\operatorname{ad}}b) δ f ( f ∗ b ) = f ∗ ( δ ad b ) f f f f ∗ f^{*} f ∗ H f n H^{n}_{f} H f n n ≥ 2 n\ge2 n ≥ 2 H f 1 H^{1}_{f} H f 1 (36) .
Definition 3.5. A formal deformation of f f f
f t = f + ∑ r ≥ 1 t r f r ∈ Hom ( g , h ) [ [ t ] ] f_{t}=f+\sum_{r\ge1}t^{r}f_{r}\in\operatorname{Hom}(\mathfrak{g},\mathfrak{h})[[t]] f t = f + r ≥ 1 ∑ t r f r ∈ Hom ( g , h ) [[ t ]] that satisfies the morphism identity over K [ [ t ] ] \mathbb{K}[[t]] K [[ t ]] N N N t N + 1 t^{N+1} t N + 1
f t ′ = B t f t A t − 1 , A 0 = id g , B 0 = id h , f'_{t}=B_{t}f_{t}A_{t}^{-1},\qquad A_{0}=\operatorname{id}_{\mathfrak{g}},\qquad B_{0}=\operatorname{id}_{\mathfrak{h}}, f t ′ = B t f t A t − 1 , A 0 = id g , B 0 = id h , where A t , B t A_{t},B_{t} A t , B t t t t
Proposition 3.6. The coefficient f 1 f_{1} f 1 Z f 1 Z^{1}_{f} Z f 1 Z f 1 Z^{1}_{f} Z f 1 K [ t ] / ( t 2 ) \mathbb{K}[t]/(t^{2}) K [ t ] / ( t 2 ) H f 1 H^{1}_{f} H f 1
Proof. The coefficient of t t t f t f_t f t
μ ( f 1 x , f y , f z ) + μ ( f x , f 1 y , f z ) + μ ( f x , f y , f 1 z ) − f 1 π ( x , y , z ) = 0. \mu(f_1x,f_y,f_z)+\mu(f_x,f_1y,f_z)+\mu(f_x,f_y,f_1z)-f_1\pi(x,y,z)=0. μ ( f 1 x , f y , f z ) + μ ( f x , f 1 y , f z ) + μ ( f x , f y , f 1 z ) − f 1 π ( x , y , z ) = 0. Hence f 1 ∈ Z f 1 f_1\in Z_f^1 f 1 ∈ Z f 1 f t ′ = B t f t A t − 1 f'_t=B_tf_tA_t^{-1} f t ′ = B t f t A t − 1 t t t
f 1 ′ = f 1 + d B t d t ∣ t = 0 ∘ f − f ∘ d A t d t ∣ t = 0 . f'_1=f_1+\left.\frac{dB_t}{dt}\right|_{t=0}\circ f-f\circ\left.\frac{dA_t}{dt}\right|_{t=0}. f 1 ′ = f 1 + d t d B t t = 0 ∘ f − f ∘ d t d A t t = 0 . The two derivatives are inner derivations by the definition of formal inner equivalence. Formula (31) therefore shows that f 1 ′ − f 1 ∈ B f 1 f'_1-f_1\in B_f^1 f 1 ′ − f 1 ∈ B f 1 f + t f 1 f+tf_1 f + t f 1 t 2 t^2 t 2 Lemma 3.4 , every first coboundary has the form ad U ∘ f \operatorname{ad}_U\circ f ad U ∘ f U ∈ ∧ 2 h U\in\wedge^2\mathfrak{h} U ∈ ∧ 2 h exp ( t ad U ) \exp(t\operatorname{ad}_U) exp ( t ad U ) t 2 t^2 t 2
Theorem 3.7. Let f t ( N ) = ∑ i = 0 N t i f i f_t^{(N)}=\sum_{i=0}^{N}t^if_i f t ( N ) = ∑ i = 0 N t i f i f 0 = f f_0=f f 0 = f N ≥ 1 N\geq1 N ≥ 1
O N + 1 ( x , y , z ) = ∑ i + j + k = N + 1 0 ≤ i , j , k ≤ N μ ( f i x , f j y , f k z ) . (38) O_{N+1}(x,y,z)=\sum_{\substack{i+j+k=N+1\\0\leq i,j,k\leq N}}\mu(f_ix,f_jy,f_kz). \tag*{(38)} O N + 1 ( x , y , z ) = i + j + k = N + 1 0 ≤ i , j , k ≤ N ∑ μ ( f i x , f j y , f k z ) . (38) Then O N + 1 ∈ Z f 2 O_{N+1}\in Z_f^2 O N + 1 ∈ Z f 2 N + 1 N+1 N + 1 [ O N + 1 ] = 0 [O_{N+1}]=0 [ O N + 1 ] = 0 H f 2 H_f^2 H f 2
Proof. Since the morphism identity holds through order N N N
μ ( ∑ i = 0 N t i f i x , ∑ j = 0 N t j f j y , ∑ k = 0 N t k f k z ) − ∑ i = 0 N t i f i π ( x , y , z ) = t N + 1 ∑ i + j + k = N + 1 0 ≤ i , j , k ≤ N μ ( f i x , f j y , f k z ) + O ( t N + 2 ) . \mu\left(\sum_{i=0}^{N}t^if_ix,\sum_{j=0}^{N}t^jf_jy,\sum_{k=0}^{N}t^kf_kz\right)-\sum_{i=0}^{N}t^if_i\pi(x,y,z)=t^{N+1}\sum_{\substack{i+j+k=N+1\\0\leq i,j,k\leq N}}\mu(f_ix,f_jy,f_kz)+O(t^{N+2}). μ ( i = 0 ∑ N t i f i x , j = 0 ∑ N t j f j y , k = 0 ∑ N t k f k z ) − i = 0 ∑ N t i f i π ( x , y , z ) = t N + 1 i + j + k = N + 1 0 ≤ i , j , k ≤ N ∑ μ ( f i x , f j y , f k z ) + O ( t N + 2 ) . The left-hand side is skew in x , y x,y x , y x , y , z x,y,z x , y , z t N + 1 t^{N+1} t N + 1 C f 2 C_f^2 C f 2
Apply Lemma 2.5 over K [ [ t ] ] \mathbb{K}[[t]] K [[ t ]] ∑ i = 0 N t i f i \sum_{i=0}^{N}t^if_i ∑ i = 0 N t i f i t N + 1 t^{N+1} t N + 1 (35) gives, for x , y , u , v , w ∈ g x,y,u,v,w\in\mathfrak{g} x , y , u , v , w ∈ g
0 = ∑ i + j + k = N + 1 0 ≤ i , j , k ≤ N { μ ( f x , f y , μ ( f i u , f j v , f k w ) ) − μ ( f u , f v , μ ( f i x , f j y , f k w ) ) − μ ( μ ( f i x , f j y , f k u ) , f v , f w ) − μ ( f u , μ ( f i x , f j y , f k v ) , f w ) + μ ( f i x , f j y , f k π ( u , v , w ) ) − μ ( f i u , f j v , f k π ( x , y , w ) ) − μ ( f i π ( x , y , u ) , f j v , f k w ) − μ ( f i u , f j π ( x , y , v ) , f k w ) } . \begin{aligned}
0={}&\sum_{\substack{i+j+k=N+1\\0\leq i,j,k\leq N}}\bigl\{\mu(f_x,f_y,\mu(f_iu,f_jv,f_kw))\\
&-\mu(f_u,f_v,\mu(f_ix,f_jy,f_kw))-\mu(\mu(f_ix,f_jy,f_ku),f_v,f_w)\\
&-\mu(f_u,\mu(f_ix,f_jy,f_kv),f_w)+\mu(f_ix,f_jy,f_k\pi(u,v,w))\\
&-\mu(f_iu,f_jv,f_k\pi(x,y,w))-\mu(f_i\pi(x,y,u),f_jv,f_kw)\\
&-\mu(f_iu,f_j\pi(x,y,v),f_kw)\bigr\}.
\end{aligned} 0 = i + j + k = N + 1 0 ≤ i , j , k ≤ N ∑ { μ ( f x , f y , μ ( f i u , f j v , f k w )) − μ ( f u , f v , μ ( f i x , f j y , f k w )) − μ ( μ ( f i x , f j y , f k u ) , f v , f w ) − μ ( f u , μ ( f i x , f j y , f k v ) , f w ) + μ ( f i x , f j y , f k π ( u , v , w )) − μ ( f i u , f j v , f k π ( x , y , w )) − μ ( f i π ( x , y , u ) , f j v , f k w ) − μ ( f i u , f j π ( x , y , v ) , f k w ) } . Thus the trilinear sum in (38) is a second cocycle. Adding t N + 1 f N + 1 t^{N+1}f_{N+1} t N + 1 f N + 1 N + 1 N+1 N + 1
∑ i + j + k = N + 1 0 ≤ i , j , k ≤ N μ ( f i x , f j y , f k z ) + μ ( f N + 1 x , f y , f z ) + μ ( f x , f N + 1 y , f z ) + μ ( f x , f y , f N + 1 z ) − f N + 1 π ( x , y , z ) . \begin{aligned}
&\sum_{\substack{i+j+k=N+1\\0\leq i,j,k\leq N}}\mu(f_ix,f_jy,f_kz)\\
&\quad+\mu(f_{N+1}x,f_y,f_z)+\mu(f_x,f_{N+1}y,f_z)+\mu(f_x,f_y,f_{N+1}z)-f_{N+1}\pi(x,y,z).
\end{aligned} i + j + k = N + 1 0 ≤ i , j , k ≤ N ∑ μ ( f i x , f j y , f k z ) + μ ( f N + 1 x , f y , f z ) + μ ( f x , f N + 1 y , f z ) + μ ( f x , f y , f N + 1 z ) − f N + 1 π ( x , y , z ) . This expression vanishes for some f N + 1 ∈ Hom ( g , h ) f_{N+1}\in\operatorname{Hom}(\mathfrak{g},\mathfrak{h}) f N + 1 ∈ Hom ( g , h ) im δ f 1 \operatorname{im}\delta_f^1 im δ f 1 H f 2 H_f^2 H f 2
Corollary 3.8. If H f 2 = 0 H_f^2=0 H f 2 = 0
Proof. Every first cocycle gives a deformation modulo t 2 t^2 t 2 Proposition 3.6 . At each subsequent order, Theorem 3.7 places
( x , y , z ) ⟼ ∑ i + j + k = N + 1 0 ≤ i , j , k ≤ N μ ( f i x , f j y , f k z ) (x,y,z) \longmapsto\sum_{\substack{i+j+k=N+1\\0\leq i,j,k\leq N}} \mu(f_i x,f_j y,f_k z) ( x , y , z ) ⟼ i + j + k = N + 1 0 ≤ i , j , k ≤ N ∑ μ ( f i x , f j y , f k z ) in ker δ f 2 \ker\delta_f^2 ker δ f 2 H f 2 = 0 H_f^2=0 H f 2 = 0 im δ f 1 \operatorname{im}\delta_f^1 im δ f 1 f N + 1 f_{N+1} f N + 1 K [ [ t ] ] \mathbb{K}[[t]] K [[ t ]]
Theorem 3.9. If H f 1 = 0 H_f^1=0 H f 1 = 0 f f f
Proof. Suppose the coefficients of orders 1 , … , r − 1 1,\ldots,r-1 1 , … , r − 1 t r t^r t r
μ ( f r x , f y , f z ) + μ ( f x , f r y , f z ) + μ ( f x , f y , f r z ) − f r π ( x , y , z ) = 0. \mu(f_r x,f y,f z)+\mu(f x,f_r y,f z)+\mu(f x,f y,f_r z)-f_r\pi(x,y,z)=0. μ ( f r x , f y , f z ) + μ ( f x , f r y , f z ) + μ ( f x , f y , f r z ) − f r π ( x , y , z ) = 0. Since H f 1 = 0 H_f^1=0 H f 1 = 0 Lemma 3.4 gives an element U r ∈ ⋀ 2 h U_r\in\bigwedge^2\mathfrak{h} U r ∈ ⋀ 2 h
f r x = ad U r f x ( x ∈ g ) . f_r x=\operatorname{ad}_{U_r}f x \qquad(x\in\mathfrak{g}). f r x = ad U r f x ( x ∈ g ) . Postcomposition with exp ( − t r ad U r ) \exp(-t^r\operatorname{ad}_{U_r}) exp ( − t r ad U r ) r r r f r − ad U r ∘ f = 0 f_r-\operatorname{ad}_{U_r}\circ f=0 f r − ad U r ∘ f = 0 t t t f f f
Example 3.10. Let g = R 2 \mathfrak{g}=\mathbb{R}^2 g = R 2 h = R 2 \mathfrak{h}=\mathbb{R}^2 h = R 2
μ ( u , v , w ) = ⟨ v , w ⟩ u − ⟨ u , w ⟩ v . (39) \mu(u,v,w)=\langle v,w\rangle u-\langle u,w\rangle v. \tag*{(39)} μ ( u , v , w ) = ⟨ v , w ⟩ u − ⟨ u , w ⟩ v . (39) This is a Lie triple system: it has the required skew and cyclic symmetries, and each ad u ∧ v \operatorname{ad}_{u\wedge v} ad u ∧ v (39) . Take f = 0 f=0 f = 0 δ f 1 = δ f 2 = 0 \delta_f^1=\delta_f^2=0 δ f 1 = δ f 2 = 0 δ f 0 = 0 \delta_f^0=0 δ f 0 = 0
The identity map a : g → h a:\mathfrak{g}\to\mathfrak{h} a : g → h t a ta t a t 3 t^3 t 3
S 0 , μ ( t a ) ( x , y , z ) = t 3 μ ( a x , a y , a z ) . S_{0,\mu}(ta)(x,y,z)=t^3\mu(ax,ay,az). S 0 , μ ( t a ) ( x , y , z ) = t 3 μ ( a x , a y , a z ) . For an orthonormal basis e 1 , e 2 e_1,e_2 e 1 , e 2 ( e 1 , e 2 , e 1 ) (e_1,e_2,e_1) ( e 1 , e 2 , e 1 ) − e 2 -e_2 − e 2 δ f 1 = 0 \delta_f^1=0 δ f 1 = 0 t 2 b t^2b t 2 b a a a
Here C f 2 C_f^2 C f 2 ( e 1 , e 2 , e 1 ) (e_1,e_2,e_1) ( e 1 , e 2 , e 1 ) ( e 1 , e 2 , e 2 ) (e_1,e_2,e_2) ( e 1 , e 2 , e 2 ) H f 2 = C f 2 ≠ 0 H_f^2=C_f^2\ne0 H f 2 = C f 2 = 0 (15) produces this nonzero third-order obstruction.
We now compare this cohomology and the formal deformation problem with those for Lie algebra homomorphisms. First consider the universal enveloping algebras. Put Φ = U ( f ) \Phi=\mathcal{U}(f) Φ = U ( f )
C C E , g r n ( Φ ) = Hom g r ( ∧ n U ( g ) , U ( h ) ) , C C E , g r 0 ( Φ ) = U ( h ) 0 . C_{\mathrm{CE},\mathrm{gr}}^n(\Phi)=\operatorname{Hom}_{\mathrm{gr}}(\wedge^n\mathcal{U}(\mathfrak{g}),\mathcal{U}(\mathfrak{h})),\qquad C_{\mathrm{CE},\mathrm{gr}}^0(\Phi)=\mathcal{U}(\mathfrak{h})_0. C CE , gr n ( Φ ) = Hom gr ( ∧ n U ( g ) , U ( h )) , C CE , gr 0 ( Φ ) = U ( h ) 0 . The subscript g r \mathrm{gr} gr v ⋅ w = [ Φ v , w ] v\cdot w=[\Phi v,w] v ⋅ w = [ Φ v , w ]
( d Φ v ) ( X ) = [ Φ X , v ] , ( d Φ A ) ( X , Y ) = [ Φ X , A ( Y ) ] + [ A ( X ) , Φ Y ] − A ( [ X , Y ] ) . (40) (d_\Phi v)(X)=[\Phi X,v],\qquad(d_\Phi A)(X,Y)=[\Phi X,A(Y)]+[A(X),\Phi Y]-A([X,Y]). \tag*{(40)} ( d Φ v ) ( X ) = [ Φ X , v ] , ( d Φ A ) ( X , Y ) = [ Φ X , A ( Y )] + [ A ( X ) , Φ Y ] − A ([ X , Y ]) . (40) Denote its cohomology by H C E , g r n ( Φ ) H_{\mathrm{CE},\mathrm{gr}}^n(\Phi) H CE , gr n ( Φ )
Theorem 3.11. Restriction to the odd component induces an isomorphism
H C E , g r 1 ( U ( f ) ) ≅ H f 1 . (41) H^1_{\mathrm{CE},\mathrm{gr}}(\mathcal{U}(f)) \cong H^1_f. \tag*{(41)} H CE , gr 1 ( U ( f )) ≅ H f 1 . (41) Moreover, f t ↦ U ( f t ) f_t \mapsto\mathcal{U}(f_t) f t ↦ U ( f t ) f f f U ( f ) \mathcal{U}(f) U ( f ) f f f U ( f ) \mathcal{U}(f) U ( f )
Proof. Let A A A U ( f ) \mathcal{U}(f) U ( f ) U ( f ) + ε A \mathcal{U}(f)+\varepsilon A U ( f ) + ε A K [ ε ] / ( ε 2 ) \mathbb{K}[\varepsilon]/(\varepsilon^2) K [ ε ] / ( ε 2 ) ε \varepsilon ε
μ ( A x , f y , f z ) + μ ( f x , A y , f z ) + μ ( f x , f y , A z ) − A π ( x , y , z ) = 0. \mu(Ax,fy,fz)+\mu(fx,Ay,fz)+\mu(fx,fy,Az)-A\pi(x,y,z)=0. μ ( A x , f y , f z ) + μ ( f x , A y , f z ) + μ ( f x , f y , A z ) − A π ( x , y , z ) = 0. Thus the odd restriction of A A A ker δ f 1 \ker\delta^1_f ker δ f 1
Conversely, for a ∈ ker δ f 1 a\in\ker\delta^1_f a ∈ ker δ f 1 f + ε a f+\varepsilon a f + ε a ε 2 \varepsilon^2 ε 2
U ( f + ε a ) ( x ) = f x + ε a ( x ) , U ( f + ε a ) ( [ x , y ] ) = [ f x , f y ] + ε ( [ a ( x ) , f y ] + [ f x , a ( y ) ] ) . (42) \mathcal{U}(f+\varepsilon a)(x)=fx+\varepsilon a(x),\qquad\mathcal{U}(f+\varepsilon a)([x,y])=[fx,fy]+\varepsilon\bigl([a(x),fy]+[fx,a(y)]\bigr). \tag*{(42)} U ( f + ε a ) ( x ) = f x + ε a ( x ) , U ( f + ε a ) ([ x , y ]) = [ f x , f y ] + ε ( [ a ( x ) , f y ] + [ f x , a ( y )] ) . (42) Its coefficient of ε \varepsilon ε a a a
Every even element of U ( h ) \mathcal{U}(\mathfrak{h}) U ( h )
[ ∑ i [ u i , v i ] , f x ] = ∑ i [ [ u i , v i ] , f x ] = ∑ i [ u i , v i , f x ] . \left[\sum_i [u_i,v_i],fx\right]=\sum_i [[u_i,v_i],fx]=\sum_i [u_i,v_i,fx]. [ i ∑ [ u i , v i ] , f x ] = i ∑ [[ u i , v i ] , f x ] = i ∑ [ u i , v i , f x ] . Lemma 3.4 identifies these maps with B f 1 B^1_f B f 1 B f 1 B^1_f B f 1
The same presentation argument applies over K [ t ] / ( t N + 1 ) \mathbb{K}[t]/(t^{N+1}) K [ t ] / ( t N + 1 ) t t t ad [ u , v ] \operatorname{ad}_{[u,v]} ad [ u , v ] x ↦ [ u , v , x ] x\mapsto[u,v,x] x ↦ [ u , v , x ] ad u ∧ v \operatorname{ad}_{u\wedge v} ad u ∧ v
Consequently, H f 1 = 0 H^1_f=0 H f 1 = 0 U ( f ) \mathcal{U}(f) U ( f ) H f 2 = 0 H^2_f=0 H f 2 = 0 U ( f ) \mathcal{U}(f) U ( f ) H f 2 H^2_f H f 2
For the second comparison, let f : a → b f:\mathfrak{a}\to\mathfrak{b} f : a → b d f d_f d f x ⋅ u = [ f x , u ] b x\cdot u=[f x,u]_{\mathfrak{b}} x ⋅ u = [ f x , u ] b a : a → b a:\mathfrak{a}\to\mathfrak{b} a : a → b
( d f a ) ( x , y ) = [ f x , a y ] b + [ a x , f y ] b − a [ x , y ] a . (43) (d_fa)(x,y)=[fx,ay]_{\mathfrak{b}}+[ax,fy]_{\mathfrak{b}}-a[x,y]_{\mathfrak{a}}. \tag*{(43)} ( d f a ) ( x , y ) = [ f x , a y ] b + [ a x , f y ] b − a [ x , y ] a . (43) In the following statements, H f n H^n_f H f n
Proposition 3.12. For every a : a → b a:\mathfrak{a}\to\mathfrak{b} a : a → b
δ f 1 a = T f ( d f a ) . (44) \delta^1_fa=\mathcal{T}_f(d_fa). \tag*{(44)} δ f 1 a = T f ( d f a ) . (44) Hence Z C E 1 ( f ) ⊆ Z f 1 Z^1_{\mathrm{CE}}(f)\subseteq Z^1_f Z CE 1 ( f ) ⊆ Z f 1 b ↦ T f b b\mapsto\mathcal{T}_fb b ↦ T f b
H C E 2 ( f ) ⟶ H f 2 . (45) H^2_{\mathrm{CE}}(f)\longrightarrow H^2_f. \tag*{(45)} H CE 2 ( f ) ⟶ H f 2 . (45) If Eq. (25) holds, then the first cocycle spaces are equal and the map in Eq. (45) is injective. If, in addition, b \mathfrak{b} b
H C E 1 ( f ) ≅ H f 1 . (46) H_{\mathrm{CE}}^{1}(f) \cong H_{f}^{1}. \tag*{(46)} H CE 1 ( f ) ≅ H f 1 . (46) For every Lie algebra homomorphism f f f
B C E 1 ( f ) = { [ v , f ( ⋅ ) ] : v ∈ b } , B f 1 = { [ v , f ( ⋅ ) ] : v ∈ [ b , b ] } . (47) B_{\mathrm{CE}}^{1}(f)=\{[v,f(\cdot)]:v\in\mathfrak{b}\},\qquad B_{f}^{1}=\{[v,f(\cdot)]:v\in[\mathfrak{b},\mathfrak{b}]\}. \tag*{(47)} B CE 1 ( f ) = {[ v , f ( ⋅ )] : v ∈ b } , B f 1 = {[ v , f ( ⋅ )] : v ∈ [ b , b ]} . (47) Proof. For a : a → b a:\mathfrak{a}\to\mathfrak{b} a : a → b
( δ f 1 a ) ( x , y , z ) = [ [ a x , f y ] b , f z ] b + [ [ f x , a y ] b , f z ] b + [ [ f x , f y ] b , a z ] b − a [ [ x , y ] a , z ] a = [ [ f x , a y ] b + [ a x , f y ] b − a [ x , y ] a , f z ] b + [ f [ x , y ] a , a z ] b + [ a [ x , y ] a , f z ] b − a [ [ x , y ] a , z ] a . \begin{aligned}
(\delta_{f}^{1}a)(x,y,z)=[[ax,fy]_{\mathfrak{b}},fz]_{\mathfrak{b}}+[[fx,ay]_{\mathfrak{b}},fz]_{\mathfrak{b}}+[[fx,fy]_{\mathfrak{b}},az]_{\mathfrak{b}}-a[[x,y]_{\mathfrak{a}},z]_{\mathfrak{a}} \\
&=[ [fx,ay]_{\mathfrak{b}}+[ax,fy]_{\mathfrak{b}}-a[x,y]_{\mathfrak{a}},fz]_{\mathfrak{b}}+[f[x,y]_{\mathfrak{a}},az]_{\mathfrak{b}}+[a[x,y]_{\mathfrak{a}},fz]_{\mathfrak{b}}-a[[x,y]_{\mathfrak{a}},z]_{\mathfrak{a}}.
\end{aligned} ( δ f 1 a ) ( x , y , z ) = [[ a x , f y ] b , f z ] b + [[ f x , a y ] b , f z ] b + [[ f x , f y ] b , a z ] b − a [[ x , y ] a , z ] a = [[ f x , a y ] b + [ a x , f y ] b − a [ x , y ] a , f z ] b + [ f [ x , y ] a , a z ] b + [ a [ x , y ] a , f z ] b − a [[ x , y ] a , z ] a . This is the full expression in (2.23) applied to the binary differential in (3.18), proving (3.19). Let b \mathfrak{b} b a ⊕ b \mathfrak{a}\oplus\mathfrak{b} a ⊕ b
[ ( x , u ) , ( y , v ) ] = ( [ x , y ] a , [ f x , v ] b − [ f y , u ] b + b ( x , y ) ) . [(x,u),(y,v)]=([x,y]_{\mathfrak{a}},[fx,v]_{\mathfrak{b}}-[fy,u]_{\mathfrak{b}}+b(x,y)). [( x , u ) , ( y , v )] = ([ x , y ] a , [ f x , v ] b − [ f y , u ] b + b ( x , y )) . The representation identity and the second cocycle equation give the Jacobi identity for this bracket. Its iterated bracket is
[ ( x , u ) , ( y , v ) , ( z , w ) ] = ( [ [ x , y ] a , z ] a , [ [ u , f y ] b , f z ] b + [ [ f x , v ] b , f z ] b + [ [ f x , f y ] b , w ] b + [ b ( x , y ) , f z ] b + b ( [ x , y ] a , z ) ) . \begin{aligned}
[(x,u),(y,v),(z,w)]=([[x,y]_{\mathfrak{a}},z]_{\mathfrak{a}},[[u,fy]_{\mathfrak{b}},fz]_{\mathfrak{b}}+[[fx,v]_{\mathfrak{b}},fz]_{\mathfrak{b}} \\
&+[[fx,fy]_{\mathfrak{b}},w]_{\mathfrak{b}}+[b(x,y),fz]_{\mathfrak{b}}+b([x,y]_{\mathfrak{a}},z)).
\end{aligned} [( x , u ) , ( y , v ) , ( z , w )] = ([[ x , y ] a , z ] a , [[ u , f y ] b , f z ] b + [[ f x , v ] b , f z ] b + [[ f x , f y ] b , w ] b + [ b ( x , y ) , f z ] b + b ([ x , y ] a , z )) . The skew and cyclic identities show that the trilinear map
( x , y , z ) ⟼ [ b ( x , y ) , f z ] b + b ( [ x , y ] a , z ) (x,y,z)\longmapsto[b(x,y),fz]_{\mathfrak{b}}+b([x,y]_{\mathfrak{a}},z) ( x , y , z ) ⟼ [ b ( x , y ) , f z ] b + b ([ x , y ] a , z ) belongs to C f 2 C_{f}^{2} C f 2
δ f 2 ( [ b ( ⋅ , ⋅ ) , f ⋅ ] b + b ( [ ⋅ , ⋅ ] a , ⋅ ) ) = 0. \delta_{f}^{2}([b(\cdot,\cdot),f\cdot]_{\mathfrak{b}}+b([\cdot,\cdot]_{\mathfrak{a}},\cdot))=0. δ f 2 ([ b ( ⋅ , ⋅ ) , f ⋅ ] b + b ([ ⋅ , ⋅ ] a , ⋅ )) = 0. The terms are those of (3.10) with the displayed trilinear expression substituted throughout. Therefore second Lie cocycles give second Lie triple cocycles. The first differential identity already proved shows that second coboundaries also map to second coboundaries. This proves (3.20).
Under (2.25), Lemma 2.8 and (3.19) give equality of the first cocycle spaces. If the class of a second Lie cocycle maps to zero, there is a : a → b a:\mathfrak{a}\to\mathfrak{b} a : a → b
T f ( b ( ⋅ , ⋅ ) − [ f ⋅ , a ⋅ ] b − [ a ⋅ , f ⋅ ] b + a [ ⋅ , ⋅ ] a ) = 0. \mathcal{T}_{f}(b(\cdot,\cdot)-[f\cdot,a\cdot]_{\mathfrak{b}}-[a\cdot,f\cdot]_{\mathfrak{b}}+a[\cdot,\cdot]_{\mathfrak{a}})=0. T f ( b ( ⋅ , ⋅ ) − [ f ⋅ , a ⋅ ] b − [ a ⋅ , f ⋅ ] b + a [ ⋅ , ⋅ ] a ) = 0. Injectivity implies
b ( x , y ) = [ f x , a y ] b + [ a x , f y ] b − a [ x , y ] a , b(x,y)=[fx,ay]_{\mathfrak{b}}+[ax,fy]_{\mathfrak{b}}-a[x,y]_{\mathfrak{a}}, b ( x , y ) = [ f x , a y ] b + [ a x , f y ] b − a [ x , y ] a , so the original class is zero. This proves injectivity in degree two. Finally, a Lie triple inner action on the target is x ↦ [ [ u , v ] b , f x ] b x\mapsto[[u,v]_{\mathfrak{b}},fx]_{\mathfrak{b}} x ↦ [[ u , v ] b , f x ] b x ↦ [ v , f x ] b x\mapsto[v,fx]_{\mathfrak{b}} x ↦ [ v , f x ] b b \mathfrak{b} b
For a finite-order Lie homomorphism deformation f t ( N ) = ∑ i = 0 N t i f i f_{t}^{(N)}=\sum_{i=0}^{N}t^{i}f_{i} f t ( N ) = ∑ i = 0 N t i f i
Ω N + 1 ( x , y ) = ∑ i + j = N + 1 0 ≤ i , j ≤ N [ f i x , f j y ] b . \Omega_{N+1}(x,y)=\sum_{\substack{i+j=N+1\\0\leq i,j\leq N}}[f_{i}x,f_{j}y]_{\mathfrak{b}}. Ω N + 1 ( x , y ) = i + j = N + 1 0 ≤ i , j ≤ N ∑ [ f i x , f j y ] b . For an arbitrary linear map q q q d q d_{q} d q x ⋅ u = [ q x , u ] b x\cdot u=[qx,u]_{\mathfrak{b}} x ⋅ u = [ q x , u ] b d q F ( q ) = 0 d_{q}F(q)=0 d q F ( q ) = 0 t N + 1 t^{N+1} t N + 1 d f Ω N + 1 = 0 d_{f}\Omega_{N+1}=0 d f Ω N + 1 = 0 t N + 1 a t^{N+1}a t N + 1 a Ω N + 1 + d f a \Omega_{N+1}+d_{f}a Ω N + 1 + d f a (24) gives
O N + 1 = T f Ω N + 1 . (48) \mathcal{O}_{N+1}=\mathcal{T}_{f}\Omega_{N+1}. \tag*{(48)} O N + 1 = T f Ω N + 1 . (48) Indeed, F ( f t ( N ) ) F(f_t^{(N)}) F ( f t ( N ) ) N N N T f T_f T f N + 1 N+1 N + 1 (45) takes the Lie obstruction class to the Lie triple obstruction class. Under Eq. (25) , a Lie finite-order deformation extends by one order in the Lie category if and only if it extends in the Lie triple category. Applying the same argument to the first nonzero coefficient of a binary defect shows that every formal Lie triple deformation based at f f f (47) applied to source and target. Formal inner rigidity is therefore equivalent in the two categories under these hypotheses.
Local rigidity and stability of morphisms
In this section, g \mathfrak{g} g h \mathfrak{h} h
Definition 4.1. A smooth deformation of f f f f t f_t f t f 0 = f f_0=f f 0 = f f t ′ = B t f t A t − 1 f'_t=B_tf_tA_t^{-1} f t ′ = B t f t A t − 1 A 0 = B 0 = i d A_0=B_0=\mathrm{id} A 0 = B 0 = id
Here Inn ( L ) \operatorname{Inn}(L) Inn ( L ) exp ( D ) \exp(D) exp ( D ) D ∈ InnDer ( L ) D\in\operatorname{InnDer}(L) D ∈ InnDer ( L ) D D D InnDer ( L ) \operatorname{InnDer}(L) InnDer ( L )
Proposition 4.2. For a smooth deformation f t f_t f t f ˙ 0 \dot{f}_0 f ˙ 0 Z f 1 Z_f^1 Z f 1 H f 1 H_f^1 H f 1 f f f B f 1 B_f^1 B f 1
Proof. Differentiating the morphism identity at t = 0 t=0 t = 0
μ ( f ˙ 0 x , f y , f z ) + μ ( f x , f ˙ 0 y , f z ) + μ ( f x , f y , f ˙ 0 z ) − f ˙ 0 π ( x , y , z ) = 0. \mu(\dot{f}_0x,f y,f z)+\mu(f x,\dot{f}_0y,f z)+\mu(f x,f y,\dot{f}_0z)-\dot{f}_0\pi(x,y,z)=0. μ ( f ˙ 0 x , f y , f z ) + μ ( f x , f ˙ 0 y , f z ) + μ ( f x , f y , f ˙ 0 z ) − f ˙ 0 π ( x , y , z ) = 0. Thus f ˙ 0 ∈ ker δ f 1 \dot{f}_0\in\ker\delta_f^1 f ˙ 0 ∈ ker δ f 1 f t ′ = B t f t A t − 1 f'_t=B_tf_tA_t^{-1} f t ′ = B t f t A t − 1
f ˙ 0 ′ − f ˙ 0 = d B t d t ∣ t = 0 ∘ f − f ∘ d A t d t ∣ t = 0 ∈ im δ f 0 . \dot{f}'_0-\dot{f}_0=\left.\frac{\mathrm{d}B_t}{\mathrm{d}t}\right|_{t=0}\circ f-f\circ\left.\frac{\mathrm{d}A_t}{\mathrm{d}t}\right|_{t=0}\in\operatorname{im}\delta_f^0. f ˙ 0 ′ − f ˙ 0 = d t d B t t = 0 ∘ f − f ∘ d t d A t t = 0 ∈ im δ f 0 . Conversely, for X ∈ ⋀ 2 g X\in\bigwedge^2\mathfrak{g} X ∈ ⋀ 2 g U ∈ ⋀ 2 h U\in\bigwedge^2\mathfrak{h} U ∈ ⋀ 2 h
t ⟼ exp ( t ad U ) ∘ f ∘ exp ( − t ad X ) t\longmapsto\exp(t\operatorname{ad}U)\circ f\circ\exp(-t\operatorname{ad}X) t ⟼ exp ( t ad U ) ∘ f ∘ exp ( − t ad X ) has derivative ad U ∘ f − f ∘ ad X \operatorname{ad}U\circ f-f\circ\operatorname{ad}X ad U ∘ f − f ∘ ad X (31) , proving the tangent-space assertion. Lemma 3.4 shows that taking X = 0 X=0 X = 0 □ \square □
Theorem 4.3. If H f 1 = 0 H_f^1=0 H f 1 = 0 U U U f f f Hom ( g , h ) \operatorname{Hom}(\mathfrak{g},\mathfrak{h}) Hom ( g , h )
D : U ⟶ InnDer ( h ) , D ( f ) = 0 , D:U\longrightarrow\operatorname{InnDer}(\mathfrak{h}),\qquad D(f)=0, D : U ⟶ InnDer ( h ) , D ( f ) = 0 , such that every morphism q ∈ U q\in U q ∈ U
q = exp ( D ( q ) ) f . (49) q=\exp(D(q))f. \tag*{(49)} q = exp ( D ( q )) f . (49) In particular, f f f
Proof. Choose a vector-space complement N N N im δ f 0 \operatorname{im}\delta_f^0 im δ f 0 Hom ( g , h ) \operatorname{Hom}(\mathfrak{g},\mathfrak{h}) Hom ( g , h ) a \mathfrak{a} a
{ D ∈ InnDer ( h ) : D ∘ f = 0 } \{D\in\operatorname{InnDer}(\mathfrak{h}):D\circ f=0\} { D ∈ InnDer ( h ) : D ∘ f = 0 } in InnDer ( h ) \operatorname{InnDer}(\mathfrak{h}) InnDer ( h ) D ↦ D ∘ f D \mapsto D \circ f D ↦ D ∘ f a \mathfrak{a} a im δ f 0 \operatorname{im}\delta_f^0 im δ f 0
a × N ⟶ Hom ( g , h ) , ( D , n ) ↦ exp ( D ) ∘ ( f + n ) \mathfrak{a} \times N \longrightarrow\operatorname{Hom}(\mathfrak{g},\mathfrak{h}), \qquad(D,n) \mapsto\exp(D) \circ(f+n) a × N ⟶ Hom ( g , h ) , ( D , n ) ↦ exp ( D ) ∘ ( f + n ) has derivative ( D , n ) ↦ D ∘ f + n (D,n) \mapsto D \circ f+n ( D , n ) ↦ D ∘ f + n ( 0 , 0 ) (0,0) ( 0 , 0 ) D ( q ) D(q) D ( q ) n ( q ) n(q) n ( q )
q = exp ( D ( q ) ) ∘ ( f + n ( q ) ) , D ( f ) = 0 , n ( f ) = 0. q=\exp(D(q))\circ(f+n(q)), \qquad D(f)=0,\qquad n(f)=0. q = exp ( D ( q )) ∘ ( f + n ( q )) , D ( f ) = 0 , n ( f ) = 0. For every target automorphism B B B q q q
μ ( B q x , B q y , B q z ) − B q π ( x , y , z ) = B ( μ ( q x , q y , q z ) − q π ( x , y , z ) ) . \mu(Bqx,Bqy,Bqz)-Bq\pi(x,y,z)=B\bigl(\mu(qx,qy,qz)-q\pi(x,y,z)\bigr). μ ( B q x , B q y , B q z ) − B q π ( x , y , z ) = B ( μ ( q x , q y , q z ) − q π ( x , y , z ) ) . Hence f + n ( q ) f+n(q) f + n ( q ) q q q
( d S π , μ ) f ( a ) ( x , y , z ) = μ ( a x , f y , f z ) + μ ( f x , a y , f z ) + μ ( f x , f y , a z ) − a π ( x , y , z ) = ( δ f 1 a ) ( x , y , z ) . (50) \begin{aligned}
(dS_{\pi,\mu})_f(a)(x,y,z)={}&\mu(ax,fy,fz)+\mu(fx,ay,fz)\\
&+\mu(fx,fy,az)-a\pi(x,y,z)=(\delta_f^1a)(x,y,z). \tag*{(50)}
\end{aligned} ( d S π , μ ) f ( a ) ( x , y , z ) = μ ( a x , f y , f z ) + μ ( f x , a y , f z ) + μ ( f x , f y , a z ) − aπ ( x , y , z ) = ( δ f 1 a ) ( x , y , z ) . (50) The hypothesis H f 1 = 0 H_f^1=0 H f 1 = 0 ker δ f 1 = im δ f 0 \ker\delta_f^1=\operatorname{im}\delta_f^0 ker δ f 1 = im δ f 0 N N N c > 0 c>0 c > 0 n ∈ N n\in N n ∈ N
∥ μ ( n ⋅ , f ⋅ , f ⋅ ) + μ ( f ⋅ , n ⋅ , f ⋅ ) + μ ( f ⋅ , f ⋅ , n ⋅ ) − n π ( ⋅ , ⋅ , ⋅ ) ∥ ≥ c ∥ n ∥ . \left\|\mu(n\mathbin{\cdot},f\mathbin{\cdot},f\mathbin{\cdot})+\mu(f\mathbin{\cdot},n\mathbin{\cdot},f\mathbin{\cdot})+\mu(f\mathbin{\cdot},f\mathbin{\cdot},n\mathbin{\cdot})-n\pi(\mathbin{\cdot},\mathbin{\cdot},\mathbin{\cdot})\right\|\ge c\|n\|. ∥ μ ( n ⋅ , f ⋅ , f ⋅ ) + μ ( f ⋅ , n ⋅ , f ⋅ ) + μ ( f ⋅ , f ⋅ , n ⋅ ) − nπ ( ⋅ , ⋅ , ⋅ ) ∥ ≥ c ∥ n ∥. Taylor expansion consequently gives, for a constant C > 0 C>0 C > 0 n ∈ N n\in N n ∈ N
∥ μ ( ( f + n ) ⋅ , ( f + n ) ⋅ , ( f + n ) ⋅ ) − ( f + n ) π ( ⋅ , ⋅ , ⋅ ) ∥ ≥ c ∥ n ∥ − C ∥ n ∥ 2 . \left\|\mu((f+n)\mathbin{\cdot},(f+n)\mathbin{\cdot},(f+n)\mathbin{\cdot})-(f+n)\pi(\mathbin{\cdot},\mathbin{\cdot},\mathbin{\cdot})\right\|\ge c\|n\|-C\|n\|^2. ∥ μ (( f + n ) ⋅ , ( f + n ) ⋅ , ( f + n ) ⋅ ) − ( f + n ) π ( ⋅ , ⋅ , ⋅ ) ∥ ≥ c ∥ n ∥ − C ∥ n ∥ 2 . Thus f + n f+n f + n n = 0 n=0 n = 0 n ( q ) = 0 n(q)=0 n ( q ) = 0 q = exp ( D ( q ) ) ∘ f q=\exp(D(q))\circ f q = exp ( D ( q )) ∘ f D D D exp ( − D ( f t ) ) \exp(-D(f_t)) exp ( − D ( f t ))
The next elementary lemma is a local form of the exactness argument used in [1 ] . We include its proof to make the parameter dependence explicit.
Lemma 4.4. Let V , E , F V,E,F V , E , F s s s x 0 ∈ V x_0\in V x 0 ∈ V E E E B ( x ) : E → F B(x):E\to F B ( x ) : E → F x x x
s ( x 0 ) = 0 , B ( x ) s ( x ) = 0 , im ( d s ) x 0 = ker B ( x 0 ) . s(x_0)=0,\qquad B(x)s(x)=0,\qquad\operatorname{im}(ds)_{x_0}=\ker B(x_0). s ( x 0 ) = 0 , B ( x ) s ( x ) = 0 , im ( d s ) x 0 = ker B ( x 0 ) . Then s − 1 ( 0 ) s^{-1}(0) s − 1 ( 0 ) x 0 x_0 x 0 ker ( d s ) x 0 \ker(ds)_{x_0} ker ( d s ) x 0 s p s_p s p B p B_p B p p p p s p 0 = s s_{p_0}=s s p 0 = s B p 0 = B B_{p_0}=B B p 0 = B B p ( x ) s p ( x ) = 0 B_p(x)s_p(x)=0 B p ( x ) s p ( x ) = 0 P P P p 0 p_0 p 0 p ∈ P p\in P p ∈ P p 0 p_0 p 0 x ( p ) x(p) x ( p ) x 0 x_0 x 0 x ( p ) x(p) x ( p )
Proof. Choose a vector-space complement C C C
E = im ( d s ) x 0 ⊕ C . E=\operatorname{im}(ds)_{x_0}\oplus C. E = im ( d s ) x 0 ⊕ C . Since ker B ( x 0 ) = im ( d s ) x 0 \ker B(x_0)=\operatorname{im}(ds)_{x_0} ker B ( x 0 ) = im ( d s ) x 0 B ( x 0 ) ∣ C B(x_0)|_C B ( x 0 ) ∣ C B p ( x ) ∣ C B_p(x)|_C B p ( x ) ∣ C p p p x x x W W W
V = ker ( d s ) x 0 ⊕ W . V=\ker(ds)_{x_0}\oplus W. V = ker ( d s ) x 0 ⊕ W . Then ( d s ) x 0 ∣ W : W → im ( d s ) x 0 (ds)_{x_0}|_W:W\to\operatorname{im}(ds)_{x_0} ( d s ) x 0 ∣ W : W → im ( d s ) x 0 E E E im ( d s ) x 0 \operatorname{im}(ds)_{x_0} im ( d s ) x 0 C C C
pr im ( d s ) x 0 s p ( x 0 + k + w ) = 0 , k ∈ ker ( d s ) x 0 , w ∈ W , \operatorname{pr}_{\operatorname{im}(ds)_{x_0}}s_p(x_0+k+w)=0,\qquad k\in\ker(ds)_{x_0},\quad w\in W, pr im ( d s ) x 0 s p ( x 0 + k + w ) = 0 , k ∈ ker ( d s ) x 0 , w ∈ W , has a unique solution w = w ( p , k ) w = w(p,k) w = w ( p , k ) ( p , k ) (p,k) ( p , k ) w ( p 0 , 0 ) = 0 w(p_0,0)=0 w ( p 0 , 0 ) = 0
s p ( x 0 + k + w ( p , k ) ) ∈ C . s_p(x_0+k+w(p,k))\in C. s p ( x 0 + k + w ( p , k )) ∈ C . For p ∈ P p\in\mathcal{P} p ∈ P
B p ( x 0 + k + w ( p , k ) ) s p ( x 0 + k + w ( p , k ) ) = 0. B_p(x_0+k+w(p,k))s_p(x_0+k+w(p,k))=0. B p ( x 0 + k + w ( p , k )) s p ( x 0 + k + w ( p , k )) = 0. Injectivity of B p ( x 0 + k + w ( p , k ) ) ∣ C B_p(x_0+k+w(p,k))|_C B p ( x 0 + k + w ( p , k )) ∣ C s p ( x 0 + k + w ( p , k ) ) = 0 s_p(x_0+k+w(p,k))=0 s p ( x 0 + k + w ( p , k )) = 0 p = p 0 p=p_0 p = p 0 s s s k ↦ w ( p 0 , k ) k\mapsto w(p_0,k) k ↦ w ( p 0 , k ) ker ( d s ) x 0 \ker(ds)_{x_0} ker ( d s ) x 0 k ∈ ker ( d s ) x 0 k\in\ker(ds)_{x_0} k ∈ ker ( d s ) x 0
pr im ( d s ) x 0 ( d s ) x 0 ( k + ( d k w ) ( p 0 , 0 ) k ) = 0. \operatorname{pr}_{\operatorname{im}(ds)_{x_0}}(ds)_{x_0}\left(k+(d_kw)_{(p_0,0)}k\right)=0. pr im ( d s ) x 0 ( d s ) x 0 ( k + ( d k w ) ( p 0 , 0 ) k ) = 0. The derivative vanishes on k k k W W W ( d k w ) ( p 0 , 0 ) k = 0 (d_kw)_{(p_0,0)}k=0 ( d k w ) ( p 0 , 0 ) k = 0 p ↦ x 0 + w ( p , 0 ) p\mapsto x_0+w(p,0) p ↦ x 0 + w ( p , 0 ) s p s_p s p p ∈ P p\in\mathcal{P} p ∈ P
Let Mor ( g , h ) \operatorname{Mor}(\mathfrak{g},\mathfrak{h}) Mor ( g , h ) (16) for the fixed products π , μ \pi,\mu π , μ
Theorem 4.5. If H f 2 = 0 H_f^2=0 H f 2 = 0 Mor ( g , h ) \operatorname{Mor}(\mathfrak{g},\mathfrak{h}) Mor ( g , h ) f f f
T f Mor ( g , h ) = Z f 1 , dim f Mor ( g , h ) = dim Z f 1 . T_f\operatorname{Mor}(\mathfrak{g},\mathfrak{h})=Z_f^1,\qquad\dim_f\operatorname{Mor}(\mathfrak{g},\mathfrak{h})=\dim Z_f^1. T f Mor ( g , h ) = Z f 1 , dim f Mor ( g , h ) = dim Z f 1 . Every a ∈ Z f 1 a\in Z_f^1 a ∈ Z f 1 f f f
Proof. Apply Lemma 4.4 to the polynomial map
Hom ( g , h ) ⟶ C 1 ( g , h ) , q ⟼ ( ( x , y , z ) ⟼ μ ( q x , q y , q z ) − q π ( x , y , z ) ) , \operatorname{Hom}(\mathfrak{g},\mathfrak{h})\longrightarrow C^1(\mathfrak{g},\mathfrak{h}),\qquad q\longmapsto\bigl((x,y,z)\longmapsto\mu(qx,qy,qz)-q\pi(x,y,z)\bigr), Hom ( g , h ) ⟶ C 1 ( g , h ) , q ⟼ ( ( x , y , z ) ⟼ μ ( q x , q y , q z ) − q π ( x , y , z ) ) , and to the family of linear maps from C 1 ( g , h ) C^1(\mathfrak{g},\mathfrak{h}) C 1 ( g , h ) C 2 ( g , h ) C^2(\mathfrak{g},\mathfrak{h}) C 2 ( g , h )
b ⟼ P [ π , b ^ ] B + 1 2 P [ [ [ μ , q ~ b ] B , q ~ b ] B , b ^ ] B . b\longmapsto P[\pi,\widehat{b}]_B+\frac{1}{2}P[[[\mu,\widetilde{q}b]_B,\widetilde{q}b]_B,\widehat{b}]_B. b ⟼ P [ π , b ] B + 2 1 P [[[ μ , q b ] B , q b ] B , b ] B . The compatibility required by the lemma is exactly Lemma 2.5 . At q = f q=f q = f a a a
( x , y , z ) ⟼ μ ( a x , f y , f z ) + μ ( f x , a y , f z ) + μ ( f x , f y , a z ) − a π ( x , y , z ) , (x,y,z)\longmapsto\mu(ax,fy,fz)+\mu(fx,ay,fz)+\mu(fx,fy,az)-a\pi(x,y,z), ( x , y , z ) ⟼ μ ( a x , f y , f z ) + μ ( f x , a y , f z ) + μ ( f x , f y , a z ) − aπ ( x , y , z ) , which is δ f 1 a \delta_f^1a δ f 1 a (32) , the displayed family of linear maps at q = f q=f q = f b b b − δ f 2 b -\delta_f^2b − δ f 2 b
im δ f 1 = ker δ f 2 , \operatorname{im}\delta_f^1=\ker\delta_f^2, im δ f 1 = ker δ f 2 , which follows from H f 2 = 0 H_f^2=0 H f 2 = 0 ker δ f 1 = Z f 1 \ker\delta_f^1=Z_f^1 ker δ f 1 = Z f 1
Definition 4.6. A morphism f : ( g , π ) → ( h , μ ) f:(\mathfrak{g},\pi)\to(\mathfrak{h},\mu) f : ( g , π ) → ( h , μ ) U U U f f f W W W ( π , μ ) (\pi,\mu) ( π , μ ) ( π ′ , μ ′ ) ∈ W (\pi',\mu')\in W ( π ′ , μ ′ ) ∈ W f ′ ∈ U f'\in U f ′ ∈ U ( g , π ′ ) (\mathfrak{g},\pi') ( g , π ′ ) ( h , μ ′ ) (\mathfrak{h},\mu') ( h , μ ′ )
Theorem 4.7. If H f 2 = 0 H_f^2=0 H f 2 = 0 f f f π t , μ t \pi_t,\mu_t π t , μ t ( π 0 , μ 0 ) = ( π , μ ) (\pi_0,\mu_0)=(\pi,\mu) ( π 0 , μ 0 ) = ( π , μ ) f t f_t f t f 0 = f f_0=f f 0 = f
Proof. Use the ambient parameter vector space
C 1 ( g , g ) ⊕ C 1 ( h , h ) . C^{1}(\mathfrak{g},\mathfrak{g}) \oplus C^{1}(\mathfrak{h},\mathfrak{h}). C 1 ( g , g ) ⊕ C 1 ( h , h ) . For arbitrary ( π ′ , μ ′ ) (\pi',\mu') ( π ′ , μ ′ )
q ⟼ ( ( x , y , z ) ⟼ μ ′ ( q x , q y , q z ) − q π ′ ( x , y , z ) ) q \longmapsto\bigl((x,y,z) \longmapsto\mu'(qx,qy,qz)-q\pi'(x,y,z)\bigr) q ⟼ ( ( x , y , z ) ⟼ μ ′ ( q x , q y , q z ) − q π ′ ( x , y , z ) ) and the family of linear maps
b ⟼ P [ π ′ , b ˉ ] B + 1 2 P [ [ μ ′ , q ˉ ] B , q ˉ ] B , b ˉ ] B . b \longmapsto\mathop{\mathrm{P}}[\pi',\bar{b}]_{\mathrm{B}}+\frac{1}{2}\mathop{\mathrm{P}}\bigl[[\mu',\bar{q}]_{\mathrm{B}},\bar{q}\bigr]_{\mathrm{B}},\bar{b}\bigr]_{\mathrm{B}}. b ⟼ P [ π ′ , b ˉ ] B + 2 1 P [ [ μ ′ , q ˉ ] B , q ˉ ] B , b ˉ ] B . Both depend polynomially on ( π ′ , μ ′ , q ) (\pi',\mu',q) ( π ′ , μ ′ , q ) π ′ \pi' π ′ μ ′ \mu' μ ′ Lemma 2.5 gives the compatibility identity for these two maps. At ( π , μ , f ) (\pi,\mu,f) ( π , μ , f ) δ f 1 \delta_f^1 δ f 1 − δ f 2 -\delta_f^2 − δ f 2 Theorem 4.5 . The condition H f 2 = 0 H_f^2=0 H f 2 = 0 Lemma 4.4 .
The parameter part of that lemma gives a smooth choice q ( π ′ , μ ′ ) q(\pi',\mu') q ( π ′ , μ ′ ) q ( π , μ ) = f q(\pi,\mu)=f q ( π , μ ) = f
μ ′ ( q ( π ′ , μ ′ ) x , q ( π ′ , μ ′ ) y , q ( π ′ , μ ′ ) z ) = q ( π ′ , μ ′ ) π ′ ( x , y , z ) \mu'\bigl(q(\pi',\mu')x,q(\pi',\mu')y,q(\pi',\mu')z\bigr)=q(\pi',\mu')\pi'(x,y,z) μ ′ ( q ( π ′ , μ ′ ) x , q ( π ′ , μ ′ ) y , q ( π ′ , μ ′ ) z ) = q ( π ′ , μ ′ ) π ′ ( x , y , z ) whenever the two parameter tensors are Lie triple products. Continuity of q q q π t , μ t \pi_t,\mu_t π t , μ t t ↦ q ( π t , μ t ) t\mapsto q(\pi_t,\mu_t) t ↦ q ( π t , μ t ) f f f
Corollary 4.8. Under the hypothesis of Theorem 4.7 , f f f f f f
Proof. For perturbations of the target product alone, apply Theorem 4.7 to pairs ( π , μ ′ ) (\pi,\mu') ( π , μ ′ ) ( π ′ , μ ) (\pi',\mu) ( π ′ , μ ) f f f f f f
Example 4.9. Let T = R 2 T=\mathbb{R}^{2} T = R 2 f = id T f=\operatorname{id}_{T} f = id T
J = ( 0 1 − 1 0 ) , [ x , y , z ] = ω ( x , y ) J z , J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}, \qquad[x,y,z]=\omega(x,y)Jz, J = ( 0 − 1 1 0 ) , [ x , y , z ] = ω ( x , y ) J z , where ω ( e 1 , e 2 ) = 1 \omega(e_{1},e_{2})=1 ω ( e 1 , e 2 ) = 1 A = ( a b c d ) A=\begin{pmatrix}a&b\\c&d\end{pmatrix} A = ( a c b d )
( δ f 1 A ) ( e 1 , e 2 , z ) = ( ( tr A ) J + J A − A J ) z = ( b + c 2 d − 2 a − b − c ) z . (51) (\delta_f^1A)(e_1,e_2,z)=\bigl((\operatorname{tr}A)J+JA-AJ\bigr)z=
\begin{pmatrix}b+c&2d\\-2a&-b-c\end{pmatrix}z. \tag*{(51)} ( δ f 1 A ) ( e 1 , e 2 , z ) = ( ( tr A ) J + J A − A J ) z = ( b + c − 2 a 2 d − b − c ) z . (51) Thus Z f 1 = R J Z_f^1=\mathbb{R}J Z f 1 = R J InnDer ( T ) = R J \operatorname{InnDer}(T)=\mathbb{R}J InnDer ( T ) = R J B f 1 = R J B_f^1=\mathbb{R}J B f 1 = R J H f 1 = 0 H_f^1=0 H f 1 = 0
Every b ∈ C f 2 b\in C_f^2 b ∈ C f 2 b ( x , y , z ) = ω ( x , y ) K z b(x,y,z)=\omega(x,y)Kz b ( x , y , z ) = ω ( x , y ) K z K ∈ M 2 ( R ) K\in M_2(\mathbb{R}) K ∈ M 2 ( R )
( δ f 2 b ) ( x , y , u , v , w ) = − ω ( x , y ) ω ( u , v ) ( tr K ) J w . (52) (\delta_f^2b)(x,y,u,v,w)=-\omega(x,y)\omega(u,v)(\operatorname{tr}K)Jw. \tag*{(52)} ( δ f 2 b ) ( x , y , u , v , w ) = − ω ( x , y ) ω ( u , v ) ( tr K ) J w . (52) Consequently, Z f 2 Z_f^2 Z f 2 tr K = 0 \operatorname{tr}K=0 tr K = 0 B f 2 B_f^2 B f 2 H f 2 = 0 H_f^2=0 H f 2 = 0 J J J exp ( t J ) \exp(tJ) exp ( t J )
Example 4.10. Let g = R e \mathfrak{g}=\mathbb{R}e g = R e h \mathfrak{h} h f : g → h f:\mathfrak{g}\to\mathfrak{h} f : g → h [ u , w , u ] = 0 [u,w,u]=0 [ u , w , u ] = 0 C f 2 = 0 C_f^2=0 C f 2 = 0 H f 2 = 0 H_f^2=0 H f 2 = 0 h = T \mathfrak{h}=T h = T f ( e ) = e 1 f(e)=e_1 f ( e ) = e 1
Z f 1 ≅ R 2 , B f 1 = R e 2 , dim H f 1 = 1. Z_f^1\cong\mathbb{R}^2,\qquad B_f^1=\mathbb{R}e_2,\qquad\dim H_f^1=1. Z f 1 ≅ R 2 , B f 1 = R e 2 , dim H f 1 = 1. The family f t ( e ) = ( 1 + t ) e 1 f_t(e)=(1+t)e_1 f t ( e ) = ( 1 + t ) e 1 f f f T T T
We next apply these results to the universal enveloping Lie algebra homomorphism. In the following statements, a graded deformation preserves the fixed Z 2 \mathbb{Z}_2 Z 2
Theorem 4.11. Let g \mathfrak{g} g h \mathfrak{h} h Φ = U ( f ) \Phi=\mathcal{U}(f) Φ = U ( f ) Φ \Phi Φ f f f f f f Φ \Phi Φ
If H f 1 = 0 H^1_f=0 H f 1 = 0 Φ \Phi Φ H f 2 = 0 H^2_f=0 H f 2 = 0
Proof. The even component U ( g ) 0 \mathcal{U}(\mathfrak{g})_0 U ( g ) 0 ⋀ 2 g \bigwedge^2\mathfrak{g} ⋀ 2 g g \mathfrak{g} g x i , y i ∈ g x_i,y_i\in\mathfrak{g} x i , y i ∈ g [ x i , y i ] [x_i,y_i] [ x i , y i ] U ( g ) 0 \mathcal{U}(\mathfrak{g})_0 U ( g ) 0 q : g → h q:\mathfrak{g}\to\mathfrak{h} q : g → h
x ⟼ q x ( x ∈ g ) , [ x i , y i ] ⟼ [ q x i , q y i ] x\longmapsto qx\quad(x\in\mathfrak{g}),\qquad[x_i,y_i]\longmapsto[qx_i,qy_i] x ⟼ q x ( x ∈ g ) , [ x i , y i ] ⟼ [ q x i , q y i ] on the odd space and the chosen even basis define a grading-preserving linear map on the source envelope. Its coefficients depend polynomially on the coefficients of q q q
These polynomial maps identify smooth curves on the two loci. Whenever one locus is a smooth submanifold, their compositions restrict to the identity and their differentials are inverse on the corresponding tangent spaces; the same local parametrizations consequently give the submanifold structure on the other locus.
For even inner derivations, the restriction formula is
ad ∑ i [ u i , v i ] ∣ h ( x ) = ∑ i [ u i , v i , x ] = ( ∑ i ad u i ∧ v i ) ( x ) . \operatorname{ad}_{\sum_i [u_i,v_i]}\big|_{\mathfrak{h}}(x)=\sum_i [u_i,v_i,x]=\left(\sum_i\operatorname{ad}_{u_i\wedge v_i}\right)(x). ad ∑ i [ u i , v i ] h ( x ) = i ∑ [ u i , v i , x ] = ( i ∑ ad u i ∧ v i ) ( x ) . Every Lie triple inner derivation arises in this way. Exponentiation therefore identifies the inner orbits on the two morphism spaces, proving the rigidity equivalence. The remaining assertions follow from Theorems 4.3, 4.5 and 3.11. □ \square □
For stability, the products on the envelopes are also allowed to vary. A graded Lie homomorphism is called graded stable if it persists under every sufficiently small pair of Lie bracket perturbations preserving the fixed grading on its source and target.
Theorem 4.12. Let f : g → h f:\mathfrak{g}\to\mathfrak{h} f : g → h H f 2 = 0 H^2_f=0 H f 2 = 0 U ( f ) \mathcal{U}(f) U ( f ) U ( f ) \mathcal{U}(f) U ( f )
Proof. By the recognition theorem for universal envelopes [11 ] , a graded Lie algebra generated by its odd component is the universal envelope of that odd Lie triple system when its second graded Chevalley–Eilenberg cohomology with every trivial even coefficient module vanishes. For the source universal envelope, the corresponding conditions are
[ U ( g ) 1 , U ( g ) 1 ] = U ( g ) 0 , H C E , g r 2 ( U ( g ) , R e v e n ) = 0. (53) [\mathcal{U}(\mathfrak{g})_1,\mathcal{U}(\mathfrak{g})_1]=\mathcal{U}(\mathfrak{g})_0,\qquad H^2_{\mathrm{CE,gr}}\bigl(\mathcal{U}(\mathfrak{g}),\mathbb{R}_{\mathrm{even}}\bigr)=0. \tag*{(53)} [ U ( g ) 1 , U ( g ) 1 ] = U ( g ) 0 , H CE , gr 2 ( U ( g ) , R even ) = 0. (53) The coefficient module here is trivial and concentrated in even degree. Both conditions hold for U ( g ) \mathcal{U}(\mathfrak{g}) U ( g ) L = U ( g ) L=\mathcal{U}(\mathfrak{g}) L = U ( g ) L = L 0 ⊕ L 1 L=L_0\oplus L_1 L = L 0 ⊕ L 1 M M M L L L
C C E , g r n ( L , M ) ≅ C C E , g r n ( L , R e v e n ) ⊗ M . C_{\mathrm{CE,gr}}^n(L,M)\cong C_{\mathrm{CE,gr}}^n(L,\mathbb{R}_{\mathrm{even}})\otimes M. C CE , gr n ( L , M ) ≅ C CE , gr n ( L , R even ) ⊗ M . Under this identification the differential is d n ⊗ id M d^n\otimes\operatorname{id}_M d n ⊗ id M R \mathbb{R} R R e v e n \mathbb{R}_{\mathrm{even}} R even M M M
Both conditions persist under sufficiently small graded Lie bracket perturbations on the fixed vector space L L L ⋀ 2 L 1 \bigwedge^2L_1 ⋀ 2 L 1 L 0 L_0 L 0 [ L 1 , L 1 ] ′ = L 0 [L_1,L_1]'=L_0 [ L 1 , L 1 ] ′ = L 0
For the second, consider the fixed finite-dimensional spaces
C C E , g r n ( L , R e v e n ) = Hom ( ( ⋀ n L ) 0 , R ) , n = 1 , 2 , 3. C_{\mathrm{CE,gr}}^n(L,\mathbb{R}_{\mathrm{even}})=\operatorname{Hom}((\bigwedge^nL)_0,\mathbb{R}),\qquad n=1,2,3. C CE , gr n ( L , R even ) = Hom (( ⋀ n L ) 0 , R ) , n = 1 , 2 , 3. Here ( ⋀ n L ) 0 (\bigwedge^nL)_0 ( ⋀ n L ) 0 d 1 , d 2 d^1,d^2 d 1 , d 2 d ′ 1 , d ′ 2 d^{\prime1},d^{\prime2} d ′1 , d ′2
rank d 1 + rank d 2 = dim C C E , g r 2 ( L , R e v e n ) . \operatorname{rank}d^1+\operatorname{rank}d^2=\dim C_{\mathrm{CE,gr}}^2(L,\mathbb{R}_{\mathrm{even}}). rank d 1 + rank d 2 = dim C CE , gr 2 ( L , R even ) . For every nearby bracket satisfying the Jacobi identity, d ′ 2 d ′ 1 = 0 d^{\prime2}d^{\prime1}=0 d ′2 d ′1 = 0
dim C C E , g r 2 ( L , R e v e n ) = rank d ′ 1 + rank d ′ 2 ≤ rank d ′ 1 + rank d ′ 2 ≤ dim C C E , g r 2 ( L , R e v e n ) . \begin{aligned}
\dim C_{\mathrm{CE,gr}}^2(L,\mathbb{R}_{\mathrm{even}})=\operatorname{rank}d^{\prime1}+\operatorname{rank}d^{\prime2} \\
&\leq\operatorname{rank}d^{\prime1}+\operatorname{rank}d^{\prime2}\leq\dim C_{\mathrm{CE,gr}}^2(L,\mathbb{R}_{\mathrm{even}}).
\end{aligned} dim C CE , gr 2 ( L , R even ) = rank d ′1 + rank d ′2 ≤ rank d ′1 + rank d ′2 ≤ dim C CE , gr 2 ( L , R even ) . Equality follows, so im d ′ 1 = ker d ′ 2 \operatorname{im}d^{\prime1}=\ker d^{\prime2} im d ′1 = ker d ′2
More explicitly, let L ′ L' L ′ L L L T ′ = L 1 T'=L_1 T ′ = L 1 [ x , y , z ] ′ = [ [ x , y ] ′ , z ] ′ [x,y,z]'=[[x,y]',z]' [ x , y , z ] ′ = [[ x , y ] ′ , z ] ′ L 1 L_1 L 1
p ′ : U ( T ′ ) ⟶ L ′ , p ′ ( x ) = x , p ′ ( [ x , y ] ) = [ x , y ] ′ . p'\colon\mathcal{U}(T')\longrightarrow L',\qquad p'(x)=x,\quad p'([x,y])=[x,y]'. p ′ : U ( T ′ ) ⟶ L ′ , p ′ ( x ) = x , p ′ ([ x , y ]) = [ x , y ] ′ . It is surjective because [ L 1 , L 1 ] ′ = L 0 [L_1,L_1]'=L_0 [ L 1 , L 1 ] ′ = L 0 p ′ p' p ′ ker p ′ \ker p' ker p ′ R \mathbb{R} R H C E , g r 2 ( L ′ , ker p ′ ) H^2_{\mathrm{CE,gr}}(L',\ker p') H CE , gr 2 ( L ′ , ker p ′ ) s ′ s' s ′ p ′ s ′ = id L ′ p's'=\operatorname{id}_{L'} p ′ s ′ = id L ′ [11 ] . Since s ′ s' s ′ U ( T ′ ) \mathcal{U}(T') U ( T ′ ) s ′ s' s ′ p ′ p' p ′ T ′ T' T ′
Now perturb the source and target graded brackets on the fixed spaces U ( g ) \mathcal{U}(\mathfrak{g}) U ( g ) U ( h ) \mathcal{U}(\mathfrak{h}) U ( h ) g \mathfrak{g} g h \mathfrak{h} h q : g → h q\colon\mathfrak{g}\to\mathfrak{h} q : g → h
q ( [ [ x , y ] U ( g ) , z ] U ( g ) ) = [ [ q x , q y ] U ( h ) , q z ] U ( h ) . q([[x,y]_{\mathcal{U}(\mathfrak{g})},z]_{\mathcal{U}(\mathfrak{g})})=[[qx,qy]_{\mathcal{U}(\mathfrak{h})},qz]_{\mathcal{U}(\mathfrak{h})}. q ([[ x , y ] U ( g ) , z ] U ( g ) ) = [[ q x , q y ] U ( h ) , q z ] U ( h ) . The universal property first extends q q q q ~ : U ( T ′ ) → ( U ( h ) , [ , ] ′ ) \widetilde q\colon\mathcal{U}(T')\to(\mathcal{U}(\mathfrak{h}),[\, ,\, ]') q : U ( T ′ ) → ( U ( h ) , [ , ] ′ ) q ^ = q ~ ∘ ( p ′ ) − 1 \widehat q=\widetilde q\circ(p')^{-1} q = q ∘ ( p ′ ) − 1 q q q U ( f ) \mathcal{U}(f) U ( f )
To see its closeness and smooth dependence, choose x i , y i ∈ g x_i,y_i\in\mathfrak{g} x i , y i ∈ g [ x i , y i ] U ( g ) [x_i,y_i]_{\mathcal{U}(\mathfrak{g})} [ x i , y i ] U ( g )
[ q x i , q y i ] U ( h ) . [qx_i,qy_i]_{\mathcal{U}(\mathfrak{h})}. [ q x i , q y i ] U ( h ) . More explicitly, for a ∈ L 0 a\in L_0 a ∈ L 0 a = ∑ i c i ′ [ x i , y i ] U ( g ) a=\sum_i c_i'[x_i,y_i]_{\mathcal{U}(\mathfrak{g})} a = ∑ i c i ′ [ x i , y i ] U ( g ) c i ′ c_i' c i ′ a a a x ∈ g x\in\mathfrak{g} x ∈ g
q ^ ( a + x ) = ∑ i c i ′ [ q x i , q y i ] U ( h ) + q x . \widehat{q}(a+x)=\sum_i c_i'[qx_i,qy_i]_{\mathcal{U}(\mathfrak{h})}+qx. q ( a + x ) = i ∑ c i ′ [ q x i , q y i ] U ( h ) + q x . The basis change, the target brackets and the odd map supplied by Theorem 4.7 depend smoothly on the parameters. Hence the whole graded extension depends smoothly on the perturbed brackets and equals U ( f ) \mathcal{U}(f) U ( f ) H f 2 H_f^2 H f 2
Theorems 4.11 and 4.12 compare the local Lie triple problem with grading-preserving Lie deformations. The corresponding inner equivalences are generated by even Lie inner derivations. For prescribed ungraded Lie algebras, the comparison takes a different form.
Theorem 4.13. Let f : a → b f:\mathfrak{a}\to\mathfrak{b} f : a → b (25) . Then the spaces of Lie algebra homomorphisms and associated Lie triple morphisms coincide in a neighborhood of f f f f f f f f f
If b \mathfrak{b} b H f 2 = 0 H_f^2=0 H f 2 = 0 H C E 2 ( f ) = 0 H_{\mathrm{CE}}^2(f)=0 H CE 2 ( f ) = 0 b \mathfrak{b} b H f 1 = 0 H_f^1=0 H f 1 = 0 H C E 1 ( f ) = 0 H_{\mathrm{CE}}^1(f)=0 H CE 1 ( f ) = 0
Proof. Lemma 2.8 makes the linear map
b ⟼ ( ( x , y , z ) ⟼ [ b ( x , y ) , f z ] b + b ( [ x , y ] a , z ) ) b\longmapsto\bigl((x,y,z)\longmapsto[b(x,y),fz]_{\mathfrak{b}}+b([x,y]_{\mathfrak{a}},z)\bigr) b ⟼ ( ( x , y , z ) ⟼ [ b ( x , y ) , f z ] b + b ([ x , y ] a , z ) ) injective. Injectivity is open for linear maps between fixed finite-dimensional spaces. Thus, for all nearby q q q
b ⟼ ( ( x , y , z ) ⟼ [ b ( x , y ) , q z ] b + b ( [ x , y ] a ′ , z ) ) b\longmapsto\bigl((x,y,z)\longmapsto[b(x,y),qz]_{\mathfrak{b}}+b([x,y]_{\mathfrak{a}}',z)\bigr) b ⟼ ( ( x , y , z ) ⟼ [ b ( x , y ) , q z ] b + b ([ x , y ] a ′ , z ) ) remains injective in a common neighborhood. Expanding the identity from Proposition 2.7 with the perturbed brackets gives
[ [ q x , q y ] b ′ , q z ] b ′ − q [ [ x , y ] a ′ , z ] a ′ = [ [ q x , q y ] b ′ − q [ x , y ] a ′ , q z ] b ′ + [ q [ x , y ] a ′ , q z ] b ′ − q [ [ x , y ] a ′ , z ] a ′ . [[qx,qy]_{\mathfrak{b}}',qz]_{\mathfrak{b}}'-q[[x,y]_{\mathfrak{a}}',z]_{\mathfrak{a}}'=[[qx,qy]_{\mathfrak{b}}'-q[x,y]_{\mathfrak{a}}',qz]_{\mathfrak{b}}'+[q[x,y]_{\mathfrak{a}}',qz]_{\mathfrak{b}}'-q[[x,y]_{\mathfrak{a}}',z]_{\mathfrak{a}}'. [[ q x , q y ] b ′ , q z ] b ′ − q [[ x , y ] a ′ , z ] a ′ = [[ q x , q y ] b ′ − q [ x , y ] a ′ , q z ] b ′ + [ q [ x , y ] a ′ , q z ] b ′ − q [[ x , y ] a ′ , z ] a ′ . By the displayed injectivity, the left-hand side vanishes for all x , y , z x,y,z x , y , z
[ q x , q y ] b ′ − q [ x , y ] a ′ = 0 ( x , y ∈ a ) . [qx,qy]_{\mathfrak{b}}'-q[x,y]_{\mathfrak{a}}'=0\qquad(x,y\in\mathfrak{a}). [ q x , q y ] b ′ − q [ x , y ] a ′ = 0 ( x , y ∈ a ) . This proves the uniform local equality of the morphism spaces and the equivalence of stability along the induced family of triple products.
The source is perfect by hypothesis. If the target is also perfect, every Lie inner action on either space is a sum of actions z ↦ [ [ u , v ] , z ] z\mapsto[[u,v],z] z ↦ [[ u , v ] , z ] H f 2 = 0 H_f^2=0 H f 2 = 0 H C E 2 ( f ) = 0 H_{\mathrm{CE}}^2(f)=0 H CE 2 ( f ) = 0
This stability equivalence concerns perturbations of triple products induced by binary Lie brackets. The injective map in Eq. (45) carries Lie obstruction classes to Lie triple obstruction classes.
Example 4.14. Let n \mathfrak{n} n p , q , z p,q,z p , q , z [ p , q ] = z [p,q]=z [ p , q ] = z
f t ( p ) = p , f t ( q ) = q , f t ( z ) = ( 1 + t ) z f_t(p)=p,\qquad f_t(q)=q,\qquad f_t(z)=(1+t)z f t ( p ) = p , f t ( q ) = q , f t ( z ) = ( 1 + t ) z form a smooth Lie triple deformation of id n \operatorname{id}_{\mathfrak{n}} id n t ≠ 0 t\ne0 t = 0 [ f t ( p ) , f t ( q ) ] = z ≠ f t ( [ p , q ] ) [f_t(p),f_t(q)]=z\ne f_t([p,q]) [ f t ( p ) , f t ( q )] = z = f t ([ p , q ]) ad p \operatorname{ad}_p ad p ad q \operatorname{ad}_q ad q (47) .
Triple subsystems and the graph comparison
In this section, we study triple subsystems through their normal representations and graph defects. We establish local rigidity, smoothness and stability, and compare their cohomology with morphism cohomology. Finally, we relate triple graph subsystems to graded Lie graph subalgebras and compare the normal differentials for Lie subalgebras and their associated triple subsystems.
Let E E E s ⊆ E \mathfrak{s}\subseteq E s ⊆ E E / s E/\mathfrak{s} E / s u ‾ = u + s \overline{u}=u+\mathfrak{s} u = u + s
Proposition 5.1. The maps
θ N ( x , y ) u ‾ = [ u , x , y ] ‾ , D N ( x , y ) u ‾ = [ x , y , u ] ‾ ( x , y ∈ s ) (54) \theta_N(x,y)\overline{u}=\overline{[u,x,y]},\qquad D_N(x,y)\overline{u}=\overline{[x,y,u]}\qquad(x,y\in\mathfrak{s}) \tag*{(54)} θ N ( x , y ) u = [ u , x , y ] , D N ( x , y ) u = [ x , y , u ] ( x , y ∈ s ) (54) define a representation of s \mathfrak{s} s E / s E/\mathfrak{s} E / s
Proof. For x , y , v ∈ s x,y,v\in\mathfrak{s} x , y , v ∈ s u ∈ E u\in E u ∈ E s \mathfrak{s} s
[ u + v , x , y ] ‾ = [ u , x , y ] + [ v , x , y ] ‾ = [ u , x , y ] ‾ , \overline{[u+v,x,y]}=\overline{[u,x,y]+[v,x,y]}=\overline{[u,x,y]}, [ u + v , x , y ] = [ u , x , y ] + [ v , x , y ] = [ u , x , y ] , [ x , y , u + v ] ‾ = [ x , y , u ] + [ x , y , v ] ‾ = [ x , y , u ] ‾ . \overline{[x,y,u+v]}=\overline{[x,y,u]+[x,y,v]}=\overline{[x,y,u]}. [ x , y , u + v ] = [ x , y , u ] + [ x , y , v ] = [ x , y , u ] . Thus the two quotient operators are well defined. Restrict the adjoint representation of E E E s \mathfrak{s} s u ↦ [ u , x , y ] u\mapsto[u,x,y] u ↦ [ u , x , y ] x , y ∈ s x,y\in\mathfrak{s} x , y ∈ s s \mathfrak{s} s E / s E/\mathfrak{s} E / s
[ u , y , x ] ‾ − [ u , x , y ] ‾ = [ x , y , u ] ‾ , \overline{[u,y,x]}-\overline{[u,x,y]}=\overline{[x,y,u]}, [ u , y , x ] − [ u , x , y ] = [ x , y , u ] , which is the required identity between θ N \theta_N θ N D N D_N D N □ \square □
For n ≥ 1 n\ge1 n ≥ 1
C N n ( s , E ) = C n − 1 ( s , E / s ) . C_N^n(\mathfrak{s},E)=C^{n-1}(\mathfrak{s},E/\mathfrak{s}). C N n ( s , E ) = C n − 1 ( s , E / s ) . Let ∂ n \partial^n ∂ n H N n ( s , E ) H_N^n(\mathfrak{s},E) H N n ( s , E )
H N 1 ( s , E ) = Z N 1 ( s , E ) = ker ∂ 1 . H_N^1(\mathfrak{s},E)=Z_N^1(\mathfrak{s},E)=\ker\partial^1. H N 1 ( s , E ) = Z N 1 ( s , E ) = ker ∂ 1 . In particular,
( ∂ 1 a ) ( x , y , z ) = θ N ( y , z ) a ( x ) − θ N ( x , z ) a ( y ) + D N ( x , y ) a ( z ) − a ( [ x , y , z ] ) . (55) (\partial^1a)(x,y,z)=\theta_N(y,z)a(x)-\theta_N(x,z)a(y)+D_N(x,y)a(z)-a([x,y,z]). \tag*{(55)} ( ∂ 1 a ) ( x , y , z ) = θ N ( y , z ) a ( x ) − θ N ( x , z ) a ( y ) + D N ( x , y ) a ( z ) − a ([ x , y , z ]) . (55) For comparison with inner orbits, augment this complex by
C ~ N 0 ( s , E ) = ⋀ 2 E , ( ∂ 0 X ) ( x ) = ad X x ‾ . (56) \widetilde{C}_N^0(\mathfrak{s},E)=\bigwedge\nolimits^2 E,\qquad(\partial^0X)(x)=\overline{\operatorname{ad}_Xx}. \tag*{(56)} C N 0 ( s , E ) = ⋀ 2 E , ( ∂ 0 X ) ( x ) = ad X x . (56) A derivation calculation as in Theorem 3.3 gives ∂ 1 ∂ 0 = 0 \partial^1\partial^0=0 ∂ 1 ∂ 0 = 0 H ~ N n \widetilde{H}_N^n H N n
H ~ N 1 = Z N 1 / im ∂ 0 , H ~ N n = H N n ( n ≥ 2 ) . \widetilde{H}_N^1=Z_N^1/\operatorname{im}\partial^0,\qquad\widetilde{H}_N^n=H_N^n\quad(n\ge2). H N 1 = Z N 1 / im ∂ 0 , H N n = H N n ( n ≥ 2 ) . Suppose now that E E E k = dim s k=\dim\mathfrak{s} k = dim s s \mathfrak{s} s k k k Gr k ( E ) \operatorname{Gr}_{k}(E) Gr k ( E ) s \mathfrak{s} s E = s ⊕ m E=\mathfrak{s}\oplus\mathfrak{m} E = s ⊕ m m \mathfrak{m} m E / s E/\mathfrak{s} E / s k k k
s a = { x + a ( x ) : x ∈ s } , a ∈ Hom ( s , m ) . \mathfrak{s}_{a}=\{x+a(x):x\in\mathfrak{s}\},\qquad a\in\operatorname{Hom}(\mathfrak{s},\mathfrak{m}). s a = { x + a ( x ) : x ∈ s } , a ∈ Hom ( s , m ) . For a trilinear tensor ν \nu ν
S ν ( a ) ( x , y , z ) = pr m ν ( x + a x , y + a y , z + a z ) − a pr s ν ( x + a x , y + a y , z + a z ) . (57) S_{\nu}(a)(x,y,z)=\operatorname{pr}_{\mathfrak{m}}\nu(x+ax,y+ay,z+az)-a\operatorname{pr}_{\mathfrak{s}}\nu(x+ax,y+ay,z+az). \tag*{(57)} S ν ( a ) ( x , y , z ) = pr m ν ( x + a x , y + a y , z + a z ) − a pr s ν ( x + a x , y + a y , z + a z ) . (57) This is a cochain in C 1 ( s , m ) C^{1}(\mathfrak{s},\mathfrak{m}) C 1 ( s , m ) s a \mathfrak{s}_{a} s a ν \nu ν
Proposition 5.2. The derivative of a smooth subsystem deformation belongs to Z N 1 ( s , E ) Z^{1}_{N}(\mathfrak{s},E) Z N 1 ( s , E )
( d S ν ) 0 = ∂ 1 (58) (dS_{\nu})_{0}=\partial^{1} \tag*{(58)} ( d S ν ) 0 = ∂ 1 (58) when ν \nu ν E E E s \mathfrak{s} s
Proof. For a ∈ Hom ( s , m ) a\in\operatorname{Hom}(\mathfrak{s},\mathfrak{m}) a ∈ Hom ( s , m ) x , y , z ∈ s x,y,z\in\mathfrak{s} x , y , z ∈ s
( d S ν ) 0 ( a ) ( x , y , z ) = pr m ( [ a x , y , z ] + [ x , a y , z ] + [ x , y , a z ] ) − a ( [ x , y , z ] ) . (dS_{\nu})_{0}(a)(x,y,z)=\operatorname{pr}_{\mathfrak{m}}\bigl([ax,y,z]+[x,ay,z]+[x,y,az]\bigr)-a([x,y,z]). ( d S ν ) 0 ( a ) ( x , y , z ) = pr m ( [ a x , y , z ] + [ x , a y , z ] + [ x , y , a z ] ) − a ([ x , y , z ]) . Under the identification of m \mathfrak{m} m E / s E/\mathfrak{s} E / s a t a_{t} a t a 0 = 0 a_{0}=0 a 0 = 0
pr m [ x + a t x , y + a t y , z + a t z ] − a t pr s [ x + a t x , y + a t y , z + a t z ] = 0. \operatorname{pr}_{\mathfrak{m}}[x+a_{t}x,y+a_{t}y,z+a_{t}z]-a_{t}\operatorname{pr}_{\mathfrak{s}}[x+a_{t}x,y+a_{t}y,z+a_{t}z]=0. pr m [ x + a t x , y + a t y , z + a t z ] − a t pr s [ x + a t x , y + a t y , z + a t z ] = 0. Differentiating at t = 0 t=0 t = 0
pr m ( [ a ˙ 0 x , y , z ] + [ x , a ˙ 0 y , z ] + [ x , y , a ˙ 0 z ] ) − a ˙ 0 ( [ x , y , z ] ) = 0. \operatorname{pr}_{\mathfrak{m}}\bigl([\dot{a}_{0}x,y,z]+[x,\dot{a}_{0}y,z]+[x,y,\dot{a}_{0}z]\bigr)-\dot{a}_{0}([x,y,z])=0. pr m ( [ a ˙ 0 x , y , z ] + [ x , a ˙ 0 y , z ] + [ x , y , a ˙ 0 z ] ) − a ˙ 0 ([ x , y , z ]) = 0. Hence ∂ 1 a ˙ 0 = 0 \partial^{1}\dot{a}_{0}=0 ∂ 1 a ˙ 0 = 0 Hom ( s , E / s ) \operatorname{Hom}(\mathfrak{s},E/\mathfrak{s}) Hom ( s , E / s )
We construct the compatibility map for (5.4) explicitly. Let
P s : C ∙ ( E , E ) ⟶ C ∙ ( s , m ) P_{\mathfrak{s}}:C^{\bullet}(E,E)\longrightarrow C^{\bullet}(\mathfrak{s},\mathfrak{m}) P s : C ∙ ( E , E ) ⟶ C ∙ ( s , m ) be restriction to s \mathfrak{s} s m \mathfrak{m} m s \mathfrak{s} s m \mathfrak{m} m m \mathfrak{m} m ker P s \ker P_{\mathfrak{s}} ker P s a a a a ^ \hat{a} a ^ E E E a ^ ∣ m = 0 \hat{a}|_{\mathfrak{m}}=0 a ^ ∣ m = 0
ν a = ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) , (59) \nu_{a}=(\operatorname{id}-\hat{a})\nu(\operatorname{id}+\hat{a},\operatorname{id}+\hat{a},\operatorname{id}+\hat{a}), \tag*{(59)} ν a = ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) , (59) B ν , a ( b ) = P s [ ν a , b ^ ] B , b ∈ C 1 ( s , m ) . (60) \mathcal{B}_{\nu,a}(b)=P_{\mathfrak{s}}[\nu_{a},\hat{b}]_{\mathcal{B}},\qquad b\in C^{1}(\mathfrak{s},\mathfrak{m}). \tag*{(60)} B ν , a ( b ) = P s [ ν a , b ^ ] B , b ∈ C 1 ( s , m ) . (60) Lemma 5.3. If ν \nu ν E E E
B ν , a ( S ν ( a ) ) = 0 (61) \mathcal{B}_{\nu,a}(S_{\nu}(a))=0 \tag*{(61)} B ν , a ( S ν ( a )) = 0 (61) for every a a a a = 0 a=0 a = 0 B ν , 0 = − ∂ 2 \mathcal{B}_{\nu,0}=-\partial^{2} B ν , 0 = − ∂ 2
Proof. Conjugation by id + a ^ \operatorname{id}+\hat{a} id + a ^ ker P s \ker P_{\mathfrak{s}} ker P s s \mathfrak{s} s m \mathfrak{m} m
0 = P s [ ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) , ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) ] B = 2 P s [ ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) , P s ( ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) ) ] B . \begin{aligned}
0=P_{\mathfrak{s}}\left[(\operatorname{id}-\hat{a})\nu(\operatorname{id}+\hat{a},\operatorname{id}+\hat{a},\operatorname{id}+\hat{a}),(\operatorname{id}-\hat{a})\nu(\operatorname{id}+\hat{a},\operatorname{id}+\hat{a},\operatorname{id}+\hat{a})\right]_{\mathcal{B}} \\
&=2P_{\mathfrak{s}}\left[(\operatorname{id}-\hat{a})\nu(\operatorname{id}+\hat{a},\operatorname{id}+\hat{a},\operatorname{id}+\hat{a}),P_{\mathfrak{s}}\bigl((\operatorname{id}-\hat{a})\nu(\operatorname{id}+\hat{a},\operatorname{id}+\hat{a},\operatorname{id}+\hat{a})\bigr)\right]_{\mathcal{B}}.
\end{aligned} 0 = P s [ ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) , ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) ] B = 2 P s [ ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) , P s ( ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) ) ] B . On inputs x , y , z ∈ s x,y,z\in\mathfrak{s} x , y , z ∈ s
pr m ν ( x + a x , y + a y , z + a z ) − a pr s ν ( x + a x , y + a y , z + a z ) . \operatorname{pr}_{\mathfrak{m}}\nu(x+ax,y+ay,z+az)-a\operatorname{pr}_{\mathfrak{s}}\nu(x+ax,y+ay,z+az). pr m ν ( x + a x , y + a y , z + a z ) − a pr s ν ( x + a x , y + a y , z + a z ) . Thus the last equality is precisely (5.8), by (5.7).
At the original law and a = 0 a=0 a = 0 b ∈ C 1 ( s , m ) b\in C^{1}(\mathfrak{s},\mathfrak{m}) b ∈ C 1 ( s , m )
( P s [ ν , b ^ ] B ) ( x , y , u , v , w ) = pr m [ b ( x , y , u ) , v , w ] + pr m [ u , b ( x , y , v ) , w ] + pr m [ u , v , b ( x , y , w ) ] − pr m [ x , y , b ( u , v , w ) ] + b ( [ x , y , u ] , v , w ) + b ( u , [ x , y , v ] , w ) + b ( u , v , [ x , y , w ] ) − b ( x , y , [ u , v , w ] ) = − ( ∂ 2 b ) ( x , y , u , v , w ) . (P_{\mathfrak{s}}[\nu,\widehat{b}]_{\mathrm{B}})(x,y,u,v,w)=\operatorname{pr}_{\mathfrak{m}}[b(x,y,u),v,w]+\operatorname{pr}_{\mathfrak{m}}[u,b(x,y,v),w]
+\operatorname{pr}_{\mathfrak{m}}[u,v,b(x,y,w)]-\operatorname{pr}_{\mathfrak{m}}[x,y,b(u,v,w)]+b([x,y,u],v,w)+b(u,[x,y,v],w)
+b(u,v,[x,y,w])-b(x,y,[u,v,w])=-(\partial^{2}b)(x,y,u,v,w). ( P s [ ν , b ] B ) ( x , y , u , v , w ) = pr m [ b ( x , y , u ) , v , w ] + pr m [ u , b ( x , y , v ) , w ] + pr m [ u , v , b ( x , y , w )] − pr m [ x , y , b ( u , v , w )] + b ([ x , y , u ] , v , w ) + b ( u , [ x , y , v ] , w ) + b ( u , v , [ x , y , w ]) − b ( x , y , [ u , v , w ]) = − ( ∂ 2 b ) ( x , y , u , v , w ) . This proves the asserted value at a = 0 a=0 a = 0 n − 1 n-1 n − 1 ( − 1 ) n − 1 (-1)^{n-1} ( − 1 ) n − 1 (54) .
Remark 5.4. If s \mathfrak{s} s ν \nu ν P s ν = 0 P_{\mathfrak{s}}\nu=0 P s ν = 0 L ∞ [ 1 ] L_{\infty}[1] L ∞ [ 1 ] C ∙ ( s , m ) C^{\bullet}(\mathfrak{s},\mathfrak{m}) C ∙ ( s , m )
m j ( b 1 , … , b j ) = P s [ ⋯ [ [ ν , b ^ 1 ] B , b ^ 2 ] B , … , b ^ j ] B . m_j(b_1,\ldots,b_j)=P_{\mathfrak{s}}[\cdots[[\nu,\widehat{b}_1]_{\mathrm{B}},\widehat{b}_2]_{\mathrm{B}},\ldots,\widehat{b}_j]_{\mathrm{B}}. m j ( b 1 , … , b j ) = P s [ ⋯ [[ ν , b 1 ] B , b 2 ] B , … , b j ] B . For a degree-zero map a a a S ν ( a ) S_{\nu}(a) S ν ( a ) a a a
− a pr s ν ( a x , a y , a z ) . -a\operatorname{pr}_{\mathfrak{s}}\nu(ax,ay,az). − a pr s ν ( a x , a y , a z ) . For morphism graphs, the ambient law is a direct product and the complementary target summand is an ideal. Hence pr s ν ( a x , a y , a z ) = 0 \operatorname{pr}_{\mathfrak{s}}\nu(ax,ay,az)=0 pr s ν ( a x , a y , a z ) = 0 (59) applies to every vector-space complement m \mathfrak{m} m
Definition 5.5. A k k k s \mathfrak{s} s ( E , ν ) (E,\nu) ( E , ν ) U U U s \mathfrak{s} s Gr k ( E ) \operatorname{Gr}_{k}(E) Gr k ( E ) ν ′ \nu' ν ′ E E E k k k U U U
Theorem 5.6. If H N 2 ( s , E ) = 0 H^{2}_{N}(\mathfrak{s},E)=0 H N 2 ( s , E ) = 0 k k k E E E s \mathfrak{s} s dim Z N 1 ( s , E ) \dim Z^{1}_{N}(\mathfrak{s},E) dim Z N 1 ( s , E ) s \mathfrak{s} s
Proof. In the Grassmannian graph chart, apply Lemma 4.4 to the polynomial map from Hom ( s , m ) \operatorname{Hom}(\mathfrak{s},\mathfrak{m}) Hom ( s , m ) C 1 ( s , m ) C^{1}(\mathfrak{s},\mathfrak{m}) C 1 ( s , m )
a ⟼ ( ( x , y , z ) ⟼ pr m ν ( x + a x , y + a y , z + a z ) − a pr s ν ( x + a x , y + a y , z + a z ) ) , a\longmapsto\bigl((x,y,z)\longmapsto\operatorname{pr}_{\mathfrak{m}}\nu(x+ax,y+ay,z+az)-a\operatorname{pr}_{\mathfrak{s}}\nu(x+ax,y+ay,z+az)\bigr), a ⟼ ( ( x , y , z ) ⟼ pr m ν ( x + a x , y + a y , z + a z ) − a pr s ν ( x + a x , y + a y , z + a z ) ) , and to the family of linear maps from C 1 ( s , m ) C^{1}(\mathfrak{s},\mathfrak{m}) C 1 ( s , m ) C 2 ( s , m ) C^{2}(\mathfrak{s},\mathfrak{m}) C 2 ( s , m )
b ⟼ P s [ ( id − a ^ ) ν ( id + a ^ , id + a ^ , id + a ^ ) , b ^ ] B . b\longmapsto P_{\mathfrak{s}}\bigl[(\operatorname{id}-\widehat{a})\nu(\operatorname{id}+\widehat{a},\operatorname{id}+\widehat{a},\operatorname{id}+\widehat{a}),\widehat{b}\bigr]_{\mathrm{B}}. b ⟼ P s [ ( id − a ) ν ( id + a , id + a , id + a ) , b ] B . Lemma 5.3 supplies their compatibility. The derivative of the polynomial map at zero sends a a a
( x , y , z ) ⟼ pr m ( [ a x , y , z ] + [ x , a y , z ] + [ x , y , a z ] ) − a ( [ x , y , z ] ) , (x,y,z)\longmapsto\operatorname{pr}_{\mathfrak{m}}\bigl([ax,y,z]+[x,ay,z]+[x,y,az]\bigr)-a([x,y,z]), ( x , y , z ) ⟼ pr m ( [ a x , y , z ] + [ x , a y , z ] + [ x , y , a z ] ) − a ([ x , y , z ]) , which is ∂ 1 a \partial^{1}a ∂ 1 a b b b − ∂ 2 b -\partial^{2}b − ∂ 2 b im ∂ 1 = ker ∂ 2 \operatorname{im}\partial^{1}=\ker\partial^{2} im ∂ 1 = ker ∂ 2 H N 2 ( s , E ) = 0 H^{2}_{N}(\mathfrak{s},E)=0 H N 2 ( s , E ) = 0 ker ∂ 1 \ker\partial^{1} ker ∂ 1 dim Z N 1 ( s , E ) \dim Z^{1}_{N}(\mathfrak{s},E) dim Z N 1 ( s , E )
For stability, allow ν ′ \nu' ν ′ ν \nu ν ν ′ \nu' ν ′ ( ν ′ , a ) (\nu',a) ( ν ′ , a ) ν ′ \nu' ν ′ a ( ν ′ ) a(\nu') a ( ν ′ ) a ( ν ) = 0 a(\nu)=0 a ( ν ) = 0
pr m ν ′ ( x + a ( ν ′ ) x , y + a ( ν ′ ) y , z + a ( ν ′ ) z ) = a ( ν ′ ) pr s ν ′ ( x + a ( ν ′ ) x , y + a ( ν ′ ) y , z + a ( ν ′ ) z ) . \operatorname{pr}_{\mathfrak{m}}\nu'(x+a(\nu')x,y+a(\nu')y,z+a(\nu')z)=a(\nu')\operatorname{pr}_{\mathfrak{s}}\nu'(x+a(\nu')x,y+a(\nu')y,z+a(\nu')z). pr m ν ′ ( x + a ( ν ′ ) x , y + a ( ν ′ ) y , z + a ( ν ′ ) z ) = a ( ν ′ ) pr s ν ′ ( x + a ( ν ′ ) x , y + a ( ν ′ ) y , z + a ( ν ′ ) z ) . Its graph is the required nearby subsystem. Continuity gives stability, and composition with a smooth curve of ambient laws gives a smooth curve of subsystems.
Theorem 5.7. If H ~ N 1 ( s , E ) = 0 \widetilde{H}^{1}_{N}(\mathfrak{s},E)=0 H N 1 ( s , E ) = 0 k k k s \mathfrak{s} s E E E s \mathfrak{s} s
Proof. For X ∈ ∧ 2 E X \in\wedge^{2}E X ∈ ∧ 2 E t ↦ exp ( t ad X ) s t \mapsto\exp(t\operatorname{ad}X)\mathfrak{s} t ↦ exp ( t ad X ) s
x ⟼ ad X x + s . x \longmapsto\operatorname{ad}X\,x+\mathfrak{s}. x ⟼ ad X x + s . Thus the tangent space to the inner orbit is im ∂ 0 \operatorname{im}\partial^{0} im ∂ 0 H ~ N 1 ( s , E ) = 0 \widetilde{H}^{1}_{N}(\mathfrak{s},E)=0 H N 1 ( s , E ) = 0 ker ∂ 1 \ker\partial^{1} ker ∂ 1
Choose a complement to im ∂ 0 \operatorname{im}\partial^{0} im ∂ 0 Hom ( s , m ) \operatorname{Hom}(\mathfrak{s},\mathfrak{m}) Hom ( s , m ) InnDer ( E ) \operatorname{InnDer}(E) InnDer ( E )
a ⟼ ( ( x , y , z ) ⟼ pr m ( [ a x , y , z ] + [ x , a y , z ] + [ x , y , a z ] ) − a ( [ x , y , z ] ) ) a\longmapsto\left((x,y,z)\longmapsto\operatorname{pr}_{\mathfrak{m}}\bigl([ax,y,z]+[x,ay,z]+[x,y,az]\bigr)-a([x,y,z])\right) a ⟼ ( ( x , y , z ) ⟼ pr m ( [ a x , y , z ] + [ x , a y , z ] + [ x , y , a z ] ) − a ([ x , y , z ]) ) is injective. The finite-dimensional Taylor estimate used in Theorem 4.3 therefore yields constants c , C > 0 c,C>0 c , C > 0 a a a
∥ pr m ν ( ⋅ + a ⋅ , ⋅ + a ⋅ , ⋅ + a ⋅ ) − a pr s ν ( ⋅ + a ⋅ , ⋅ + a ⋅ , ⋅ + a ⋅ ) ∥ ≥ c ∥ a ∥ − C ∥ a ∥ 2 . \left\|\operatorname{pr}_{\mathfrak{m}}\nu(\cdot+a\cdot,\cdot+a\cdot,\cdot+a\cdot)-a\operatorname{pr}_{\mathfrak{s}}\nu(\cdot+a\cdot,\cdot+a\cdot,\cdot+a\cdot)\right\|\geq c\lVert a\rVert-C\lVert a\rVert^{2}. ∥ pr m ν ( ⋅ + a ⋅ , ⋅ + a ⋅ , ⋅ + a ⋅ ) − a pr s ν ( ⋅ + a ⋅ , ⋅ + a ⋅ , ⋅ + a ⋅ ) ∥ ≥ c ∥ a ∥ − C ∥ a ∥ 2 . Hence a sufficiently small transverse graph is a subsystem only when a = 0 a=0 a = 0 s \mathfrak{s} s
Triple subsystems and the graph comparison
Proposition 5.8. Let f : ( g , π ) → ( h , μ ) f:(\mathfrak{g},\pi)\to(\mathfrak{h},\mu) f : ( g , π ) → ( h , μ )
G f = { ( x , f x ) : x ∈ g } ⊆ g ⊕ h . G_f=\{(x,fx):x\in\mathfrak{g}\}\subseteq\mathfrak{g}\oplus\mathfrak{h}. G f = {( x , f x ) : x ∈ g } ⊆ g ⊕ h . Then G f G_f G f f f f dim g \dim\mathfrak{g} dim g G f G_f G f
Proof. The direct product of ( x , f x ) , ( y , f y ) , ( z , f z ) (x,fx),(y,fy),(z,fz) ( x , f x ) , ( y , f y ) , ( z , f z )
( π ( x , y , z ) , μ ( f x , f y , f z ) ) = ( π ( x , y , z ) , f π ( x , y , z ) ) , \bigl(\pi(x,y,z),\mu(fx,fy,fz)\bigr)=\bigl(\pi(x,y,z),f\pi(x,y,z)\bigr), ( π ( x , y , z ) , μ ( f x , f y , f z ) ) = ( π ( x , y , z ) , f π ( x , y , z ) ) , which belongs to the graph of f f f μ ( f x , f y , f z ) = f π ( x , y , z ) \mu(fx,fy,fz)=f\pi(x,y,z) μ ( f x , f y , f z ) = f π ( x , y , z ) f f f 0 ⊕ h 0\oplus\mathfrak{h} 0 ⊕ h dim g \dim\mathfrak{g} dim g q : g → h q:\mathfrak{g}\to\mathfrak{h} q : g → h
μ ( q x , q y , q z ) = q π ( x , y , z ) . \mu(qx,qy,qz)=q\pi(x,y,z). μ ( q x , q y , q z ) = q π ( x , y , z ) . The Grassmannian graph chart and its inverse are smooth, proving the local identification.
There are canonical vector-space identifications
j f : g ⟶ G f , j f ( x ) = ( x , f x ) , Q f : ( g ⊕ h ) / G f ⟶ h , Q f ( ( x , u ) ) = u − f x . (62) j_f:\mathfrak{g}\longrightarrow G_f,\qquad j_f(x)=(x,fx),\qquad Q_f:(\mathfrak{g}\oplus\mathfrak{h})/G_f\longrightarrow\mathfrak{h},\qquad Q_f((x,u))=u-fx. \tag*{(62)} j f : g ⟶ G f , j f ( x ) = ( x , f x ) , Q f : ( g ⊕ h ) / G f ⟶ h , Q f (( x , u )) = u − f x . (62) Theorem 5.9. For every n ≥ 1 n\geq1 n ≥ 1
( Ξ n a ) ( j f x 1 , … , j f x 2 n − 1 ) = ( 0 , a ( x 1 , … , x 2 n − 1 ) ) . (63) (\Xi_n a)(j_fx_1,\ldots,j_fx_{2n-1})=(0,a(x_1,\ldots,x_{2n-1})). \tag*{(63)} ( Ξ n a ) ( j f x 1 , … , j f x 2 n − 1 ) = ( 0 , a ( x 1 , … , x 2 n − 1 )) . (63) Then Ξ n \Xi_n Ξ n G f G_f G f ∧ ∙ ( g ⊕ h ) \wedge^{\bullet}(\mathfrak{g}\oplus\mathfrak{h}) ∧ ∙ ( g ⊕ h )
H f n ≅ H ~ N n ( G f , g ⊕ h ) ( n ≥ 1 ) . (64) H_f^n\cong\widetilde{H}^{n}_{N}(G_f,\mathfrak{g}\oplus\mathfrak{h})\qquad(n\geq1). \tag*{(64)} H f n ≅ H N n ( G f , g ⊕ h ) ( n ≥ 1 ) . (64) For the unaugmented normal complex, the comparison is
H f n ≅ H N n ( G f , g ⊕ h ) ( n ≥ 2 ) , (65) H_f^n\cong H^{n}_{N}(G_f,\mathfrak{g}\oplus\mathfrak{h})\qquad(n\geq2), \tag*{(65)} H f n ≅ H N n ( G f , g ⊕ h ) ( n ≥ 2 ) , (65) 0 ⟶ B f 1 → Ξ 1 H N 1 ( G f , g ⊕ h ) ⟶ H f 1 ⟶ 0. (66) 0\longrightarrow B_f^1\xrightarrow{\Xi_1}H^{1}_{N}(G_f,\mathfrak{g}\oplus\mathfrak{h})\longrightarrow H_f^1\longrightarrow0. \tag*{(66)} 0 ⟶ B f 1 Ξ 1 H N 1 ( G f , g ⊕ h ) ⟶ H f 1 ⟶ 0. (66) Proof. The identification ( x , f x ) ↦ x (x,fx) \mapsto x ( x , f x ) ↦ x g \mathfrak{g} g ( z , u ) + G f ↦ u − f z (z,u)+G_f \mapsto u-fz ( z , u ) + G f ↦ u − f z (63) is an isomorphism in each positive degree. These identifications preserve the cochain conditions in the last three variables. For x , y , z ∈ g x,y,z \in\mathfrak{g} x , y , z ∈ g u ∈ h u \in\mathfrak{h} u ∈ h
θ N ( ( x , f x ) , ( y , f y ) ) ( z , u ) ‾ = ( π ( z , x , y ) , μ ( u , f x , f y ) ) ‾ = ( 0 , μ ( u , f x , f y ) − f π ( z , x , y ) ) ‾ = ( 0 , μ ( u − f z , f x , f y ) ) ‾ . \begin{aligned}
\theta_N((x,fx),(y,fy))\overline{(z,u)}=\overline{(\pi(z,x,y),\mu(u,fx,fy))}=\overline{(0,\mu(u,fx,fy)-f\pi(z,x,y))} \\
&=\overline{(0,\mu(u-fz,fx,fy))}.
\end{aligned} θ N (( x , f x ) , ( y , f y )) ( z , u ) = ( π ( z , x , y ) , μ ( u , f x , f y )) = ( 0 , μ ( u , f x , f y ) − f π ( z , x , y )) = ( 0 , μ ( u − f z , f x , f y )) . Thus the quotient identification carries this normal action to the coefficient action in (29) . The map x ↦ ( x , f x ) x \mapsto(x,fx) x ↦ ( x , f x ) (33) gives
( δ n Ξ n a ) ( ( x 1 , f x 1 ) , … , ( x 2 n + 1 , f x 2 n + 1 ) ) = ( 0 , ( δ f n a ) ( x 1 , … , x 2 n + 1 ) ) = ( Ξ n + 1 δ f n a ) ( ( x 1 , f x 1 ) , … , ( x 2 n + 1 , f x 2 n + 1 ) ) . \begin{aligned}
(\delta^n\Xi_na)((x_1,fx_1),\ldots,(x_{2n+1},fx_{2n+1}))=(0,(\delta^n_fa)(x_1,\ldots,x_{2n+1})) \\
&=(\Xi_{n+1}\delta^n_fa)((x_1,fx_1),\ldots,(x_{2n+1},fx_{2n+1})).
\end{aligned} ( δ n Ξ n a ) (( x 1 , f x 1 ) , … , ( x 2 n + 1 , f x 2 n + 1 )) = ( 0 , ( δ f n a ) ( x 1 , … , x 2 n + 1 )) = ( Ξ n + 1 δ f n a ) (( x 1 , f x 1 ) , … , ( x 2 n + 1 , f x 2 n + 1 )) . Hence the positive-degree complexes are identified.
For the incoming maps in degree one, use the canonical decomposition
Λ 2 ( g ⊕ h ) = Λ 2 g ⊕ ( g ⊗ h ) ⊕ Λ 2 h . \Lambda^2(\mathfrak{g}\oplus\mathfrak{h})=\Lambda^2\mathfrak{g}\oplus(\mathfrak{g}\otimes\mathfrak{h})\oplus\Lambda^2\mathfrak{h}. Λ 2 ( g ⊕ h ) = Λ 2 g ⊕ ( g ⊗ h ) ⊕ Λ 2 h . Mixed pairs act trivially in the direct product. For X ∈ Λ 2 g X\in\Lambda^2\mathfrak{g} X ∈ Λ 2 g U ∈ Λ 2 h U\in\Lambda^2\mathfrak{h} U ∈ Λ 2 h ( x , f x ) (x,fx) ( x , f x )
( ad X x , ad U f x ) ‾ = ( 0 , ad U f x − f ad X x ) ‾ = ( 0 , ( δ f 0 ( X , U ) ) ( x ) ) ‾ . \overline{(\operatorname{ad}_Xx,\operatorname{ad}_Ufx)}=\overline{(0,\operatorname{ad}_Ufx-f\operatorname{ad}_Xx)}=\overline{(0,(\delta_f^0(X,U))(x))}. ( ad X x , ad U f x ) = ( 0 , ad U f x − f ad X x ) = ( 0 , ( δ f 0 ( X , U )) ( x )) . Every ambient inner derivative has this form, and every pair ( X , U ) (X,U) ( X , U ) Ξ 1 \Xi_1 Ξ 1 im δ f 0 \operatorname{im}\delta_f^0 im δ f 0 im ∂ 0 \operatorname{im}\partial^0 im ∂ 0 (64) . The first unaugmented normal cohomology is the first cocycle space itself. Quotienting by the displayed boundary space gives (66) . The groups in degrees at least two are unchanged, proving (65) . □
Example 5.10. For the identity of the two-dimensional system in Example 4.9, H f 1 = 0 H_f^1=0 H f 1 = 0
H N 1 ( G f , T ⊕ T ) ≅ Z f 1 = R J . H_N^1(G_f,T\oplus T)\cong Z_f^1=\mathbb{R}J. H N 1 ( G f , T ⊕ T ) ≅ Z f 1 = R J . Thus the first unaugmented normal group has dimension one. Its quotient by the one-dimensional ambient inner orbit tangent space is zero, in agreement with (64) .
Theorem 5.9 and Proposition 5.8 show that the morphism and graph-subsystem rigidity criteria agree after augmentation. Their smoothness and stability criteria agree through the degree-two cohomology isomorphism. Under a perturbation of the two direct product factors, a nearby graph subsystem remains transverse to the target summand, so it again determines a morphism. For a general ambient perturbation of g ⊕ h \mathfrak{g}\oplus\mathfrak{h} g ⊕ h
We finish by relating the graph construction to Lie subalgebras and Lie algebra homomorphisms. For a triple subsystem s ⊆ E \mathfrak{s}\subseteq E s ⊆ E U ( s ) → U ( E ) \mathcal{U}(\mathfrak{s})\to\mathcal{U}(E) U ( s ) → U ( E ) s \mathfrak{s} s
[ s , s ] ( E ) ⊕ s . (67) [\mathfrak{s},\mathfrak{s}]_{(E)}\oplus\mathfrak{s}. \tag*{(67)} [ s , s ] ( E ) ⊕ s . (67) The kernel is contained in the even component and is central: it acts trivially on the odd generators, and these generate U ( s ) \mathcal{U}(\mathfrak{s}) U ( s ) (67) is the quotient of U ( s ) \mathcal{U}(\mathfrak{s}) U ( s )
Proposition 5.11. Let Φ = U ( f ) \Phi=\mathcal{U}(f) Φ = U ( f )
Γ Φ = { ( X , Φ X ) : X ∈ U ( g ) } ⊆ U ( g ) ⊕ U ( h ) . \Gamma_\Phi=\{(X,\Phi X):X\in\mathcal{U}(\mathfrak{g})\}\subseteq\mathcal{U}(\mathfrak{g})\oplus\mathcal{U}(\mathfrak{h}). Γ Φ = {( X , Φ X ) : X ∈ U ( g )} ⊆ U ( g ) ⊕ U ( h ) . Then Γ Φ \Gamma_\Phi Γ Φ
( Γ Φ ) 1 = G f , ( Γ Φ ) 0 = span { ( [ x , y ] , [ f x , f y ] ) : x , y ∈ g } . (68) (\Gamma_\Phi)_1=G_f,\qquad(\Gamma_\Phi)_0=\operatorname{span}\{([x,y],[fx,fy]):x,y\in\mathfrak{g}\}. \tag*{(68)} ( Γ Φ ) 1 = G f , ( Γ Φ ) 0 = span {([ x , y ] , [ f x , f y ]) : x , y ∈ g } . (68) In finite dimension, the map G q ↦ Γ U ( q ) G_q \mapsto\Gamma_{\mathcal{U}(q)} G q ↦ Γ U ( q ) Γ Φ \Gamma_\Phi Γ Φ
Proof. For X , Y ∈ U ( g ) X,Y \in\mathcal{U}(\mathfrak{g}) X , Y ∈ U ( g )
[ ( X , U ( f ) X ) , ( Y , U ( f ) Y ) ] = ( [ X , Y ] , [ U ( f ) X , U ( f ) Y ] ) = ( [ X , Y ] , U ( f ) [ X , Y ] ) . [(X,\mathcal{U}(f)X),(Y,\mathcal{U}(f)Y)] = ([X,Y],[\mathcal{U}(f)X,\mathcal{U}(f)Y]) = ([X,Y],\mathcal{U}(f)[X,Y]). [( X , U ( f ) X ) , ( Y , U ( f ) Y )] = ([ X , Y ] , [ U ( f ) X , U ( f ) Y ]) = ([ X , Y ] , U ( f ) [ X , Y ]) . Thus the graph is a Lie subalgebra, and it is graded because U ( f ) \mathcal{U}(f) U ( f ) ( x , f x ) (x,fx) ( x , f x ) x ∈ g x \in\mathfrak{g} x ∈ g
[ ( x , f x ) , ( y , f y ) ] = ( [ x , y ] , [ f x , f y ] ) . [(x,fx),(y,fy)] = ([x,y],[fx,fy]). [( x , f x ) , ( y , f y )] = ([ x , y ] , [ f x , f y ]) . These brackets span the even component, since the odd generators generate U ( g ) \mathcal{U}(\mathfrak{g}) U ( g )
In finite dimension, a sufficiently close graded plane of the same graded dimensions, transverse to the target summand, is the graph of a unique parity-preserving linear map. Closure of the graph under the bracket is exactly preservation of that bracket. Proposition 2.6 identifies these graded Lie homomorphisms with the universal extensions of Lie triple morphisms. Proposition 5.8 identifies their odd graphs with the nearby triple graph subsystems. The polynomial extension and linear restriction from Theorem 4.11, together with the smooth Grassmannian graph charts, give the stated local identification. □ \square □
Consequently, formal and local inner rigidity of a triple graph, with the augmentation in Eq. (56) , corresponds to inner rigidity of its graded Lie graph under even ambient inner automorphisms. Projection onto the source identifies the even graph with U ( g ) 0 \mathcal{U}(\mathfrak{g})_0 U ( g ) 0 U ( f ) \mathcal{U}(f) U ( f )
There is a further direct comparison when the triple systems come from prescribed Lie algebras. Let e \mathfrak{e} e s ⊆ e \mathfrak{s} \subseteq\mathfrak{e} s ⊆ e
ρ N ( x ) u ‾ = [ x , u ] , x ∈ s , u ‾ ∈ e / s . \rho_N(x)\overline{u}=[x,u], \qquad x\in\mathfrak{s},\quad\overline{u}\in\mathfrak{e}/\mathfrak{s}. ρ N ( x ) u = [ x , u ] , x ∈ s , u ∈ e / s . Its Chevalley–Eilenberg differential in degree one is
( d N a ) ( x , y ) = ρ N ( x ) a ( y ) − ρ N ( y ) a ( x ) − a ( [ x , y ] ) . (69) (d_Na)(x,y)=\rho_N(x)a(y)-\rho_N(y)a(x)-a([x,y]). \tag*{(69)} ( d N a ) ( x , y ) = ρ N ( x ) a ( y ) − ρ N ( y ) a ( x ) − a ([ x , y ]) . (69) For an alternating map b : ∧ 2 s → e / s b:\wedge^2\mathfrak{s}\to\mathfrak{e}/\mathfrak{s} b : ∧ 2 s → e / s
( T N b ) ( x , y , z ) = b ( [ x , y ] , z ) − ρ N ( z ) b ( x , y ) . (T_Nb)(x,y,z)=b([x,y],z)-\rho_N(z)b(x,y). ( T N b ) ( x , y , z ) = b ([ x , y ] , z ) − ρ N ( z ) b ( x , y ) . Proposition 5.12. For the associated Lie triple subsystem, the normal representation satisfies
θ N ( x , y ) = ρ N ( y ) ρ N ( x ) , D N ( x , y ) = ρ N ( [ x , y ] ) . (70) \theta_N(x,y)=\rho_N(y)\rho_N(x),\qquad D_N(x,y)=\rho_N([x,y]). \tag*{(70)} θ N ( x , y ) = ρ N ( y ) ρ N ( x ) , D N ( x , y ) = ρ N ([ x , y ]) . (70) Its first coboundary operator is related to the Lie normal differential by
∂ 1 a = T N ( d N a ) . (71) \partial^1a=T_N(d_Na). \tag*{(71)} ∂ 1 a = T N ( d N a ) . (71) Thus every infinitesimal Lie subalgebra deformation is an infinitesimal triple subsystem deformation. For a Lie homomorphism f : a → b f:\mathfrak{a}\to\mathfrak{b} f : a → b f f f (71) becomes Eq. (44) under the graph identifications.
Proof. For x , y ∈ s x,y\in\mathfrak{s} x , y ∈ s u ∈ e u\in\mathfrak{e} u ∈ e
ρ N ( y ) ρ N ( x ) ( u + s ) = [ y , [ x , u ] ] + s = [ [ u , x ] , y ] + s , ρ N ( [ x , y ] ) ( u + s ) = [ [ x , y ] , u ] + s . \rho_N(y)\rho_N(x)(u+\mathfrak{s})=[y,[x,u]]+\mathfrak{s}=[[u,x],y]+\mathfrak{s},\qquad\rho_N([x,y])(u+\mathfrak{s})=[[x,y],u]+\mathfrak{s}. ρ N ( y ) ρ N ( x ) ( u + s ) = [ y , [ x , u ]] + s = [[ u , x ] , y ] + s , ρ N ([ x , y ]) ( u + s ) = [[ x , y ] , u ] + s . These are exactly the two normal Lie triple actions in (54) . Expanding the right-hand side of (71) gives
ρ N ( [ x , y ] ) a ( z ) − ρ N ( z ) a ( [ x , y ] ) − a ( [ [ x , y ] , z ] ) − ρ N ( z ) ( ρ N ( x ) a ( y ) − ρ N ( y ) a ( x ) − a ( [ x , y ] ) ) = ρ N ( z ) ρ N ( y ) a ( x ) − ρ N ( z ) ρ N ( x ) a ( y ) + ρ N ( [ x , y ] ) a ( z ) − a ( [ [ x , y ] , z ] ) . \begin{aligned}
\rho_N([x,y])a(z)-\rho_N(z)a([x,y])-a([[x,y],z])-\rho_N(z)\bigl(\rho_N(x)a(y)-\rho_N(y)a(x)-a([x,y])\bigr) \\
= \rho_N(z)\rho_N(y)a(x)-\rho_N(z)\rho_N(x)a(y)+\rho_N([x,y])a(z)-a([[x,y],z]).
\end{aligned} ρ N ([ x , y ]) a ( z ) − ρ N ( z ) a ([ x , y ]) − a ([[ x , y ] , z ]) − ρ N ( z ) ( ρ N ( x ) a ( y ) − ρ N ( y ) a ( x ) − a ([ x , y ]) ) = ρ N ( z ) ρ N ( y ) a ( x ) − ρ N ( z ) ρ N ( x ) a ( y ) + ρ N ([ x , y ]) a ( z ) − a ([[ x , y ] , z ]) . The two action identities just proved identify this expression with
θ N ( y , z ) a ( x ) − θ N ( x , z ) a ( y ) + D N ( x , y ) a ( z ) − a ( [ [ x , y ] , z ] ) , \theta_N(y,z)a(x)-\theta_N(x,z)a(y)+D_N(x,y)a(z)-a([[x,y],z]), θ N ( y , z ) a ( x ) − θ N ( x , z ) a ( y ) + D N ( x , y ) a ( z ) − a ([[ x , y ] , z ]) , which is the first normal Lie triple differential. In particular, a vanishing first Lie normal differential gives a vanishing first Lie triple normal differential.
For the graph of f f f
ρ N ( ( x , f x ) ) ( y , u ) = ( [ x , y ] a , [ f x , u ] b ) = ( 0 , [ f x , u ] b − f [ x , y ] a ) = ( 0 , [ f x , u − f y ] b ) . \rho_N((x,fx))(y,u)=([x,y]_{\mathfrak a},[fx,u]_{\mathfrak b})=(0,[fx,u]_{\mathfrak b}-f[x,y]_{\mathfrak a})=(0,[fx,u-fy]_{\mathfrak b}). ρ N (( x , f x )) ( y , u ) = ([ x , y ] a , [ f x , u ] b ) = ( 0 , [ f x , u ] b − f [ x , y ] a ) = ( 0 , [ f x , u − f y ] b ) . Hence the quotient identification ( y , u ) + G f ↦ u − f y (y,u)+G_f\mapsto u-fy ( y , u ) + G f ↦ u − f y u ↦ [ f x , u ] b u\mapsto[fx,u]_{\mathfrak b} u ↦ [ f x , u ] b ( x , f x ) ↦ x (x,fx)\mapsto x ( x , f x ) ↦ x (44) . □
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, P. R. of China
Email address: tengwen@mail.gufe.edu.cn
Department of Mathematics, Rajagiri University Rajagni 733134, West Bengal, India
Email address: ripanjumaths@gmail.com
