Introduction
A linear map n : g → g n:g\to g n : g → g ( g , [ − , − ] ) (g,[-,-]) ( g , [ − , − ])
[ n x , n y ] = n ( [ n x , y ] + [ x , n y ] − n [ x , y ] ) , x , y ∈ g . (1) [nx,ny]=n\left([nx,y]+[x,ny]-n[x,y]\right), \qquad x,y\in g. \tag*{(1)} [ n x , n y ] = n ( [ n x , y ] + [ x , n y ] − n [ x , y ] ) , x , y ∈ g . (1) This condition originates in the theory of vector-valued differential forms [15 , 25 ] and in the Newlander–Nirenberg theorem [23 ] . It represents the infinitesimal condition governing linear deformations of a Lie bracket [17 , 32 ] , and it constitutes one of the two halves of Poisson–Nijenhuis and bi-Hamiltonian structures [14 , 9 , 31 ] . The cohomology and deformation theory of Nijenhuis Lie algebras were developed in [11 , 28 , 30 ] , while their equivariant versions were studied in [16 ] .
Independently, Lie algebras possess two standard higher analogues. Categorification replaces equalities by coherent isomorphisms, whereas homotopification replaces them by higher homotopies. These two procedures meet in the theorem of Baez and Crans [3 ] , which states that the category of Lie 2-algebras is equivalent to the category of 2-term L ∞ L_\infty L ∞ n n n n n n L ∞ L_\infty L ∞ [18 , 19 ] , that is, an L ∞ L_\infty L ∞ 0 , 1 , … , n − 1 0,1,\ldots,n-1 0 , 1 , … , n − 1 [4 , 5 , 10 , 20 ] .
Operators on higher Lie structures have traditionally been studied one level at a time. Rota–Baxter operators on Lie 2-algebras were introduced in [33 ] , and Nijenhuis operators on Lie 2-algebras were introduced by Das [13 ] , who proved that the category NijLie 2 _2 2 L ∞ L_\infty L ∞ NijLie 3 _3 3 [24 ] . At each level, the definition relies on an explicit list of component identities that grows rapidly in complexity: two identities at level one, three at level two, and seven at level three. However, the general pattern is not immediately transparent from these lists. Indeed, the transition from level two to level three introduces genuinely new phenomena: a 3-term L ∞ L_\infty L ∞ l 4 l_4 l 4 g 1 × g 1 → g 2 \mathfrak{g}_1 \times\mathfrak{g}_1 \to\mathfrak{g}_2 g 1 × g 1 → g 2
The rest of this paper is organized as follows. In § 2 , we recall the essential background on L ∞ L_\infty L ∞ n n n § 3 introduces the k k k § 4 , we define Nijenhuis operators on arbitrary and semistrict L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ § 5 studies the category NijLie n _n n n n n NijLie m ↪ _m \hookrightarrow m ↪ NijLie n _n n m ≤ n m \le n m ≤ n § 6 develops the theory of n n n § 7 investigates the semidirect product construction and proves that it establishes an equivalence between the category of such representations and split abelian NijLie n _n n § 8 and the appendices provide concluding remarks, document the explicit unshifted component identities up to arity five, and outline the computer algebra verifications.
Main Results
This paper establishes a unified framework for Nijenhuis operators on arbitrary L ∞ L_\infty L ∞
For a multilinear operation μ \mu μ k k k n n n k k k
Tors n ( μ ) ( a 1 , … , a k ) = ∑ S ⊆ { 1 , … , k } ( − 1 ) k − ∣ S ∣ n k − ∣ S ∣ μ ( a 1 S , … , a k S ) , a i S = { n ( a i ) , i ∈ S , a i , i ∉ S . (2) \operatorname{Tors}_{n}(\mu)(a_1,\ldots,a_k)=\sum_{S\subseteq\{1,\ldots,k\}}(-1)^{k-|S|}n^{k-|S|}\mu(a_1^S,\ldots,a_k^S),\qquad a_i^S=\begin{cases}n(a_i),&i\in S,\\a_i,&i\notin S.\end{cases} \tag*{(2)} Tors n ( μ ) ( a 1 , … , a k ) = S ⊆ { 1 , … , k } ∑ ( − 1 ) k − ∣ S ∣ n k − ∣ S ∣ μ ( a 1 S , … , a k S ) , a i S = { n ( a i ) , a i , i ∈ S , i ∈ / S . (2) recovering Tors n ( l 2 ) = 0 \operatorname{Tors}_{n}(l_2)=0 Tors n ( l 2 ) = 0 (1) . By Lemma 3.3, Tors n ( μ ) = ∂ 1 ∂ 2 ⋯ ∂ k μ \operatorname{Tors}_{n}(\mu)=\partial_1\partial_2\cdots\partial_k\mu Tors n ( μ ) = ∂ 1 ∂ 2 ⋯ ∂ k μ ∂ i μ = μ ( … , n a i , … ) − n μ ( … , a i , … ) \partial_i\mu=\mu(\ldots,na_i,\ldots)-n\mu(\ldots,a_i,\ldots) ∂ i μ = μ ( … , n a i , … ) − n μ ( … , a i , … ) k k k Tors λ n = λ k Tors n \operatorname{Tors}_{\lambda n}=\lambda^k\operatorname{Tors}_{n} Tors λn = λ k Tors n
Definition 1.1. A Nijenhuis operator on ( V , ℓ ) (V,\ell) ( V , ℓ ) N = ( N q ) q ≥ 1 N=(N_q)_{q\geq1} N = ( N q ) q ≥ 1 N q : S q V → V N_q:S^qV\to V N q : S q V → V N 1 = n N_1=n N 1 = n deg N q = q − 1 \deg N_q=q-1 deg N q = q − 1 m ≥ 1 m\geq1 m ≥ 1
endomorphism N N N L ∞ L_\infty L ∞ ( V , ℓ N ) (V,\ell^N) ( V , ℓ N ) ( V , ℓ ) (V,\ell) ( V , ℓ ) ℓ k N \ell_k^N ℓ k N (15) .
Torsion form (Theorem 4.6): Tors n ( ℓ m ) + Λ m N = 0 \operatorname{Tors}_n(\ell_m)+\Lambda_m^N=0 Tors n ( ℓ m ) + Λ m N = 0 Λ m N \Lambda_m^N Λ m N (24) aggregates partition sums and higher-component substitutions: Λ 2 N = d N 2 \Lambda_2^N=dN_2 Λ 2 N = d N 2 Λ 3 N \Lambda_3^N Λ 3 N Λ \Lambda Λ [12 ] and d N 3 dN_3 d N 3 Λ 4 N \Lambda_4^N Λ 4 N Ξ \Xi Ξ [24 ] ; Λ 5 N \Lambda_5^N Λ 5 N
By Lemma 2.3, n n n L ∞ L_\infty L ∞ ℓ k = 0 \ell_k=0 ℓ k = 0 k > n + 1 k>n+1 k > n + 1 N q = 0 N_q=0 N q = 0 q > n q>n q > n n + 1 n+1 n + 1 N i j L i e m ↪ N i j L i e n \mathbf{NijLie}_m\hookrightarrow\mathbf{NijLie}_n NijLie m ↪ NijLie n m ≤ n m\le n m ≤ n
N i j L i e 1 ↪ N i j L i e 2 ↪ N i j L i e 3 ↪ ⋯ ↪ N i j L i e ∞ = lim n N i j L i e n , \mathbf{NijLie}_1\hookrightarrow\mathbf{NijLie}_2\hookrightarrow\mathbf{NijLie}_3\hookrightarrow\cdots\hookrightarrow\mathbf{NijLie}_\infty=\lim_{n}\mathbf{NijLie}_n, NijLie 1 ↪ NijLie 2 ↪ NijLie 3 ↪ ⋯ ↪ NijLie ∞ = n lim NijLie n , which unifies classical Nijenhuis Lie algebras, Nijenhuis Lie 2-algebras [12 ] , and Nijenhuis Lie 3-algebras [24 ] within the category of Nijenhuis L ∞ L_\infty L ∞
Two Corrections at Level Three
Theorem 4.6 corrects two identities in the N i j L i e 3 NijLie3 N ij L i e 3 [24 ] :
The arity-three identity on g 0 × 3 \mathfrak{g}_0^{\times3} g 0 × 3 d N 3 ( x , y , z ) dN_3(x,y,z) d N 3 ( x , y , z ) [24 ] , (N5), corresponding to the single-block partition P = { { 1 , 2 , 3 } } P=\{\{1,2,3\}\} P = {{ 1 , 2 , 3 }}
Tors n ( ℓ 3 ) ( x , y , z ) + Λ 3 ( x , y , z ) + d N 3 ( x , y , z ) = 0 , x , y , z ∈ g 0 . (3) \operatorname{Tors}_n(\ell_3)(x,y,z)+\Lambda_3(x,y,z)+dN_3(x,y,z)=0,\qquad x,y,z\in\mathfrak{g}_0. \tag*{(3)} Tors n ( ℓ 3 ) ( x , y , z ) + Λ 3 ( x , y , z ) + d N 3 ( x , y , z ) = 0 , x , y , z ∈ g 0 . (3) In the expression for Ξ \Xi Ξ N 2 N_2 N 2 ℓ 3 \ell_3 ℓ 3 n n n n n n [24 ] , (3.22):
ℓ 3 ( n x k , n x l , N 2 ( x i , x j ) ) − n ℓ 3 ( x k , x l , N 2 ( x i , x j ) ) . (4) \ell_3(nx_k,nx_l,N_2(x_i,x_j))-n\ell_3(x_k,x_l,N_2(x_i,x_j)). \tag*{(4)} ℓ 3 ( n x k , n x l , N 2 ( x i , x j )) − n ℓ 3 ( x k , x l , N 2 ( x i , x j )) . (4) Consequently, the twisted operation ℓ 4 N \ell_4^N ℓ 4 N [24 ] , (3.23) is modified to:
ℓ 4 N = ℓ 4 ( n ) + A ‾ 4 ( N ) − B ‾ 4 ( N ) . (5) \ell_4^N=\ell_4^{(n)}+\overline{A}_4(N)-\overline{B}_4(N). \tag*{(5)} ℓ 4 N = ℓ 4 ( n ) + A 4 ( N ) − B 4 ( N ) . (5) Both corrections are verified symbolically in Appendix § 10 .
Throughout this paper, K \mathbb{K} K K \mathbb{K} K
Preliminaries on L ∞ L_\infty L ∞
Two Conventions
We employ two equivalent conventions, outlined below.
The unshifted convention (standard in [3 , 12 , 24 ] ) considers a graded vector space g = ⨁ i ≥ 0 g i \mathfrak{g}=\bigoplus_{i\ge0}\mathfrak{g}_i g = ⨁ i ≥ 0 g i μ : g ⊗ k → g \mu:\mathfrak{g}^{\otimes k}\to\mathfrak{g} μ : g ⊗ k → g p p p μ ( … , a , b , … ) = − μ ( … , b , a , … ) \mu(\ldots,a,b,\ldots)=-\mu(\ldots,b,a,\ldots) μ ( … , a , b , … ) = − μ ( … , b , a , … ) L ∞ L_\infty L ∞ ℓ k \ell_k ℓ k k − 2 k-2 k − 2 (6) .
The shifted convention relies on the décalage isomorphism V = g [ − 1 ] V=\mathfrak{g}[-1] V = g [ − 1 ] V i + 1 = g i V_{i+1}=\mathfrak{g}_i V i + 1 = g i μ \mu μ p p p graded symmetric map μ ^ : S k V → V \hat{\mu}:S^kV\to V μ ^ : S k V → V p + k − 1 p+k-1 p + k − 1 L ∞ L_\infty L ∞ { ℓ k } \{\ell_k\} { ℓ k } ℓ ^ k : S k V → V \hat{\ell}_k:S^kV\to V ℓ ^ k : S k V → V uniform degree − 1 -1 − 1 { N q } \{N_q\} { N q } uniform degree 0 0 0 § 2.2 onward, hats are omitted; unshifted component identities are deferred to Appendix § 9 .
Definition 2.1. An L ∞ L_{\infty} L ∞ g g g l k : g ⊗ k → g l_k:g^{\otimes k}\to g l k : g ⊗ k → g k − 2 k-2 k − 2 k ≥ 1 k\ge1 k ≥ 1 m ≥ 1 m\ge1 m ≥ 1 x 1 , … , x m ∈ g x_1,\ldots,x_m\in g x 1 , … , x m ∈ g
∑ i + j = m + 1 ( − 1 ) i ( j − 1 ) ∑ σ ∈ Unsh ( i , m − i ) ε ( σ ) l j ( l i ( x σ ( 1 ) , … , x σ ( i ) ) , x σ ( i + 1 ) , … , x σ ( m ) ) = 0 (6) \sum_{i+j=m+1}(-1)^{i(j-1)}\sum_{\sigma\in\operatorname{Unsh}(i,m-i)}\varepsilon(\sigma)l_j\left(l_i\left(x_{\sigma(1)},\ldots,x_{\sigma(i)}\right),x_{\sigma(i+1)},\ldots,x_{\sigma(m)}\right)=0 \tag*{(6)} i + j = m + 1 ∑ ( − 1 ) i ( j − 1 ) σ ∈ Unsh ( i , m − i ) ∑ ε ( σ ) l j ( l i ( x σ ( 1 ) , … , x σ ( i ) ) , x σ ( i + 1 ) , … , x σ ( m ) ) = 0 (6) where Unsh ( i , m − i ) \operatorname{Unsh}(i,m-i) Unsh ( i , m − i ) ( i , m − i ) (i,m-i) ( i , m − i ) ε ( σ ) \varepsilon(\sigma) ε ( σ ) x 1 , … , x m x_1,\ldots,x_m x 1 , … , x m d : = l 1 d:=l_1 d := l 1 [ , ] : = l 2 [\ ,\ ]:=l_2 [ , ] := l 2
The Shifted Convention
Let V V V S c V = ⨁ m ≥ 1 S m V S^cV=\bigoplus_{m\ge1}S^mV S c V = ⨁ m ≥ 1 S m V p p p ϕ : S c V → V \phi:S^cV\to V ϕ : S c V → V D ϕ D_\phi D ϕ p p p p = 0 p=0 p = 0 ϕ \phi ϕ Φ \Phi Φ S c V S^cV S c V π V Φ = ϕ \pi_V\Phi=\phi π V Φ = ϕ
Φ ( v 1 ⋯ v m ) = ∑ P ∈ Part ( m ) ε ( P ) ϕ ∣ B 1 ∣ ( v B 1 ) ⋯ ϕ ∣ B ∣ P ∣ ∣ ( v B ∣ P ∣ ) , (7) \Phi(v_1\cdots v_m)=\sum_{P\in\operatorname{Part}(m)}\varepsilon(P)\phi_{\lvert B_1\rvert}(v_{B_1})\cdots\phi_{\lvert B_{\lvert P\rvert}\rvert}(v_{B_{\lvert P\rvert}}), \tag*{(7)} Φ ( v 1 ⋯ v m ) = P ∈ Part ( m ) ∑ ε ( P ) ϕ ∣ B 1 ∣ ( v B 1 ) ⋯ ϕ ∣ B ∣ P ∣ ∣ ( v B ∣ P ∣ ) , (7) where Part ( m ) \operatorname{Part}(m) Part ( m ) { 1 , … , m } \{1,\ldots,m\} { 1 , … , m } B 1 , … , B ∣ P ∣ B_1,\ldots,B_{\lvert P\rvert} B 1 , … , B ∣ P ∣ v B : = ( v b 1 , … , v b ∣ B ∣ ) v_B:=(v_{b_1},\ldots,v_{b_{\lvert B\rvert}}) v B := ( v b 1 , … , v b ∣ B ∣ ) B = { b 1 < ⋯ < b ∣ B ∣ } B=\{b_1<\cdots<b_{\lvert B\rvert}\} B = { b 1 < ⋯ < b ∣ B ∣ } ϕ q : = ϕ ∣ S q V \phi_q:=\phi|_{S^qV} ϕ q := ϕ ∣ S q V ε ( P ) \varepsilon(P) ε ( P ) v 1 , … , v m v_1,\ldots,v_m v 1 , … , v m v B 1 , … , v B ∣ P ∣ v_{B_1},\ldots,v_{B_{\lvert P\rvert}} v B 1 , … , v B ∣ P ∣
An L ∞ L_{\infty} L ∞ V V V ( − 1 ) (-1) ( − 1 ) ℓ = ∑ k ≥ 1 l k : S c V → V \ell=\sum_{k\ge1}l_k:S^cV\to V ℓ = ∑ k ≥ 1 l k : S c V → V D ℓ 2 = 0 D_\ell^2=0 D ℓ 2 = 0
∑ k = 1 m ∑ σ ∈ Unsh ( k , m − k ) ε ( σ ) l m − k + 1 ( l k ( v σ ( 1 ) , … , v σ ( k ) ) , v σ ( k + 1 ) , … , v σ ( m ) ) = 0 (8) \sum_{k=1}^{m}\sum_{\sigma\in\operatorname{Unsh}(k,m-k)}\varepsilon(\sigma)l_{m-k+1}\left(l_k\left(v_{\sigma(1)},\ldots,v_{\sigma(k)}\right),v_{\sigma(k+1)},\ldots,v_{\sigma(m)}\right)=0 \tag*{(8)} k = 1 ∑ m σ ∈ Unsh ( k , m − k ) ∑ ε ( σ ) l m − k + 1 ( l k ( v σ ( 1 ) , … , v σ ( k ) ) , v σ ( k + 1 ) , … , v σ ( m ) ) = 0 (8) for all m ≥ 1 m\ge1 m ≥ 1 ε ( σ ) \varepsilon(\sigma) ε ( σ ) L ∞ L_{\infty} L ∞ F : ( V , ℓ ′ ) → ( V ′ , ℓ ) F:(V,\ell')\to(V',\ell) F : ( V , ℓ ′ ) → ( V ′ , ℓ ) 0 0 0 F = ∑ q ≥ 1 F q : S c V → V ′ F=\sum_{q\ge1}F_q:S^cV\to V' F = ∑ q ≥ 1 F q : S c V → V ′ F \mathbf F F D ℓ F = F D ℓ ′ D_\ell\mathbf F=\mathbf F D_{\ell'} D ℓ F = F D ℓ ′
∑ P ∈ Part ( m ) ε ( P ) l ∣ P ∣ ( F ∣ B 1 ∣ ( v B 1 ) , … , F ∣ B ∣ P ∣ ∣ ( v B ∣ P ∣ ) ) = ∑ k = 1 m ∑ σ ∈ Unsh ( k , m − k ) ε ( σ ) F m − k + 1 ( l k ′ ( v σ ( 1 ) , … ) , v σ ( k + 1 ) , … ) . (9) \begin{aligned}
\sum_{P\in\operatorname{Part}(m)}\varepsilon(P)l_{\lvert P\rvert}\left(F_{\lvert B_1\rvert}(v_{B_1}),\ldots,F_{\lvert B_{\lvert P\rvert}\rvert}(v_{B_{\lvert P\rvert}})\right) \\
&=\sum_{k=1}^{m}\sum_{\sigma\in\operatorname{Unsh}(k,m-k)}\varepsilon(\sigma)F_{m-k+1}\left(l'_k\left(v_{\sigma(1)},\ldots\right),v_{\sigma(k+1)},\ldots\right). \tag*{(9)}
\end{aligned} P ∈ Part ( m ) ∑ ε ( P ) l ∣ P ∣ ( F ∣ B 1 ∣ ( v B 1 ) , … , F ∣ B ∣ P ∣ ∣ ( v B ∣ P ∣ ) ) = k = 1 ∑ m σ ∈ Unsh ( k , m − k ) ∑ ε ( σ ) F m − k + 1 ( l k ′ ( v σ ( 1 ) , … ) , v σ ( k + 1 ) , … ) . (9) Composition of L ∞ L_{\infty} L ∞ L ∞ L_{\infty} L ∞ strict if F q = 0 F_q=0 F q = 0 q ≥ 2 q\ge2 q ≥ 2
Definition 2.2. An L ∞ L_{\infty} L ∞ g g g n n n semistrict Lie n n n [3 ] ) if g i = 0 g_i=0 g i = 0 i ≥ n i\ge n i ≥ n TermL n \operatorname{TermL}_n TermL n L ∞ L_{\infty} L ∞ n n n
In particular, 1-term, 2-term, and 3-term L ∞ L_{\infty} L ∞ [12 Def. 2.3] , and [24 Def. 2.2] , respectively.
Lemma 2.3. Let g g g n n n L ∞ L_{\infty} L ∞ μ : g ⊗ k → g \mu:g^{\otimes k}\to g μ : g ⊗ k → g p p p p > n − 1 p>n-1 p > n − 1 μ = 0 \mu=0 μ = 0
l k = 0 l_k=0 l k = 0 k > n + 1 k>n+1 k > n + 1 (8) are non-trivial only for 1 ≤ m ≤ n + 2 1\le m\le n+2 1 ≤ m ≤ n + 2
N q = 0 N_q=0 N q = 0 q > n q>n q > n Nijenhuis components { N q } \{N_q\} { N q }
F q = 0 F_q = 0 F q = 0 q > n q > n q > n L ∞ L_\infty L ∞ n n n
Proof. A map of degree p p p g i 1 ⊗ ⋯ ⊗ g i k g_{i_1} \otimes\cdots\otimes g_{i_k} g i 1 ⊗ ⋯ ⊗ g i k g ∑ i j + p g_{\sum i_j+p} g ∑ i j + p g p g_p g p g 0 ⊗ k g_0^{\otimes k} g 0 ⊗ k g p = 0 g_p = 0 g p = 0 p > n − 1 p > n-1 p > n − 1 μ = 0 \mu= 0 μ = 0 l k l_k l k k − 2 k-2 k − 2 k − 2 > n − 1 ⟹ l k = 0 k-2 > n-1 \Longrightarrow l_k = 0 k − 2 > n − 1 ⟹ l k = 0 (8) of arity m m m m − 3 m-3 m − 3 m − 3 > n − 1 m-3 > n-1 m − 3 > n − 1 N q N_q N q F q F_q F q
Remark 2.4. For n = 1 n = 1 n = 1 l 1 = 0 l_1 = 0 l 1 = 0 n = 2 n = 2 n = 2 l 1 , l 2 , l 3 l_1,l_2,l_3 l 1 , l 2 , l 3 ≤ 4 \leq4 ≤ 4 n = 3 n = 3 n = 3 l 1 , … , l 4 l_1,\ldots,l_4 l 1 , … , l 4 ≤ 5 \leq5 ≤ 5 [24 Def. 2.2] .
The Nijenhuis Torsion of a Multilinear Operation
Let g g g n : g → g n:g \to g n : g → g 0 0 0 μ : g ⊗ k → g \mu:g^{\otimes k} \to g μ : g ⊗ k → g k k k n n n
Definition 3.1. The k k k Nijenhuis torsion of μ \mu μ n n n k k k
Tors n ( μ ) ( a 1 , … , a k ) = ∑ S ⊆ { 1 , … , k } ( − 1 ) k − ∣ S ∣ n k − ∣ S ∣ μ ( a 1 S , … , a k S ) , a i S = { n ( a i ) , i ∈ S , a i , i ∉ S . (10) \operatorname{Tors}_n(\mu)(a_1,\ldots,a_k)=\sum_{S\subseteq\{1,\ldots,k\}}(-1)^{k-\lvert S\rvert}n^{k-\lvert S\rvert}\mu(a_1^S,\ldots,a_k^S),\qquad a_i^S=\begin{cases}n(a_i),&i\in S,\\a_i,&i\notin S.\end{cases} \tag*{(10)} Tors n ( μ ) ( a 1 , … , a k ) = S ⊆ { 1 , … , k } ∑ ( − 1 ) k − ∣ S ∣ n k − ∣ S ∣ μ ( a 1 S , … , a k S ) , a i S = { n ( a i ) , a i , i ∈ S , i ∈ / S . (10) and the n n n μ \mu μ
μ ( n ) ( a 1 , … , a k ) = ∑ S ⊊ { 1 , … , k } ( − 1 ) k − 1 − ∣ S ∣ n k − 1 − ∣ S ∣ μ ( a 1 S , … , a k S ) . (11) \mu^{(n)}(a_1,\ldots,a_k)=\sum_{S\subsetneq\{1,\ldots,k\}}(-1)^{k-1-\lvert S\rvert}n^{k-1-\lvert S\rvert}\mu(a_1^S,\ldots,a_k^S). \tag*{(11)} μ ( n ) ( a 1 , … , a k ) = S ⊊ { 1 , … , k } ∑ ( − 1 ) k − 1 − ∣ S ∣ n k − 1 − ∣ S ∣ μ ( a 1 S , … , a k S ) . (11) Isolating the maximal subset S = { 1 , … , k } S=\{1,\ldots,k\} S = { 1 , … , k } (10) yields the fundamental relation:
Tors n ( μ ) = μ ( n ⋅ , … , n ⋅ ) − n ∘ μ ( n ) . (12) \operatorname{Tors}_n(\mu)=\mu(n\cdot,\ldots,n\cdot)-n\circ\mu^{(n)}. \tag*{(12)} Tors n ( μ ) = μ ( n ⋅ , … , n ⋅ ) − n ∘ μ ( n ) . (12) Special cases include:
[ a , b ] n = [ , ] ( n ) ( a , b ) = [ n a , b ] + [ a , n b ] − n [ a , b ] , Tors n ( [ , ] ) ( a , b ) = [ n a , n b ] − n [ a , b ] n . (13) [a,b]_n=[\, ,\,]^{(n)}(a,b)=[na,b]+[a,nb]-n[a,b],\quad\operatorname{Tors}_n([\, ,\,])(a,b)=[na,nb]-n[a,b]_n. \tag*{(13)} [ a , b ] n = [ , ] ( n ) ( a , b ) = [ na , b ] + [ a , nb ] − n [ a , b ] , Tors n ([ , ]) ( a , b ) = [ na , nb ] − n [ a , b ] n . (13) recovering the classical condition Tors n ( [ , ] ) = 0 \operatorname{Tors}_n([\, ,\,])=0 Tors n ([ , ]) = 0 (1) .
μ ( n ) ( a , b , c ) = ∑ 2 slots μ ( n a , n b , c ) − ∑ 1 slot n μ ( n a , b , c ) + n 2 μ ( a , b , c ) , \mu^{(n)}(a,b,c)=\sum_{\text{2 slots}}\mu(na,nb,c)-\sum_{\text{1 slot}}n\mu(na,b,c)+n^2\mu(a,b,c), μ ( n ) ( a , b , c ) = 2 slots ∑ μ ( na , nb , c ) − 1 slot ∑ n μ ( na , b , c ) + n 2 μ ( a , b , c ) , Tors n ( μ ) = μ ( n a , n b , n c ) − n μ ( n ) ( a , b , c ) . \operatorname{Tors}_n(\mu)=\mu(na,nb,nc)-n\mu^{(n)}(a,b,c). Tors n ( μ ) = μ ( na , nb , n c ) − n μ ( n ) ( a , b , c ) . The algebraic structure underlying (10) is captured by elementary slot deformations:
Definition 3.2. For each 1 ≤ i ≤ k 1\leq i\leq k 1 ≤ i ≤ k i i i elementary slot deformation ∂ i \partial_i ∂ i k k k
( ∂ i μ ) ( a 1 , … , a k ) = μ ( a 1 , … , n a i , … , a k ) − n μ ( a 1 , … , a k ) . (\partial_i\mu)(a_1,\ldots,a_k)=\mu(a_1,\ldots,na_i,\ldots,a_k)-n\mu(a_1,\ldots,a_k). ( ∂ i μ ) ( a 1 , … , a k ) = μ ( a 1 , … , n a i , … , a k ) − n μ ( a 1 , … , a k ) . Lemma 3.3. The elementary slot deformations ∂ 1 , … , ∂ k \partial_1,\ldots,\partial_k ∂ 1 , … , ∂ k k k k
Tors n ( μ ) = ∂ 1 ∂ 2 ⋯ ∂ k μ . (14) \operatorname{Tors}_n(\mu)=\partial_1\partial_2\cdots\partial_k\mu. \tag*{(14)} Tors n ( μ ) = ∂ 1 ∂ 2 ⋯ ∂ k μ . (14) Consequently:
if μ \mu μ Tors n ( μ ) \operatorname{Tors}_n(\mu) Tors n ( μ ) μ \mu μ
Tors λ n ( μ ) = λ k Tors n ( μ ) \operatorname{Tors}_{\lambda n}(\mu) = \lambda^k \operatorname{Tors}_n(\mu) Tors λn ( μ ) = λ k Tors n ( μ ) λ ∈ K \lambda\in\mathbb{K} λ ∈ K
if μ \mu μ n n n W W W Tors n ( μ ) \operatorname{Tors}_n(\mu) Tors n ( μ )
Tors n ( μ ) = 0 \operatorname{Tors}_n(\mu) = 0 Tors n ( μ ) = 0 n ∘ μ = μ ∘ ( n ⊗ id ⊗ ⋯ ⊗ id ) n \circ\mu= \mu\circ(n \otimes\operatorname{id} \otimes\cdots\otimes\operatorname{id}) n ∘ μ = μ ∘ ( n ⊗ id ⊗ ⋯ ⊗ id ) ∂ i μ = 0 \partial_i\mu= 0 ∂ i μ = 0 i i i
Proof. Let σ i \sigma_i σ i n n n i i i μ ↦ μ ( … , n a i , … ) \mu\mapsto\mu(\ldots,na_i,\ldots) μ ↦ μ ( … , n a i , … ) τ \tau τ n n n μ ↦ n ∘ μ \mu\mapsto n \circ\mu μ ↦ n ∘ μ σ i σ j = σ j σ i \sigma_i\sigma_j = \sigma_j\sigma_i σ i σ j = σ j σ i τ σ i = σ i τ \tau\sigma_i = \sigma_i\tau τ σ i = σ i τ ∂ i = σ i − τ \partial_i = \sigma_i-\tau ∂ i = σ i − τ ∂ i \partial_i ∂ i
∂ 1 ⋯ ∂ k = ∏ i = 1 k ( σ i − τ ) = ∑ S ⊆ { 1 , … , k } ( − 1 ) k − ∣ S ∣ τ k − ∣ S ∣ ∏ i ∈ S σ i , \partial_1 \cdots\partial_k = \prod_{i=1}^{k}(\sigma_i-\tau) = \sum_{S\subseteq\{1,\ldots,k\}}(-1)^{k-|S|}\tau^{k-|S|}\prod_{i\in S}\sigma_i, ∂ 1 ⋯ ∂ k = i = 1 ∏ k ( σ i − τ ) = S ⊆ { 1 , … , k } ∑ ( − 1 ) k − ∣ S ∣ τ k − ∣ S ∣ i ∈ S ∏ σ i , where τ k − ∣ S ∣ ∏ i ∈ S σ i μ = n k − ∣ S ∣ μ ( a 1 S , … , a k S ) \tau^{k-|S|}\prod_{i\in S}\sigma_i\mu= n^{k-|S|}\mu(a_1^S,\ldots,a_k^S) τ k − ∣ S ∣ ∏ i ∈ S σ i μ = n k − ∣ S ∣ μ ( a 1 S , … , a k S ) (14) .
For (i), (skew-)symmetry is preserved because ∂ 1 ⋯ ∂ k \partial_1\cdots\partial_k ∂ 1 ⋯ ∂ k deg ( n ) = 0 \deg(n)=0 deg ( n ) = 0 ∂ i λ n = λ ∂ i n \partial_i^{\lambda n}=\lambda\partial_i^n ∂ i λn = λ ∂ i n ∂ i \partial_i ∂ i n n n W W W (14) and the commutativity of the deformations.
Remark 3.4. Lemma 3.3(iv) explains why the arity-1 condition in Definition 4.3 reduces simply to the requirement that n n n Tors n ( l 1 ) = ∂ 1 l 1 = l 1 n − n l 1 \operatorname{Tors}_n(l_1)=\partial_1l_1=l_1n-nl_1 Tors n ( l 1 ) = ∂ 1 l 1 = l 1 n − n l 1
Lemma 3.5. Let g \mathfrak{g} g n n n μ \mu μ k k k p p p Tors n ( μ ) \operatorname{Tors}_n(\mu) Tors n ( μ ) μ ( n ) \mu^{(n)} μ ( n ) k k k p p p p > n − 1 p>n-1 p > n − 1 Tors n ( l k ) = 0 \operatorname{Tors}_n(l_k)=0 Tors n ( l k ) = 0 k > n + 1 k>n+1 k > n + 1
Proof. Immediate from Lemma 3.3(i) and Lemma 2.3.
Nijenhuis Operators on L ∞ L_\infty L ∞
From this point onward, we adopt the shifted convention of Section 2.2: ( V , ℓ ) (V,\ell) ( V , ℓ ) L ∞ L_\infty L ∞ l k : S k V → V l_k:S^kV\to V l k : S k V → V − 1 -1 − 1
The Data
Definition 4.1. A Nijenhuis datum on V V V N = ( N q ) q ≥ 1 N=(N_q)_{q\geq1} N = ( N q ) q ≥ 1 N q : S q V → V N_q:S^qV\to V N q : S q V → V 0 0 0 n = N 1 n=N_1 n = N 1 linear part , and denote by N N N S c V S^cV S c V N N N
In the unshifted convention, a Nijenhuis datum corresponds to a family of skew-symmetric maps N q : g ⊗ q → g N_q:\mathfrak{g}^{\otimes q}\to\mathfrak{g} N q : g ⊗ q → g q − 1 q-1 q − 1 n n n N 2 N_2 N 2 N 3 N_3 N 3 n n n L ∞ L_\infty L ∞ n n n N 1 , … , N n N_1,\ldots,N_n N 1 , … , N n
n = 3 n=3 n = 3 n n n N 2 N_2 N 2 g 0 × g 0 \mathfrak{g}_0\times\mathfrak{g}_0 g 0 × g 0 g 0 × g 1 \mathfrak{g}_0\times\mathfrak{g}_1 g 0 × g 1 N 3 : g 0 × 3 → g 2 N_3:\mathfrak{g}_0^{\times3}\to\mathfrak{g}_2 N 3 : g 0 × 3 → g 2
This setup precisely recovers the structural data of [12 Def. 4.1] and [24 Def. 3.6] .
The Twisted Operations
Recall from § 2.2 that Part ( m ) \operatorname{Part}(m) Part ( m ) { 1 , … , m } \{1,\ldots,m\} { 1 , … , m }
the discrete partition , consisting entirely of singleton blocks (unique, with ∣ P ∣ = m |P|=m ∣ P ∣ = m
Part ∗ ( m ) \operatorname{Part}^{\ast}(m) Part ∗ ( m ) and at least one block of cardinality ≥ 2 \ge2 ≥ 2
Part ∘ ( m ) \operatorname{Part}^{\circ}(m) Part ∘ ( m ) ≥ 2 \ge2 ≥ 2
These three classes are mutually disjoint and exhaust Part ( m ) \operatorname{Part}(m) Part ( m ) N N N P P P B ∈ P B\in P B ∈ P
N B = N ∣ B ∣ ( v B ) , N ‾ B = { v i if B = { i } , N ∣ B ∣ ( v B ) if ∣ B ∣ ≥ 2. N_B=N_{|B|}(v_B),\qquad
\overline{N}_B=
\begin{cases}
v_i & \text{if }B=\{i\},\\
N_{|B|}(v_B) & \text{if }|B|\ge2.
\end{cases} N B = N ∣ B ∣ ( v B ) , N B = { v i N ∣ B ∣ ( v B ) if B = { i } , if ∣ B ∣ ≥ 2. where N ‾ B \overline{N}_B N B
Definition 4.2. Let N N N L ∞ L_\infty L ∞ ( V , ℓ ) (V,\ell) ( V , ℓ ) N N N ℓ m N : S m V → V \ell_m^N:S^mV\to V ℓ m N : S m V → V − 1 -1 − 1 m ≥ 1 m\ge1 m ≥ 1
ℓ m N = ℓ m ( n ) + A ‾ m ( N ) − B ‾ m ( N ) , (15) \ell_m^N=\ell_m^{(n)}+\overline{A}_m(N)-\overline{B}_m(N), \tag*{(15)} ℓ m N = ℓ m ( n ) + A m ( N ) − B m ( N ) , (15) where ℓ m ( n ) \ell_m^{(n)} ℓ m ( n ) n n n (11) of ℓ m \ell_m ℓ m
A ‾ m ( N ) ( v 1 , … , v m ) = ∑ P ∈ Part ∗ ( m ) ε ( P ) ℓ ∣ P ∣ ( N ‾ B 1 , … , N ‾ B ∣ P ∣ ) , (16) \overline{A}_m(N)(v_1,\ldots,v_m)
=
\sum_{P\in\operatorname{Part}^{\ast}(m)}
\varepsilon(P)\ell_{|P|}
\left(\overline{N}_{B_1},\ldots,\overline{N}_{B_{|P|}}\right), \tag*{(16)} A m ( N ) ( v 1 , … , v m ) = P ∈ Part ∗ ( m ) ∑ ε ( P ) ℓ ∣ P ∣ ( N B 1 , … , N B ∣ P ∣ ) , (16) B ‾ m ( N ) ( v 1 , … , v m ) = ∑ k = 2 m − 1 ∑ σ ∈ Unsh ( k , m − k ) ε ( σ ) N m − k + 1 ( ℓ k ( v σ ( 1 ) , … , v σ ( k ) ) , v σ ( k + 1 ) , … , v σ ( m ) ) . (17) \overline{B}_m(N)(v_1,\ldots,v_m)
=
\sum_{k=2}^{m-1}\sum_{\sigma\in\operatorname{Unsh}(k,m-k)}
\varepsilon(\sigma)N_{m-k+1}
\left(\ell_k(v_{\sigma(1)},\ldots,v_{\sigma(k)}),v_{\sigma(k+1)},\ldots,v_{\sigma(m)}\right). \tag*{(17)} B m ( N ) ( v 1 , … , v m ) = k = 2 ∑ m − 1 σ ∈ Unsh ( k , m − k ) ∑ ε ( σ ) N m − k + 1 ( ℓ k ( v σ ( 1 ) , … , v σ ( k ) ) , v σ ( k + 1 ) , … , v σ ( m ) ) . (17) Formula (15) provides a direct, closed expression rather than a recursion, as only the untwisted operations ℓ k \ell_k ℓ k
ℓ 1 N = ℓ 1 = d , ℓ 2 N = [ , ] n , ℓ 3 N = ℓ 3 ( n ) + ∑ ( 3 ) ( ℓ 2 ( v i , N 2 ( v j , v k ) ) − N 2 ( ℓ 2 ( v i , v j ) , v k ) ) . (18) \ell_1^N=\ell_1=d,\qquad
\ell_2^N=[\ ,\ ]_n,\qquad
\ell_3^N=\ell_3^{(n)}+
\sum_{(3)}
\left(\ell_2(v_i,N_2(v_j,v_k))-N_2(\ell_2(v_i,v_j),v_k)\right). \tag*{(18)} ℓ 1 N = ℓ 1 = d , ℓ 2 N = [ , ] n , ℓ 3 N = ℓ 3 ( n ) + ( 3 ) ∑ ( ℓ 2 ( v i , N 2 ( v j , v k )) − N 2 ( ℓ 2 ( v i , v j ) , v k ) ) . (18) where ∑ ( 3 ) \sum_{(3)} ∑ ( 3 ) Part ∗ ( 3 ) \operatorname{Part}^{\ast}(3) Part ∗ ( 3 ) ℓ 2 N \ell_2^N ℓ 2 N ℓ 3 N \ell_3^N ℓ 3 N [12 (7)] , [24 (3.21)] . For arity four, the twisted operation is ℓ 4 N = ℓ 4 ( n ) + A ‾ 4 − B ‾ 4 \ell_4^N=\ell_4^{(n)}+\overline{A}_4-\overline{B}_4 ℓ 4 N = ℓ 4 ( n ) + A 4 − B 4 A ‾ 4 \overline{A}_4 A 4 ( 2 , 1 , 1 ) (2,1,1) ( 2 , 1 , 1 ) ( 3 , 1 ) (3,1) ( 3 , 1 ) B ‾ 4 \overline{B}_4 B 4 N 3 ( ℓ 2 ( ⋅ , ⋅ ) , ⋅ ) N_3(\ell_2(\cdot,\cdot),\cdot) N 3 ( ℓ 2 ( ⋅ , ⋅ ) , ⋅ ) N 2 ( ℓ 3 ( ⋅ , ⋅ , ⋅ ) , ⋅ ) N_2(\ell_3(\cdot,\cdot,\cdot),\cdot) N 2 ( ℓ 3 ( ⋅ , ⋅ , ⋅ ) , ⋅ )
The Definition and Morphism Property
Definition 4.3. Let ( V , ℓ ) (V,\ell) ( V , ℓ ) L ∞ L_\infty L ∞ N = ( N q ) q ≥ 1 N=(N_q)_{q\ge1} N = ( N q ) q ≥ 1 Nijenhuis operator on ( V , ℓ ) (V,\ell) ( V , ℓ ) m ≥ 1 m\ge1 m ≥ 1 v 1 , … , v m ∈ V v_1,\ldots,v_m\in V v 1 , … , v m ∈ V
A ‾ m ( N ) = B ‾ m ( N ) , (19) \overline{A}_m(N)=\overline{B}_m(N), \tag*{(19)} A m ( N ) = B m ( N ) , (19) where
A m ( N ) ( v 1 , … , v m ) = ∑ P ∈ Part ( m ) ε ( P ) ℓ ∣ P ∣ ( N B 1 , … , N B ∣ P ∣ ) , (20) \mathcal{A}_m(N)(v_1,\ldots,v_m)=\sum_{P\in\operatorname{Part}(m)}\varepsilon(P)\,\ell_{\lvert P\rvert}(N_{B_1},\ldots,N_{B_{\lvert P\rvert}}), \tag*{(20)} A m ( N ) ( v 1 , … , v m ) = P ∈ Part ( m ) ∑ ε ( P ) ℓ ∣ P ∣ ( N B 1 , … , N B ∣ P ∣ ) , (20) B m ( N ) ( v 1 , … , v m ) = ∑ k = 1 m ∑ σ ∈ Unsh ( k , m − k ) ε ( σ ) N m − k + 1 ( ℓ k N ( v σ ( 1 ) , … , v σ ( k ) ) , v σ ( k + 1 ) , … , v σ ( m ) ) . (21) \mathcal{B}_m(N)(v_1,\ldots,v_m)=\sum_{k=1}^{m}\sum_{\sigma\in\operatorname{Unsh}(k,m-k)}\varepsilon(\sigma)\,N_{m-k+1}\left(\ell_k^N(v_{\sigma(1)},\ldots,v_{\sigma(k)}),v_{\sigma(k+1)},\ldots,v_{\sigma(m)}\right). \tag*{(21)} B m ( N ) ( v 1 , … , v m ) = k = 1 ∑ m σ ∈ Unsh ( k , m − k ) ∑ ε ( σ ) N m − k + 1 ( ℓ k N ( v σ ( 1 ) , … , v σ ( k ) ) , v σ ( k + 1 ) , … , v σ ( m ) ) . (21) A Nijenhuis L ∞ L_\infty L ∞ ( V , ℓ , N ) (V,\ell,N) ( V , ℓ , N ) L ∞ L_\infty L ∞ ( V , ℓ ) (V,\ell) ( V , ℓ ) n n n ( ( V , ℓ ) , N ) ((V,\ell),N) (( V , ℓ ) , N ) n n n n n n L ∞ L_\infty L ∞
Theorem 4.4. Let N N N ( V , ℓ ) (V,\ell) ( V , ℓ ) ℓ N = ∑ m ℓ m N \ell^N=\sum_m \ell_m^N ℓ N = ∑ m ℓ m N N N N
D ℓ ∘ N = N ∘ D ℓ N , (22) D_\ell\circ N=N\circ D_{\ell_N}, \tag*{(22)} D ℓ ∘ N = N ∘ D ℓ N , (22) that is, if and only if N N N L ∞ L_\infty L ∞ (9) mapping the twisted structure to the original one.
Proof. Both sides of (22) define maps S c V → S c V S^cV\to S^cV S c V → S c V N N N Δ ∘ X = ( N ⊗ X + X ⊗ N ) ∘ Δ \Delta\circ X=(N\otimes X+X\otimes N)\circ\Delta Δ ∘ X = ( N ⊗ X + X ⊗ N ) ∘ Δ π V ∘ X \pi_V\circ X π V ∘ X S m V S^mV S m V π V D ℓ N \pi_VD_{\ell_N} π V D ℓ N A m ( N ) \mathcal{A}_m(N) A m ( N ) (7) and the fact that D ℓ D_\ell D ℓ ℓ \ell ℓ π V N D ℓ N \pi_VND_{\ell_N} π V N D ℓ N B m ( N ) \mathcal{B}_m(N) B m ( N ) D ℓ N D_{\ell_N} D ℓ N ℓ N \ell^N ℓ N (22) holds if and only if A m = B m \mathcal{A}_m=\mathcal{B}_m A m = B m m m m (19) . The identification of (19) with (9) follows by reading this computation in reverse.
Remark 4.5. Theorem 4.4 captures the conceptual essence of Definition 4.3: a Nijenhuis operator is a datum that deforms the L ∞ L_\infty L ∞ L ∞ L_\infty L ∞ n n n ( g , [ , ] n ) (\mathfrak{g},[\, ,\,]_n) ( g , [ , ] n ) ( g , [ , ] ) (\mathfrak{g},[\, ,\,]) ( g , [ , ]) N 2 , N 3 , … N_2,N_3,\ldots N 2 , N 3 , … (22) does not presuppose a priori that ℓ N \ell^N ℓ N L ∞ L_\infty L ∞
Theorem 4.6. A Nijenhuis datum N N N ( V , ℓ ) (V,\ell) ( V , ℓ ) m ≥ 1 m\ge1 m ≥ 1
Tors N ( ℓ m ) + Λ m N = 0 , (23) \operatorname{Tors}_N(\ell_m)+\Lambda_m^N=0, \tag*{(23)} Tors N ( ℓ m ) + Λ m N = 0 , (23) where the correction term Λ m N \Lambda_m^N Λ m N
Λ m N = ∑ P ∈ Part ∗ ( m ) ε ( P ) ( ℓ ∣ P ∣ ( N B 1 , … , N B ∣ P ∣ ) − N ℓ ∣ P ∣ ( N ˉ B 1 , … , N ˉ B ∣ P ∣ ) ) + ∑ P ∈ Part ∘ ( m ) ε ( P ) ℓ ∣ P ∣ ( N B 1 , … , N B ∣ P ∣ ) − ∑ k = 2 m − 1 ∑ σ ∈ Unsh ( k , m − k ) ε ( σ ) ( N m − k + 1 ( ℓ k N ( v σ ( 1 ) , … ) , v σ ( k + 1 ) , … ) − n N m − k + 1 ( ℓ k ( v σ ( 1 ) , … ) , v σ ( k + 1 ) , … ) ) − ∑ σ ∈ Unsh ( 1 , m − 1 ) ε ( σ ) N m ( d v σ ( 1 ) , v σ ( 2 ) , … , v σ ( m ) ) . (24) \begin{aligned}
\Lambda_m^N={}&\sum_{P\in\operatorname{Part}^*(m)}\varepsilon(P)\left(\ell_{\lvert P\rvert}(N_{B_1},\ldots,N_{B_{\lvert P\rvert}})-N\ell_{\lvert P\rvert}(\bar{N}_{B_1},\ldots,\bar{N}_{B_{\lvert P\rvert}})\right)\\
&+\sum_{P\in\operatorname{Part}^{\circ}(m)}\varepsilon(P)\ell_{\lvert P\rvert}(N_{B_1},\ldots,N_{B_{\lvert P\rvert}})\\
&-\sum_{k=2}^{m-1}\sum_{\sigma\in\operatorname{Unsh}(k,m-k)}\varepsilon(\sigma)\left(N_{m-k+1}\left(\ell_k^N(v_{\sigma(1)},\ldots),v_{\sigma(k+1)},\ldots\right)\right.\\
&\qquad\left.-nN_{m-k+1}\left(\ell_k(v_{\sigma(1)},\ldots),v_{\sigma(k+1)},\ldots\right)\right)\\
&-\sum_{\sigma\in\operatorname{Unsh}(1,m-1)}\varepsilon(\sigma)N_m\left(dv_{\sigma(1)},v_{\sigma(2)},\ldots,v_{\sigma(m)}\right). \tag*{(24)}
\end{aligned} Λ m N = P ∈ Part ∗ ( m ) ∑ ε ( P ) ( ℓ ∣ P ∣ ( N B 1 , … , N B ∣ P ∣ ) − N ℓ ∣ P ∣ ( N ˉ B 1 , … , N ˉ B ∣ P ∣ ) ) + P ∈ Part ∘ ( m ) ∑ ε ( P ) ℓ ∣ P ∣ ( N B 1 , … , N B ∣ P ∣ ) − k = 2 ∑ m − 1 σ ∈ Unsh ( k , m − k ) ∑ ε ( σ ) ( N m − k + 1 ( ℓ k N ( v σ ( 1 ) , … ) , v σ ( k + 1 ) , … ) − n N m − k + 1 ( ℓ k ( v σ ( 1 ) , … ) , v σ ( k + 1 ) , … ) ) − σ ∈ Unsh ( 1 , m − 1 ) ∑ ε ( σ ) N m ( d v σ ( 1 ) , v σ ( 2 ) , … , v σ ( m ) ) . (24) Proof. In the expression for A m ( N ) \mathcal{A}_m(N) A m ( N ) ℓ m ( n v 1 , … , n v m ) \ell_m(nv_1,\ldots,nv_m) ℓ m ( n v 1 , … , n v m ) Part ∗ ( m ) ⊔ Part ∘ ( m ) \operatorname{Part}^*(m)\sqcup\operatorname{Part}^{\circ}(m) Part ∗ ( m ) ⊔ Part ∘ ( m ) B m ( N ) \mathcal{B}_m(N) B m ( N ) k = m k=m k = m n ∘ l m N n \circ l_m^N n ∘ l m N k = 1 k=1 k = 1 (24) with the opposite sign, since l 1 N = d l_1^N=d l 1 N = d (18) . Consequently, condition (19) is equivalent to:
l m ( n , … , n , ⋅ ) − n ∘ l m N + ∑ P ∈ Part ∗ ⊔ Part ∘ ε ( P ) l ∣ P ∣ ( N B ∙ ) − ∑ k = 2 m − 1 ∑ σ ε ( σ ) N m − k + 1 − ∑ σ ε ( σ ) N m ( d , … ) = 0. l_m(n,\ldots,n,\mathord{\cdot})-n\circ l_m^N+\sum_{P\in\operatorname{Part}^{\ast}\sqcup\operatorname{Part}^{\circ}}\varepsilon(P)l_{\lvert P\rvert}(N_{B_{\bullet}})-\sum_{k=2}^{m-1}\sum_{\sigma}\varepsilon(\sigma)N_{m-k+1}-\sum_{\sigma}\varepsilon(\sigma)N_m(d,\ldots)=0. l m ( n , … , n , ⋅ ) − n ∘ l m N + P ∈ Part ∗ ⊔ Part ∘ ∑ ε ( P ) l ∣ P ∣ ( N B ∙ ) − k = 2 ∑ m − 1 σ ∑ ε ( σ ) N m − k + 1 − σ ∑ ε ( σ ) N m ( d , … ) = 0. Substituting l m N = l m ( n ) + A ‾ m − B ‾ m l_m^N=l_m^{(n)}+\overline{A}_m-\overline{B}_m l m N = l m ( n ) + A m − B m (12) in the form l m ( n , … , n ) − n ∘ l m ( n ) = Tors n ( l m ) l_m(n,\ldots,n)-n\circ l_m^{(n)}=\operatorname{Tors}_n(l_m) l m ( n , … , n ) − n ∘ l m ( n ) = Tors n ( l m ) (23) . The terms − n A ‾ m -n\overline{A}_m − n A m + n B ‾ m +n\overline{B}_m + n B m (24) .
Remark 4.7 (Analysis of the Correction Terms). Each line of formula (24) carries a transparent structural meaning:
First line: Collects substitutions of higher components N q N_q N q q ≥ 2 q\ge2 q ≥ 2 l ∣ P ∣ l_{\lvert P\rvert} l ∣ P ∣ n v i nv_i n v i v i v_i v i n n n Theorem 4.4 demonstrates that this specific combination is uniquely forced by the morphism principle.
Second line: Collects substitutions with no free slots, carrying no outer n n n d N m dN_m d N m ≥ 2 \ge2 ≥ 2 quadratic in higher components, exemplified by l 2 ( N 2 ( v i , v j ) , N 2 ( v k , v l ) ) l_2(N_2(v_i,v_j),N_2(v_k,v_l)) l 2 ( N 2 ( v i , v j ) , N 2 ( v k , v l )) m = 4 m=4 m = 4 l 3 ( n v i , N 2 , N 2 ) l_3(nv_i,N_2,N_2) l 3 ( n v i , N 2 , N 2 ) m = 5 m=5 m = 5 Λ m N \Lambda_m^N Λ m N ⌊ m / 2 ⌋ \lfloor m/2\rfloor ⌊ m /2 ⌋
Third line: Collects substitutions of an operation into a higher component, with the inner operation being twisted in the first term and untwisted under the outer n n n
Fourth line: Represents the boundary correction N m ( d v σ ( 1 ) , … ) N_m(dv_{\sigma(1)},\ldots) N m ( d v σ ( 1 ) , … ) d N m dN_m d N m
Corollary 4.8. Let ( V , ℓ ) (V,\ell) ( V , ℓ ) n n n L ∞ L_\infty L ∞ (23) is satisfied trivially for m > n + 2 m>n+2 m > n + 2 Λ m N \Lambda_m^N Λ m N N 1 , … , N min ( m , n ) N_1,\ldots,N_{\min(m,n)} N 1 , … , N m i n ( m , n ) l 1 , … , l min ( m , n + 1 ) l_1,\ldots,l_{\min(m,n+1)} l 1 , … , l m i n ( m , n + 1 ) n n n L ∞ L_\infty L ∞ 1 ≤ m ≤ n + 2 1\le m\le n+2 1 ≤ m ≤ n + 2
Proof. Identity (23) of arity m m m S m V → V S^mV\to V S m V → V − 1 -1 − 1 g ⊗ m → g g^{\otimes m}\to g g ⊗ m → g m − 2 m-2 m − 2 m − 2 > n − 1 m-2>n-1 m − 2 > n − 1 Lemma 3.5 .
Remark 4.9. For n = 1 n=1 n = 1 m ≤ 3 m\le3 m ≤ 3 Tors n ( [ , ] ) = 0 \operatorname{Tors}_n([\ ,\ ])=0 Tors n ([ , ]) = 0 n = 2 n=2 n = 2 m ≤ 4 m\le4 m ≤ 4 [12 Def. 4.1] . For n = 3 n=3 n = 3 m ≤ 5 m\le5 m ≤ 5 § 9 ).
The Twisted Structure
Theorem 4.10. Let N N N ( V , ℓ ) (V,\ell) ( V , ℓ )
N ∘ D ℓ 2 N = 0 N\circ D_\ell^2{}_N=0 N ∘ D ℓ 2 N = 0 L ∞ L_\infty L ∞ ℓ N \ell^N ℓ N N N N
If the linear part n n n ℓ N \ell^{N} ℓ N L ∞ L_{\infty} L ∞ V V V N : ( V , ℓ N ) → ( V , ℓ ) N:(V,\ell^{N})\to(V,\ell) N : ( V , ℓ N ) → ( V , ℓ ) L ∞ L_{\infty} L ∞ n n n
If n n n N N N L ∞ L_{\infty} L ∞ ℓ N \ell^{N} ℓ N ℓ \ell ℓ
Proof. (i) By Theorem 4.4 , we have D ℓ N = N D ℓ N D_{\ell}N=ND_{\ell^{N}} D ℓ N = N D ℓ N
N ∘ D ℓ N 2 = D ℓ ∘ N ∘ D ℓ N = D ℓ 2 ∘ N = 0 , N\circ D_{\ell^{N}}^{2}=D_{\ell}\circ N\circ D_{\ell^{N}}=D_{\ell}^{2}\circ N=0, N ∘ D ℓ N 2 = D ℓ ∘ N ∘ D ℓ N = D ℓ 2 ∘ N = 0 , since D ℓ 2 = 0 D_{\ell}^{2}=0 D ℓ 2 = 0 L ∞ L_{\infty} L ∞
(ii) Assume that n n n N N N F m = ⨁ j ≤ m S j V F_m=\bigoplus_{j\leq m}S^jV F m = ⨁ j ≤ m S j V S m V = F m / F m − 1 S^mV=F_m/F_{m-1} S m V = F m / F m − 1 S m ( n ) S^m(n) S m ( n ) n n n N N N D ℓ N 2 = 0 D_{\ell^{N}}^{2}=0 D ℓ N 2 = 0 ℓ N \ell^{N} ℓ N L ∞ L_{\infty} L ∞ (22) then asserts precisely that N N N L ∞ L_{\infty} L ∞
(iii) If n n n N N N D ℓ N = N − 1 ∘ D ℓ ∘ N D_{\ell^{N}}=N^{-1}\circ D_{\ell}\circ N D ℓ N = N − 1 ∘ D ℓ ∘ N
Remark 4.11. Injectivity in (ii) is a technical convenience rather than an absolute necessity: Proposition 6.3 below establishes the same conclusion for strict objects at every level without any hypothesis on n n n ℓ N \ell^{N} ℓ N L ∞ L_{\infty} L ∞ ker N \operatorname{ker}N ker N L ∞ L_{\infty} L ∞ ℓ N \ell^{N} ℓ N ker N \operatorname{ker}N ker N
Morphisms and the Categories NijLien
Let φ \varphi φ L ∞ L_{\infty} L ∞ φ = ( φ q ) q ≥ 1 \varphi=(\varphi_q)_{q\geq1} φ = ( φ q ) q ≥ 1 φ \varphi φ p p p p p p Ψ : S c V → S c V ′ \Psi:S^cV\to S^cV' Ψ : S c V → S c V ′ Δ Ψ = ( Ψ ⊗ φ + φ ⊗ Ψ ) Δ \Delta\Psi=(\Psi\otimes\varphi+\varphi\otimes\Psi)\Delta ΔΨ = ( Ψ ⊗ φ + φ ⊗ Ψ ) Δ Ψ \Psi Ψ ψ = π V ′ Ψ = ∑ q ≥ 1 ψ q \psi=\pi_{V'}\Psi=\sum_{q\geq1}\psi_q ψ = π V ′ Ψ = ∑ q ≥ 1 ψ q ψ q : S q V → V ′ \psi_q:S^qV\to V' ψ q : S q V → V ′ p p p φ \varphi φ
Definition 4.12. Let ( V , ℓ ) , N (V,\ell),N ( V , ℓ ) , N ( V ′ , ℓ ′ ) , N ′ (V',\ell'),N' ( V ′ , ℓ ′ ) , N ′ L ∞ L_{\infty} L ∞ ( φ , ψ ) : ( V , ℓ ) , N → ( V ′ , ℓ ′ ) , N ′ (\varphi,\psi):(V,\ell),N\to(V',\ell'),N' ( φ , ψ ) : ( V , ℓ ) , N → ( V ′ , ℓ ′ ) , N ′
An L ∞ L_{\infty} L ∞ φ : ( V , ℓ ) → ( V ′ , ℓ ′ ) \varphi:(V,\ell)\to(V',\ell') φ : ( V , ℓ ) → ( V ′ , ℓ ′ ) L ∞ L_{\infty} L ∞ φ : ( V , ℓ N ) → ( V ′ , ℓ N ′ ) \varphi:(V,\ell^{N})\to(V',\ell^{N'}) φ : ( V , ℓ N ) → ( V ′ , ℓ N ′ )
D ℓ ′ ∘ φ = φ ∘ D ℓ and D ℓ N ′ ∘ φ = φ ∘ D ℓ N ; D_{\ell'}\circ\varphi=\varphi\circ D_{\ell}\quad\text{and}\quad D_{\ell^{N'}}\circ\varphi=\varphi\circ D_{\ell^{N}}; D ℓ ′ ∘ φ = φ ∘ D ℓ and D ℓ N ′ ∘ φ = φ ∘ D ℓ N ; A family ψ = ( ψ q ) q ≥ 1 \psi=(\psi_q)_{q\geq1} ψ = ( ψ q ) q ≥ 1 ψ q : S q V → V ′ \psi_q:S^qV\to V' ψ q : S q V → V ′ + 1 +1 + 1 φ \varphi φ Ψ \Psi Ψ
φ ∘ N − N ′ ∘ φ = D ℓ ′ ∘ Ψ + Ψ ∘ D ℓ N . (25) \varphi\circ N-N'\circ\varphi=D_{\ell'}\circ\Psi+\Psi\circ D_{\ell^{N}}. \tag*{(25)} φ ∘ N − N ′ ∘ φ = D ℓ ′ ∘ Ψ + Ψ ∘ D ℓ N . (25) A morphism is said to be strict if φ q = 0 \varphi_q=0 φ q = 0 q ≥ 2 q\geq2 q ≥ 2 ψ = 0 \psi=0 ψ = 0
Condition (25) expresses that φ ∘ N \varphi\circ N φ ∘ N N ′ ∘ φ N'\circ\varphi N ′ ∘ φ L ∞ L_{\infty} L ∞ ( V , ℓ N ) (V,\ell^{N}) ( V , ℓ N ) ( V ′ , ℓ ′ ) (V',\ell') ( V ′ , ℓ ′ ) L ∞ L_{\infty} L ∞ Ψ \Psi Ψ V ′ V' V ′
φ 1 n − n ′ φ 1 = d ′ ψ 1 + ψ 1 d , \varphi_1n-n'\varphi_1=d'\psi_1+\psi_1d, φ 1 n − n ′ φ 1 = d ′ ψ 1 + ψ 1 d , while its arity-two component governs the relation between φ 1 N 2 − N 2 ′ ( φ 1 , φ 1 ) \varphi_1N_2-N'_2(\varphi_1,\varphi_1) φ 1 N 2 − N 2 ′ ( φ 1 , φ 1 ) ψ 1 , ψ 2 , φ 2 \psi_1,\psi_2,\varphi_2 ψ 1 , ψ 2 , φ 2 [12 (H4)] for 2-term objects and [24 (H6)] for 3-term objects.
Proposition 4.13. Nijenhuis L ∞ L_{\infty} L ∞ Definition 4.12 form a category. The composition of ( φ , ψ ) (\varphi,\psi) ( φ , ψ ) ( φ ′ , ψ ′ ) (\varphi',\psi') ( φ ′ , ψ ′ ) L ∞ L_{\infty} L ∞ φ ′ ∘ φ \varphi'\circ\varphi φ ′ ∘ φ
Ψ ′ ∗ Ψ : = φ ′ ∘ Ψ + Ψ ′ ∘ φ , (26) \Psi' * \Psi:= \varphi' \circ\Psi+ \Psi' \circ\varphi, \tag*{(26)} Ψ ′ ∗ Ψ := φ ′ ∘ Ψ + Ψ ′ ∘ φ , (26) with the identity morphism of ( V , ℓ , N ) (V,\ell,N) ( V , ℓ , N ) ( id , 0 ) (\operatorname{id},0) ( id , 0 ) n n n N i j L i e n \mathbf{NijLie}_{n} NijLie n N i j T e r m n \mathbf{NijTerm}_{n} NijTerm n
Proof. First, Ψ ′ ∗ Ψ \Psi' * \Psi Ψ ′ ∗ Ψ φ ′ φ \varphi'\varphi φ ′ φ + 1 +1 + 1 φ ′ \varphi' φ ′ Ψ \Psi Ψ φ \varphi φ Ψ ′ \Psi' Ψ ′ φ ′ \varphi' φ ′ φ ′ ∘ φ \varphi' \circ\varphi φ ′ ∘ φ D ℓ ′ ′ φ ′ = φ ′ D ℓ ′ D_{\ell''}\varphi' = \varphi'D_{\ell'} D ℓ ′′ φ ′ = φ ′ D ℓ ′ D ℓ ′ N ′ φ = φ D ℓ N D_{\ell'}N'\varphi= \varphi D_{\ell}N D ℓ ′ N ′ φ = φ D ℓ N (25) for both morphisms, we compute:
D ℓ ′ ′ ( Ψ ′ ∗ Ψ ) + ( Ψ ′ ∗ Ψ ) D ℓ N = φ ′ ( D ℓ ′ Ψ + Ψ D ℓ N ) + ( D ℓ ′ ′ Ψ ′ + Ψ ′ D ℓ ′ ) φ = φ ′ ( φ N − N ′ φ ) + ( φ ′ N ′ − N ′ ′ φ ′ ) φ = φ ′ φ N − N ′ ′ φ ′ φ . \begin{aligned}
D_{\ell''}(\Psi' * \Psi) + (\Psi' * \Psi)D_{\ell}N = \varphi'(D_{\ell'}\Psi+ \Psi D_{\ell}N) + (D_{\ell''}\Psi' + \Psi'D_{\ell'})\varphi\\
&= \varphi'(\varphi N - N'\varphi) + (\varphi'N' - N''\varphi')\varphi= \varphi'\varphi N - N''\varphi'\varphi.
\end{aligned} D ℓ ′′ ( Ψ ′ ∗ Ψ ) + ( Ψ ′ ∗ Ψ ) D ℓ N = φ ′ ( D ℓ ′ Ψ + Ψ D ℓ N ) + ( D ℓ ′′ Ψ ′ + Ψ ′ D ℓ ′ ) φ = φ ′ ( φN − N ′ φ ) + ( φ ′ N ′ − N ′′ φ ′ ) φ = φ ′ φN − N ′′ φ ′ φ . which corresponds precisely to relation (25) for the composite morphism. Associativity of the composition ∗ * ∗ (26) , while unitality is immediate. The fullness of N i j L i e n \mathbf{NijLie}_{n} NijLie n L ∞ L_{\infty} L ∞
Definition 4.14. A Nijenhuis L ∞ L_{\infty} L ∞ ( ( V , ℓ ) , N ) ((V,\ell),N) (( V , ℓ ) , N )
strict if ℓ k = 0 \ell_k = 0 ℓ k = 0 k ≥ 3 k \ge3 k ≥ 3 N q = 0 N_q = 0 N q = 0 q ≥ 2 q \ge2 q ≥ 2
skeletal if ℓ 1 = 0 \ell_1 = 0 ℓ 1 = 0
We denote by S N i j L i e n ⊆ N i j L i e n \mathbf{SNijLie}_{n} \subseteq\mathbf{NijLie}_{n} SNijLie n ⊆ NijLie n
The Tower of Levels
Theorem 5.1. Let 1 ≤ m ≤ n 1 \le m \le n 1 ≤ m ≤ n N i j L i e n < m \mathbf{NijLie}_{n}^{<m} NijLie n < m N i j L i e n \mathbf{NijLie}_{n} NijLie n ( ( V , ℓ ) , N ) ((V,\ell),N) (( V , ℓ ) , N ) 0 , … , m − 1 0,\ldots,m-1 0 , … , m − 1
N i j L i e m ≅ N i j L i e n < m . \mathbf{NijLie}_{m} \cong\mathbf{NijLie}_{n}^{<m}. NijLie m ≅ NijLie n < m . Proof. Let F : N i j L i e m → N i j L i e n < m \mathcal{F}:\mathbf{NijLie}_{m}\to\mathbf{NijLie}_{n}^{<m} F : NijLie m → NijLie n < m ℓ k = 0 \ell_k=0 ℓ k = 0 k > m + 1 k>m+1 k > m + 1 N q = 0 N_q=0 N q = 0 q > m q>m q > m N i j L i e n < m \mathbf{NijLie}_{n}^{<m} NijLie n < m ℓ k = 0 \ell_k=0 ℓ k = 0 k > m + 1 k>m+1 k > m + 1 N q = 0 N_q=0 N q = 0 q > m q>m q > m > m − 1 >m-1 > m − 1 F \mathcal{F} F
To prove full faithfulness, let ( ( V , ℓ ) , N ) ((V,\ell),N) (( V , ℓ ) , N ) ( ( V ′ , ℓ ′ ) , N ′ ) ((V',\ell'),N') (( V ′ , ℓ ′ ) , N ′ ) N i j L i e n < m \mathbf{NijLie}_{n}^{<m} NijLie n < m ( φ , ψ ) (\varphi,\psi) ( φ , ψ ) φ q \varphi_q φ q q > m q>m q > m ψ q \psi_q ψ q q > m − 1 q>m-1 q > m − 1 N i j L i e m \mathbf{NijLie}_{m} NijLie m N i j L i e n < m \mathbf{NijLie}_{n}^{<m} NijLie n < m G : N i j L i e n < m → N i j L i e m \mathcal{G}:\mathbf{NijLie}_{n}^{<m}\to\mathbf{NijLie}_{m} G : NijLie n < m → NijLie m F ∘ G = id \mathcal{F}\circ\mathcal{G}=\operatorname{id} F ∘ G = id G ∘ F = id \mathcal{G}\circ\mathcal{F}=\operatorname{id} G ∘ F = id
Corollary 5.2. There exists a natural tower of full embeddings forming a sequence of category isomorphisms onto full subcategories:
N i j L i e 1 ↪ N i j L i e 2 ↪ N i j L i e 3 ↪ ⋯ ↪ N i j L i e ∞ : = lim n N i j L i e n ↪ N i j L ∞ , \mathrm{NijLie}_{1}\hookrightarrow\mathrm{NijLie}_{2}\hookrightarrow\mathrm{NijLie}_{3}\hookrightarrow\cdots\hookrightarrow\mathrm{NijLie}_{\infty}:=\lim_{n}\mathrm{NijLie}_{n}\hookrightarrow\mathrm{NijL}_{\infty}, NijLie 1 ↪ NijLie 2 ↪ NijLie 3 ↪ ⋯ ↪ NijLie ∞ := n lim NijLie n ↪ NijL ∞ , where N i j L ∞ \mathbf{NijL}_{\infty} NijL ∞ L ∞ L_{\infty} L ∞ § 4.5 , and N i j L i e ∞ \mathbf{NijLie}_{\infty} NijLie ∞ n n n n n n m m m n n n n ≥ m n\ge m n ≥ m
Proof. Immediate from Theorem 5.1, where the inclusions are given by the composites N i j L i e m ≅ N i j L i e n < m ⊆ N i j L i e n \mathbf{NijLie}_{m}\cong\mathbf{NijLie}_{n}^{<m}\subseteq\mathbf{NijLie}_{n} NijLie m ≅ NijLie n < m ⊆ NijLie n n n n
We now explicitly characterize the foundational floors of this tower.
Theorem 5.3. The category N i j L i e 1 \mathbf{NijLie}_{1} NijLie 1 is isomorphic to the category of classical Nijenhuis Lie algebras: an object consists of a Lie algebra ( g , [ , ] ) (\mathfrak{g},[\ ,\ ]) ( g , [ , ]) n : g → g n:\mathfrak{g}\to\mathfrak{g} n : g → g (1) , and a morphism is a Lie algebra morphism that intertwines the respective operators.
Proof. For n = 1 n=1 n = 1 N = ( n ) N=(n) N = ( n ) l 1 = 0 l_{1}=0 l 1 = 0 l k = 0 l_{k}=0 l k = 0 k ≥ 3 k\ge3 k ≥ 3 ℓ = l 2 \ell=l_{2} ℓ = l 2 Λ m N \Lambda^{N}_{m} Λ m N Part ∗ ( m ) \operatorname{Part}^{*}(m) Part ∗ ( m ) Part ∘ ( m ) \operatorname{Part}^{\circ}(m) Part ∘ ( m ) N q N_{q} N q q ≥ 2 q\ge2 q ≥ 2 l ∣ P ∣ l_{\lvert P\rvert} l ∣ P ∣ ∣ P ∣ ≥ 3 \lvert P\rvert\ge3 ∣ P ∣ ≥ 3 l 1 l_{1} l 1 N q N_{q} N q q ≥ 2 q\ge2 q ≥ 2 Λ m N = 0 \Lambda^{N}_{m}=0 Λ m N = 0 (23) simplifies to Tors n ( l m ) = 0 \operatorname{Tors}_{n}(l_{m})=0 Tors n ( l m ) = 0 m ≠ 2 m\ne2 m = 2 (1) for m = 2 m=2 m = 2 φ q = 0 \varphi_{q}=0 φ q = 0 q ≥ 2 q\ge2 q ≥ 2 ψ q = 0 \psi_{q}=0 ψ q = 0 q ≥ 1 q\ge1 q ≥ 1 φ 1 \varphi_{1} φ 1 (25) yields the intertwining relation φ 1 n = n ′ φ 1 \varphi_{1}n=n'\varphi_{1} φ 1 n = n ′ φ 1
Theorem 5.4. The category N i j L i e 2 \mathbf{NijLie}_{2} NijLie 2 is isomorphic to the category N i j T e r m 2 \mathbf{NijTerm}_{2} NijTerm 2 of 2-term Nijenhuis L ∞ L_{\infty} L ∞ algebras from [12 Def. 4.1, Prop. 4.3] , and is therefore equivalent, via [12 Thm. 4.5] , to the category of Nijenhuis Lie 2-algebras. Similarly, N i j L i e 3 \mathbf{NijLie}_{3} NijLie 3 is isomorphic to the category of Nijenhuis Lie 3-algebras introduced in [24 Def. 3.6, Prop. 3.30] , provided that the identities (N5) and (3.22) of that reference are corrected as detailed in Remark 5.5.
Proof. By Corollary 4.8, the structural data at level two consist of n = ( n 0 , n 1 ) n=(n_{0},n_{1}) n = ( n 0 , n 1 ) N 2 : g 0 × g 0 → g 1 N_{2}:\mathfrak{g}_{0}\times\mathfrak{g}_{0}\to\mathfrak{g}_{1} N 2 : g 0 × g 0 → g 1 m ≤ 4 m\le4 m ≤ 4 (23) componentwise (see Appendix § 9 ) yields:
Arity m = 1 m=1 m = 1 The condition that n n n
Arity m = 2 m=2 m = 2 The two component equations Tors n ( [ , ] ) ( x , y ) + d N 2 ( x , y ) = 0 \operatorname{Tors}_{n}([\ ,\ ])(x,y)+dN_{2}(x,y)=0 Tors n ([ , ]) ( x , y ) + d N 2 ( x , y ) = 0 Tors n ( [ , ] ) ( x , h ) + N 2 ( x , d h ) = 0 \operatorname{Tors}_{n}([\ ,\ ])(x,h)+N_{2}(x,dh)=0 Tors n ([ , ]) ( x , h ) + N 2 ( x , d h ) = 0
Arity m = 3 m=3 m = 3 The relation Tors n ( l 3 ) + Λ = 0 \operatorname{Tors}_{n}(l_{3})+\Lambda=0 Tors n ( l 3 ) + Λ = 0
Λ ( x , y , z ) = ∑ ( 3 ) ( [ n x , N 2 ( y , z ) ] − n [ x , N 2 ( y , z ) ] + N 2 ( x , [ y , z ] n ) − n N 2 ( x , [ y , z ] ) ) ; \Lambda(x,y,z)=\sum_{(3)}\left([nx,N_{2}(y,z)]-n[x,N_{2}(y,z)]+N_{2}(x,[y,z]_{n})-nN_{2}(x,[y,z])\right); Λ ( x , y , z ) = ( 3 ) ∑ ( [ n x , N 2 ( y , z )] − n [ x , N 2 ( y , z )] + N 2 ( x , [ y , z ] n ) − n N 2 ( x , [ y , z ]) ) ; These equations precisely recover conditions (5), (6), and (7) of [12 Def. 4.1] . The corresponding comparison of morphisms arises from the corestrictions of condition (25) and condition (M1) in arities 1 and 2, matching [12 Def. 4.2] under the dictionary correspondence ψ 1 ↔ φ 3 \psi_{1}\leftrightarrow\varphi_{3} ψ 1 ↔ φ 3 § 9 .
Remark 5.5 (The two corrections at level three). Expanding (24) for m = 3 m=3 m = 3 m = 4 m=4 m = 4 L ∞ L_\infty L ∞ [24 ] yields two discrepancies, both in the direction of more terms or different placement of n n n
The missing boundary term. For m = 3 m=3 m = 3 P a r t ∘ ( 3 ) Part^\circ(3) P a r t ∘ ( 3 ) { { 1 , 2 , 3 } } \{\{1,2,3\}\} {{ 1 , 2 , 3 }} (24) is l 1 ( N 3 ( v 1 , v 2 , v 3 ) ) l_1(N_3(v_1,v_2,v_3)) l 1 ( N 3 ( v 1 , v 2 , v 3 )) d N 3 ( x , y , z ) dN_3(x,y,z) d N 3 ( x , y , z ) g 0 3 \mathfrak{g}_0^3 g 0 3
Tors n ( l 3 ) ( x , y , z ) + Λ ( x , y , z ) + d N 3 ( x , y , z ) = 0 , \operatorname{Tors}_n(l_3)(x,y,z)+\Lambda(x,y,z)+dN_3(x,y,z)=0, Tors n ( l 3 ) ( x , y , z ) + Λ ( x , y , z ) + d N 3 ( x , y , z ) = 0 , whereas [24 (N5)] omits d N 3 dN_3 d N 3 d N 3 ( x , y , z ) ∈ g 1 dN_3(x,y,z)\in\mathfrak{g}_1 d N 3 ( x , y , z ) ∈ g 1 N 3 ( x , y , d h ) N_3(x,y,dh) N 3 ( x , y , d h ) [24 (N6)] , precisely as d N 2 ( x , y ) dN_2(x,y) d N 2 ( x , y ) N 2 ( x , d h ) N_2(x,dh) N 2 ( x , d h ) N 3 = 0 N_3=0 N 3 = 0
The placement of n n n For m = 4 m=4 m = 4 ( 2 , 1 , 1 ) (2,1,1) ( 2 , 1 , 1 ) P a r t ∗ ( 4 ) Part^*(4) P a r t ∗ ( 4 ) (24) ,
l 3 ( n x k , n x l , N 2 ( x i , x j ) ) − n l 3 ( x k , x l , N 2 ( x i , x j ) ) , l_3(nx_k,nx_l,N_2(x_i,x_j))-n l_3(x_k,x_l,N_2(x_i,x_j)), l 3 ( n x k , n x l , N 2 ( x i , x j )) − n l 3 ( x k , x l , N 2 ( x i , x j )) , with n n n both free slots simultaneously. The corresponding sum in [24 (3.22)] uses instead the n n n l 3 ( n x k , x l , N 2 ) + l 3 ( x k , n x l , N 2 ) − n l 3 ( x k , x l , N 2 ) l_3(nx_k,x_l,N_2)+l_3(x_k,nx_l,N_2)-n l_3(x_k,x_l,N_2) l 3 ( n x k , x l , N 2 ) + l 3 ( x k , n x l , N 2 ) − n l 3 ( x k , x l , N 2 ) Theorem 4.4 .
Consequently the auxiliary formula [24 (3.23)] for l 4 N l_4^N l 4 N l 4 N = l 4 ( n ) + Ξ 1 l_4^N=l_4^{(n)}+\Xi_1 l 4 N = l 4 ( n ) + Ξ 1 Ξ 1 \Xi_1 Ξ 1 n n n n n n l 4 N = l 4 ( n ) + Ξ 0 l_4^N=l_4^{(n)}+\Xi_0 l 4 N = l 4 ( n ) + Ξ 0 [24 Prop. 3.19] , which correctly takes R 4 = Ξ 0 \mathcal{R}_4=\Xi_0 R 4 = Ξ 0 (15) , where A ‾ m − B ‾ m \overline{A}_m-\overline{B}_m A m − B m Ξ 1 \Xi_1 Ξ 1
None of these corrections affects the level-two reduction, nor the statements of [24 ] about strict or skeletal objects, in which the higher components vanish.
Strict and skeletal objects at every level
Strict objects
Proposition 6.1. An n n n L ∞ L_\infty L ∞ l k = 0 l_k=0 l k = 0 k ≥ 3 k\ge3 k ≥ 3 ( g , d , [ , ] ) (\mathfrak{g},d,[\ ,\ ]) ( g , d , [ , ]) 0 , … , n − 1 0,\ldots,n-1 0 , … , n − 1 − 1 -1 − 1 d d d n n n
Proof. With l k = 0 l_k=0 l k = 0 k ≥ 3 k\ge3 k ≥ 3 (8) reduce to those of arity m ≤ 3 m\le3 m ≤ 3 d 2 = 0 d^2=0 d 2 = 0 d d d l 2 l_2 l 2 l 2 l_2 l 2
For n = 1 n=1 n = 1 n = 2 n=2 n = 2 [3 ] ; for n = 3 n=3 n = 3 [24 Ex. 2.6] , whose extra pairing g 1 × g 1 → g 2 \mathfrak{g}_1\times\mathfrak{g}_1\to\mathfrak{g}_2 g 1 × g 1 → g 2 ( 1 , 1 ) (1,1) ( 1 , 1 )
Theorem 6.2. A strict Nijenhuis Lie n n n n n n ( g , d , [ , ] ) (\mathfrak{g},d,[\ ,\ ]) ( g , d , [ , ]) n : g → g n:\mathfrak{g}\to\mathfrak{g} n : g → g
[ n a , n b ] = n ( [ n a , b ] + [ a , n b ] − n [ a , b ] ) (27) [na,nb]=n\bigl([na,b]+[a,nb]-n[a,b]\bigr) \tag*{(27)} [ na , nb ] = n ( [ na , b ] + [ a , nb ] − n [ a , b ] ) (27) for all homogeneous a , b ∈ g a,b\in\mathfrak{g} a , b ∈ g Tors n ( [ , ] ) = 0 \operatorname{Tors}_n([\ ,\ ])=0 Tors n ([ , ]) = 0 n 0 n_0 n 0 g 0 \mathfrak{g}_0 g 0
Proof. Let N = ( n , 0 , 0 , … ) N=(n,0,0,\ldots) N = ( n , 0 , 0 , … ) l k = 0 l_k=0 l k = 0 k ≥ 3 k\geq3 k ≥ 3 (24) contains either a component N q N_q N q q ≥ 2 q\geq2 q ≥ 2 l ∣ P ∣ l_{\lvert P\rvert} l ∣ P ∣ ∣ P ∣ ≥ 3 \lvert P\rvert\geq3 ∣ P ∣ ≥ 3 ∣ P ∣ = 1 \lvert P\rvert=1 ∣ P ∣ = 1 2 2 2 P art ∘ ( m ) P\operatorname{art}^{\circ}(m) P art ∘ ( m ) ≥ 2 \geq2 ≥ 2 N q N_q N q q ≥ 2 q\geq2 q ≥ 2 Λ m N = 0 \Lambda^N_m=0 Λ m N = 0 m m m (23) becomes Tors n ( l m ) = 0 \operatorname{Tors}_n(l_m)=0 Tors n ( l m ) = 0 Lemma 3.5 this is vacuous unless m ∈ { 1 , 2 } m\in\{1,2\} m ∈ { 1 , 2 } m = 1 m=1 m = 1 n n n m = 2 m=2 m = 2 (27) by (13) .
Proposition 6.3. Let ( ( g , d , [ , ] ) , n ) ((g,d,[\ ,\ ]),n) (( g , d , [ , ]) , n ) n n n ( g , d , [ , ] n ) (g,d,[\ ,\ ]_n) ( g , d , [ , ] n ) n n n n : ( g , d , [ , ] n ) → ( g , d , [ , ] ) n\colon(g,d,[\ ,\ ]_n)\to(g,d,[\ ,\ ]) n : ( g , d , [ , ] n ) → ( g , d , [ , ]) L ∞ L_\infty L ∞ [ , ] n [\ ,\ ]_n [ , ] n ℓ N \ell^N ℓ N Theorem 4.10 (ii) holds for strict objects without any injectivity hypothesis on n n n
Proof. That n n n (27) rewritten as n [ a , b ] n = [ n a , n b ] n[a,b]_n=[na,nb] n [ a , b ] n = [ na , nb ] ℓ N \ell^N ℓ N [ , ] n [\ ,\ ]_n [ , ] n (18) . It remains to check that [ , ] n [\ ,\ ]_n [ , ] n d d d [ , ] [\ ,\ ] [ , ] n d = d n nd=dn n d = d n
d [ a , b ] n = d [ n a , b ] + d [ a , n b ] − d n [ a , b ] = [ n d a , b ] + ( − 1 ) ∣ a ∣ [ n a , d b ] + [ d a , n b ] + ( − 1 ) ∣ a ∣ [ a , n d b ] − n [ d a , b ] − ( − 1 ) ∣ a ∣ n [ a , d b ] , \begin{aligned}
d[a,b]_n &= d[na,b]+d[a,nb]-d\,n[a,b]\\
&=[n\,da,b]+(-1)^{\lvert a\rvert}[na,db]+[da,nb]\\
&\quad+(-1)^{\lvert a\rvert}[a,n\,db]-n[da,b]-(-1)^{\lvert a\rvert}n[a,db],
\end{aligned} d [ a , b ] n = d [ na , b ] + d [ a , nb ] − d n [ a , b ] = [ n d a , b ] + ( − 1 ) ∣ a ∣ [ na , d b ] + [ d a , nb ] + ( − 1 ) ∣ a ∣ [ a , n d b ] − n [ d a , b ] − ( − 1 ) ∣ a ∣ n [ a , d b ] , which is [ d a , b ] n + ( − 1 ) ∣ a ∣ [ a , d b ] n [da,b]_n+(-1)^{\lvert a\rvert}[a,db]_n [ d a , b ] n + ( − 1 ) ∣ a ∣ [ a , d b ] n J ( a , b , c ) = [ [ a , b ] n , c ] n J(a,b,c)=[[a,b]_n,c]_n J ( a , b , c ) = [[ a , b ] n , c ] n n [ a , b ] n = [ n a , n b ] n[a,b]_n=[na,nb] n [ a , b ] n = [ na , nb ] [ , c ] n [\ ,c]_n [ , c ] n
J ( a , b , c ) = [ [ n a , n b ] , c ] + [ [ a , b ] n , n c ] − n [ [ a , b ] n , c ] . J(a,b,c)=[[na,nb],c]+[[a,b]_n,nc]-n[[a,b]_n,c]. J ( a , b , c ) = [[ na , nb ] , c ] + [[ a , b ] n , n c ] − n [[ a , b ] n , c ] . Summing over the graded cyclic permutations of ( a , b , c ) (a,b,c) ( a , b , c ) [ , ] [\ ,\ ] [ , ] ( n a , n b , c ) (na,nb,c) ( na , nb , c ) c c c n c nc n c [ a , b ] n [a,b]_n [ a , b ] n [ , ] [\ ,\ ] [ , ] n n n n n n (27) . This is the classical computation of [17 §2] , which is sign-homogeneous and therefore valid verbatim in the graded setting.
Corollary 6.4. The assignment ( ( g , d , [ , ] ) , n ) ↦ ( ( g , d , [ , ] n ) , n ) ((g,d,[\ ,\ ]),n)\mapsto((g,d,[\ ,\ ]_n),n) (( g , d , [ , ]) , n ) ↦ (( g , d , [ , ] n ) , n ) S N i j L i e n → S N i j L i e n \mathrm{SNijLie}_n\to\mathrm{SNijLie}_n SNijLie n → SNijLie n n n n
Proof. On morphisms one takes the same underlying map; a strict morphism φ 1 \varphi_1 φ 1 φ 1 n = n ′ φ 1 \varphi_1n=n'\varphi_1 φ 1 n = n ′ φ 1 φ 1 [ a , b ] n = [ φ 1 a , φ 1 b ] n ′ \varphi_1[a,b]_n=[\varphi_1a,\varphi_1b]_{n'} φ 1 [ a , b ] n = [ φ 1 a , φ 1 b ] n ′ Tors n \operatorname{Tors}_n Tors n [ , ] n [\ ,\ ]_n [ , ] n n n n n [ a , b ] n = [ n a , n b ] n[a,b]_n=[na,nb] n [ a , b ] n = [ na , nb ] n n n [ , ] n [\ ,\ ]_n [ , ] n n n n φ 1 n = n ′ φ 1 \varphi_1n=n'\varphi_1 φ 1 n = n ′ φ 1
Skeletal objects
Theorem 6.5. Let ( ( g , ℓ ) , N ) ((g,\ell),N) (( g , ℓ ) , N ) n n n d = 0 d=0 d = 0
( g 0 , [ , ] ) (g_0,[\ ,\ ]) ( g 0 , [ , ]) n 0 n_0 n 0
each g i g_i g i g 0 g_0 g 0 x ⋅ a = [ x , a ] x\cdot a=[x,a] x ⋅ a = [ x , a ] n i : g i → g i n_i\colon g_i\to g_i n i : g i → g i ( g i , n i ) (g_i,n_i) ( g i , n i ) ( g 0 , n 0 ) (g_0,n_0) ( g 0 , n 0 ) [11 ] , i.e.
[ n 0 x , n i a ] = n i ( [ n 0 x , a ] + [ x , n i a ] − n i [ x , a ] ) , x ∈ g 0 , a ∈ g i ; [n_0x,n_i a]=n_i([n_0x,a]+[x,n_i a]-n_i[x,a]),\qquad x\in g_0,\ a\in g_i; [ n 0 x , n i a ] = n i ([ n 0 x , a ] + [ x , n i a ] − n i [ x , a ]) , x ∈ g 0 , a ∈ g i ; the components of the bracket of positive bidegree are g 0 g_0 g 0 g i × g j → g i + j g_i \times g_j \to g_{i+j} g i × g j → g i + j l 3 , … , l n + 1 l_3,\ldots,l_{n+1} l 3 , … , l n + 1 N 2 , … , N n N_2,\ldots,N_n N 2 , … , N n (8) and (23) by setting d = 0 d=0 d = 0 Tors n ( l m ) = − Λ m N \operatorname{Tors}_n(l_m)=-\Lambda_m^N Tors n ( l m ) = − Λ m N 3 ≤ m ≤ n + 2 3\leq m\leq n+2 3 ≤ m ≤ n + 2 Λ m N \Lambda_m^N Λ m N d N m dN_m d N m N m ( d ⋅ , … ) N_m(d\cdot,\ldots) N m ( d ⋅ , … )
Conversely every such collection of data defines a skeletal Nijenhuis Lie n n n
Proof. With d = 0 d=0 d = 0 (8) restricted to g 0 × 3 g_0^{\times3} g 0 × 3 (23) for m = 2 m=2 m = 2 g 0 × g 0 g_0\times g_0 g 0 × g 0 Tors n ( [ , ] ) = 0 \operatorname{Tors}_{n}([\, ,\, ])=0 Tors n ([ , ]) = 0 Λ 2 N = d N 2 \Lambda_2^N=dN_2 Λ 2 N = d N 2 g 0 × g i g_0\times g_i g 0 × g i Tors n ( [ , ] ) ( x , a ) = 0 \operatorname{Tors}_n([\, ,\, ])(x,a)=0 Tors n ([ , ]) ( x , a ) = 0 (8) and (23) , and the two boundary families of (24) contain d d d
Example 6.6. Let ( g , n ) (\mathfrak{g},n) ( g , n ) ( V 1 , s 1 ) , … , ( V n − 1 , s n − 1 ) (V_1,s_1),\ldots,(V_{n-1},s_{n-1}) ( V 1 , s 1 ) , … , ( V n − 1 , s n − 1 ) [11 ] . Put g 0 = g g_0=\mathfrak{g} g 0 = g g i = V i g_i=V_i g i = V i d = 0 d=0 d = 0 g i × g j g_i\times g_j g i × g j i , j ≥ 1 i,j\geq1 i , j ≥ 1 l k = 0 l_k=0 l k = 0 k ≥ 3 k\geq3 k ≥ 3 N q = 0 N_q=0 N q = 0 q ≥ 2 q\geq2 q ≥ 2 Theorem 6.5 hold, and one obtains a skeletal Nijenhuis Lie n n n l 3 , … , N 2 , … l_3,\ldots,N_2,\ldots l 3 , … , N 2 , … Theorem 6.5 (iii), produces the general skeletal object.
Remark 6.7. At level three, Theorems 6.2 and 6.5 specialise to [24 Thm. 4.5, Thm. 4.9] ; at level two, to [12 §4] . The relation between strict Nijenhuis Lie n n n ( n − 1 ) (n-1) ( n − 1 ) [24 Rem. 4.7] : the Peiffer liftings of a crossed module of higher order are not required to be antisymmetric, whereas the corresponding components of the bracket of a strict Lie n n n Proposition 6.1 identifies the objects on our side precisely as dg Lie algebras concentrated in n n n
n-term representations up to homotopy and the semidirect product
Throughout this section ( g , [ , ] , n ) (\mathfrak{g},[\, ,\, ],n) ( g , [ , ] , n ) [11 , 28 ] that a representation of it is a triple ( V , ρ , s ) (V,\rho,s) ( V , ρ , s ) ( V , ρ ) (V,\rho) ( V , ρ ) g \mathfrak{g} g s ∈ End ( V ) s\in\operatorname{End}(V) s ∈ End ( V )
ρ ( n x ) s ( v ) = s ( ρ ( n x ) v + ρ ( x ) s ( v ) − s ( ρ ( x ) v ) ) , x ∈ g , v ∈ V . (28) \rho(nx)s(v)=s\left(\rho(nx)v+\rho(x)s(v)-s(\rho(x)v)\right),\qquad x\in\mathfrak{g},\ v\in V. \tag*{(28)} ρ ( n x ) s ( v ) = s ( ρ ( n x ) v + ρ ( x ) s ( v ) − s ( ρ ( x ) v ) ) , x ∈ g , v ∈ V . (28) We now define the n n n
Representations up to homotopy of a Lie algebra
Let V = ( V n − 1 → ∂ ⋯ → ∂ V 0 ) \mathcal{V}=(V_{n-1}\xrightarrow{\partial}\cdots\xrightarrow{\partial}V_0) V = ( V n − 1 ∂ ⋯ ∂ V 0 ) 0 , … , n − 1 0,\ldots,n-1 0 , … , n − 1 C ∙ ( g , End ( V ) ) C^\bullet(\mathfrak{g},\operatorname{End}(\mathcal{V})) C ∙ ( g , End ( V )) g \mathfrak{g} g End ( V ) \operatorname{End}(\mathcal{V}) End ( V ) p p p R = ( R j ) j ≥ 0 R=(R_j)_{j\geq0} R = ( R j ) j ≥ 0 R j ∈ C j ( g , End ( V ) ) j − p R_j\in C^j(\mathfrak{g},\operatorname{End}(\mathcal{V}))_{j-p} R j ∈ C j ( g , End ( V ) ) j − p R j : ⋀ j g → End ( V ) R_j:\bigwedge^j\mathfrak{g}\to\operatorname{End}(\mathcal{V}) R j : ⋀ j g → End ( V ) V \mathcal{V} V p − j p-j p − j C ∙ ( g , End ( V ) ) C^\bullet(\mathfrak{g},\operatorname{End}(\mathcal{V})) C ∙ ( g , End ( V )) ⌣ \smile ⌣ End ( V ) \operatorname{End}(\mathcal{V}) End ( V )
Definition 7.1. An n n n g \mathfrak{g} g V \mathcal{V} V
R = ∂ + ρ + ν + ω + ⋯ = ∑ j ≥ 0 R j ∈ C ∙ ( g , End ( V ) ) \mathcal{R}=\partial+\rho+\nu+\omega+\cdots=\sum_{j\geq0}R_j\in C^\bullet(\mathfrak{g},\operatorname{End}(\mathcal{V})) R = ∂ + ρ + ν + ω + ⋯ = j ≥ 0 ∑ R j ∈ C ∙ ( g , End ( V )) of total degree − 1 -1 − 1 R 0 = ∂ R_0=\partial R 0 = ∂ V \mathcal{V} V
δ R + R ⌣ R = 0 , (29) \delta\mathcal{R}+\mathcal{R}\smile\mathcal{R}=0, \tag*{(29)} δ R + R ⌣ R = 0 , (29) where δ \delta δ g \mathfrak{g} g R 1 = ρ : g → End ( V i ) R_1=\rho:\mathfrak{g}\to\operatorname{End}(\mathcal{V}_i) R 1 = ρ : g → End ( V i ) R 2 = ν : Λ 2 g → End ( V ) R_2=\nu:\Lambda^2\mathfrak{g}\to\operatorname{End}(\mathcal{V}) R 2 = ν : Λ 2 g → End ( V ) R 3 = ω R_3=\omega R 3 = ω R j = 0 R_j=0 R j = 0 j > n j>n j > n homomorphism V → V ′ \mathcal{V}\to\mathcal{V}' V → V ′
Φ = ∑ j ≥ 0 Φ j \Phi=\sum_{j\geq0}\Phi_j Φ = j ≥ 0 ∑ Φ j of C ∙ ( g , Hom ( V , V ′ ) ) C^\bullet(\mathfrak{g},\operatorname{Hom}(\mathcal{V},\mathcal{V}')) C ∙ ( g , Hom ( V , V ′ )) 0 0 0 Φ 0 \Phi_0 Φ 0
Φ ⌣ R = R ′ ⌣ Φ + δ Φ . (30) \Phi\smile\mathcal{R}=\mathcal{R}'\smile\Phi+\delta\Phi. \tag*{(30)} Φ ⌣ R = R ′ ⌣ Φ + δ Φ. (30) We write Rep n ( g ) \operatorname{Rep}_n(\mathfrak{g}) Rep n ( g ) Id V \operatorname{Id}_{\mathcal{V}} Id V
Proposition 7.2. Let V \mathcal{V} V n n n g \mathfrak{g} g
g ⋉ V = ( V n − 1 → ⋯ → V 1 → ( 0 , ∂ ) g ⊕ V 0 ) \mathfrak{g}\ltimes\mathcal{V}=\left(\mathcal{V}_{n-1}\to\cdots\to\mathcal{V}_1\xrightarrow{(0,\partial)}\mathfrak{g}\oplus\mathcal{V}_0\right) g ⋉ V = ( V n − 1 → ⋯ → V 1 ( 0 , ∂ ) g ⊕ V 0 ) carries a unique n n n L ∞ L_\infty L ∞
the projection π : g ⋉ V → g \pi:\mathfrak{g}\ltimes\mathcal{V}\to\mathfrak{g} π : g ⋉ V → g L ∞ L_\infty L ∞ σ : g → g ⋉ V \sigma:\mathfrak{g}\to\mathfrak{g}\ltimes\mathcal{V} σ : g → g ⋉ V L ∞ L_\infty L ∞ π σ = id \pi\sigma=\operatorname{id} π σ = id
V \mathcal{V} V V \mathcal{V} V
for x 1 , … , x k − 1 ∈ g x_1,\ldots,x_{k-1}\in\mathfrak{g} x 1 , … , x k − 1 ∈ g a ∈ V a\in\mathcal{V} a ∈ V
l k ( σ x 1 , … , σ x k − 1 , a ) = ( − 1 ) k R k − 1 ( x 1 , … , x k − 1 ) a . l_k(\sigma x_1,\ldots,\sigma x_{k-1},a)=(-1)^kR_{k-1}(x_1,\ldots,x_{k-1})a. l k ( σ x 1 , … , σ x k − 1 , a ) = ( − 1 ) k R k − 1 ( x 1 , … , x k − 1 ) a . Conversely, every n n n L ∞ L_\infty L ∞ g ⊕ V \mathfrak{g}\oplus\mathcal{V} g ⊕ V R \mathcal{R} R Rep n ( g ) \operatorname{Rep}_n(\mathfrak{g}) Rep n ( g ) L ∞ L_\infty L ∞ g \mathfrak{g} g V \mathcal{V} V
Proof. Conditions (i)–(iii) uniquely determine all multilinear operations. Specifically:
l 2 ( σ x , σ y ) = σ [ x , y ] , l k ( σ x 1 , … , σ x k ) = 0 (for k ≠ 2 ) . l_2(\sigma x,\sigma y)=\sigma[x,y],\qquad l_k(\sigma x_1,\ldots,\sigma x_k)=0\quad\text{(for }k\ne2\text{)}. l 2 ( σ x , σ y ) = σ [ x , y ] , l k ( σ x 1 , … , σ x k ) = 0 (for k = 2 ) . l k ( σ x 1 , … , σ x k − 1 , a ) = ( − 1 ) k R k − 1 ( x 1 , … , x k − 1 ) a . l_k(\sigma x_1,\ldots,\sigma x_{k-1},a)=(-1)^kR_{k-1}(x_1,\ldots,x_{k-1})a. l k ( σ x 1 , … , σ x k − 1 , a ) = ( − 1 ) k R k − 1 ( x 1 , … , x k − 1 ) a . It remains to verify that the general L ∞ L_\infty L ∞ R \mathcal{R} R
Evaluated on arguments m m m σ g \sigma\mathfrak{g} σ g L ∞ L_\infty L ∞ g \mathfrak{g} g m = 3 m=3 m = 3 m ≠ 3 m\ne3 m = 3
:::
Evaluated on m − 1 m-1 m − 1 σ g \sigma\mathfrak{g} σ g V \mathcal{V} V V \mathcal{V} V
δ R + R ⌣ R = 0 , \delta\mathcal{R}+\mathcal{R}\smile\mathcal{R}=0, δ R + R ⌣ R = 0 , where δ R \delta\mathcal{R} δ R R ⌣ R \mathcal{R}\smile\mathcal{R} R ⌣ R
Finally, the statement concerning morphisms follows by applying the identical componentwise evaluation to the defining relations of L ∞ L_\infty L ∞
Remark 7.3. The truncation parameter n n n
For n = 1 n=1 n = 1 Equation (29) reduces precisely to the classical notion of a Lie algebra representation ρ \rho ρ
For n = 2 n=2 n = 2 It recovers the system of equations (19)–(21) from [12 Def. 5.8] , with components R 1 = ρ R_1=\rho R 1 = ρ R 2 = ν R_2=\nu R 2 = ν
For n = 3 n=3 n = 3 It yields the explicit list (R1)–(R7) of [24 Def. 5.2] , setting R 1 = ( ρ 0 , ρ 1 , ρ 2 ) R_1=(\rho_0,\rho_1,\rho_2) R 1 = ( ρ 0 , ρ 1 , ρ 2 ) R 2 = ( ν , ν ˉ ) R_2=(\nu,\bar{\nu}) R 2 = ( ν , ν ˉ ) R 3 = ω R_3=\omega R 3 = ω [24 (R7)] arises naturally as the arity-four component of (29) , where the quadratic product ν ˉ ( x i , x j ) ν ( x k , x l ) \bar{\nu}(x_i,x_j)\nu(x_k,x_l) ν ˉ ( x i , x j ) ν ( x k , x l ) R ⌣ R \mathcal{R}\smile\mathcal{R} R ⌣ R
Globally, Definition 7.1 corresponds to the n n n [1 ] , whereas Proposition 7.2 extends the semidirect product construction of [27 ] to this generalized setting.
The Nijenhuis case
Definition 7.4. Let ( g , [ , ] , n ) (\mathfrak{g},[\ ,\ ],n) ( g , [ , ] , n ) n n n ( g , n ) (\mathfrak{g},n) ( g , n ) ( V , S ) (\mathcal{V},\mathcal{S}) ( V , S ) V \mathcal{V} V n n n g \mathfrak{g} g
S = ∑ j ≥ 0 S j ∈ C ∙ ( g , End ( V ) ) , S j : Λ j g → End ( V ) , \mathcal{S}=\sum_{j\geq0}S_j\in C^\bullet(\mathfrak{g},\operatorname{End}(\mathcal{V})),\qquad S_j:\Lambda^j\mathfrak{g}\to\operatorname{End}(\mathcal{V}), S = j ≥ 0 ∑ S j ∈ C ∙ ( g , End ( V )) , S j : Λ j g → End ( V ) , is an element of total degree 0 0 0
N 1 S = n ⊕ S 0 , N q S ( σ x 1 , … , σ x q − 1 , a ) = ( − 1 ) q S q − 1 ( x 1 , … , x q − 1 ) a ( q ≥ 2 ) , (31) N^S_1=n\oplus S_0,\qquad N^S_q(\sigma x_1,\ldots,\sigma x_{q-1},a)=(-1)^qS_{q-1}(x_1,\ldots,x_{q-1})a\quad(q\geq2), \tag*{(31)} N 1 S = n ⊕ S 0 , N q S ( σ x 1 , … , σ x q − 1 , a ) = ( − 1 ) q S q − 1 ( x 1 , … , x q − 1 ) a ( q ≥ 2 ) , (31) all other components vanishing, is a Nijenhuis operator on g ⋉ V \mathfrak{g}\ltimes\mathcal{V} g ⋉ V ( V , S ) → ( V ′ , S ′ ) (\mathcal{V},\mathcal{S})\to(\mathcal{V}',\mathcal{S}') ( V , S ) → ( V ′ , S ′ ) Φ \Phi Φ Θ \Theta Θ C ∙ ( g , Hom ( V , V ′ ) ) C^\bullet(\mathfrak{g},\operatorname{Hom}(\mathcal{V},\mathcal{V}')) C ∙ ( g , Hom ( V , V ′ )) + 1 +1 + 1 ( Φ , Θ ) (\Phi,\Theta) ( Φ , Θ ) n n n g ⋉ V → g ⋉ V ′ \mathfrak{g}\ltimes\mathcal{V}\to\mathfrak{g}\ltimes\mathcal{V}' g ⋉ V → g ⋉ V ′ Rep n ( g , n ) \operatorname{Rep}_n(\mathfrak{g},n) Rep n ( g , n )
Theorem 7.5. The assignment ( V , S ) ↦ ( g ⋉ V , N S ) (\mathcal{V},\mathcal{S})\mapsto(\mathfrak{g}\ltimes\mathcal{V},N^{\mathcal{S}}) ( V , S ) ↦ ( g ⋉ V , N S )
Ψ : Rep n ( g , n ) ⟶ NijLie n , \Psi:\operatorname{Rep}_n(\mathfrak{g},n)\longrightarrow\operatorname{NijLie}_n, Ψ : Rep n ( g , n ) ⟶ NijLie n , and the underlying L ∞ L_\infty L ∞ Ψ ( V , S ) \Psi(\mathcal{V},\mathcal{S}) Ψ ( V , S )
S 0 ( ρ ( n x ) a + ρ ( x ) S 0 a − S 0 ( ρ ( x ) a ) ) − ρ ( n x ) S 0 a = ∂ ( S 1 ( x ) a ) + S 1 ( x ) ∂ a , (32) S_0\left(\rho(nx)a+\rho(x)S_0a-S_0(\rho(x)a)\right)-\rho(nx)S_0a=\partial(S_1(x)a)+S_1(x)\partial a, \tag*{(32)} S 0 ( ρ ( n x ) a + ρ ( x ) S 0 a − S 0 ( ρ ( x ) a ) ) − ρ ( n x ) S 0 a = ∂ ( S 1 ( x ) a ) + S 1 ( x ) ∂ a , (32) which for n = 1 n=1 n = 1 (28) and for n = 2 , 3 n=2,3 n = 2 , 3 [12 Def. 5.12] , resp. (S1)–(S3) of [24 Def. 5.8] .
Proof. Functoriality and well-definedness hold by construction: Definition 7.4 asks precisely that N S N^{\mathsf{S}} N S Ψ \Psi Ψ (32) is obtained by evaluating (23) for m = 2 m=2 m = 2 ( σ x , a ) (\sigma x,a) ( σ x , a ) x ∈ g x\in\mathfrak{g} x ∈ g a ∈ V a\in\mathcal{V} a ∈ V V \mathcal{V} V Tors N 1 ( [ , ] ) ( σ x , a ) \operatorname{Tors}_{N_{1}}([\ ,\ ])(\sigma x,a) Tors N 1 ([ , ]) ( σ x , a ) (13) and (31) , and Λ 2 N = d N 2 + N 2 ( d ⋅ , ⋅ ) \Lambda^{N}_{2}=dN_{2}+N_{2}(d\cdot,\cdot) Λ 2 N = d N 2 + N 2 ( d ⋅ , ⋅ )
The equivalence
Definition 7.6. Let ( g , ℓ , N ) (\mathfrak{g},\ell,N) ( g , ℓ , N ) L ∞ L_{\infty} L ∞ extension of ( g , n ) (\mathfrak{g},\mathfrak{n}) ( g , n ) π : ( g , ℓ ) , N → ( g , n ) \pi\colon(\mathfrak{g},\ell),N\to(\mathfrak{g},\mathfrak{n}) π : ( g , ℓ ) , N → ( g , n ) N i j L i e n \mathsf{NijLie}_{n} NijLie n split if it admits a strict section σ : ( g , n ) → ( g , ℓ ) , N \sigma\colon(\mathfrak{g},\mathfrak{n})\to(\mathfrak{g},\ell),N σ : ( g , n ) → ( g , ℓ ) , N π σ = id \pi\sigma=\operatorname{id} π σ = id abelian if its kernel V = ker π \mathcal{V}=\ker\pi V = ker π ℓ k \ell_k ℓ k N q N_q N q V \mathcal{V} V Ext ( g , n ) \operatorname{Ext}(\mathfrak{g},\mathfrak{n}) Ext ( g , n ) N i j L i e n \mathsf{NijLie}_{n} NijLie n π \pi π σ \sigma σ
Theorem 7.7. The functor Ψ \Psi Ψ of Theorem 7.5 induces an equivalence of categories
Ψ : Rep n ( g , n ) → ≃ Ext ( g , n ) . \Psi\colon\operatorname{Rep}_{n}(\mathfrak{g},\mathfrak{n}) \xrightarrow{\ \simeq\ } \operatorname{Ext}(\mathfrak{g},\mathfrak{n}). Ψ : Rep n ( g , n ) ≃ Ext ( g , n ) . For n = 1 n=1 n = 1 n = 2 n=2 n = 2 [12 Prop. 5.14] and for n = 3 n=3 n = 3 [24 Thm. 5.13] ; the squares
Rep m ( g , n ) → Ψ N i j L i e m ↓ ↓ Rep n ( g , n ) → Ψ N i j L i e n \begin{CD}
\operatorname{Rep}_{m}(\mathfrak{g},\mathfrak{n}) @>\Psi>> \mathsf{NijLie}_{m} \\
@VVV @VVV \\
\operatorname{Rep}_{n}(\mathfrak{g},\mathfrak{n}) @>\Psi>> \mathsf{NijLie}_{n}
\end{CD} Rep m ( g , n ) ↓ ⏐ Rep n ( g , n ) Ψ Ψ NijLie m ↓ ⏐ NijLie n commute for m ≤ n m\leq n m ≤ n
Proof. Ψ \Psi Ψ Ext ( g , n ) \operatorname{Ext}(\mathfrak{g},\mathfrak{n}) Ext ( g , n ) L ∞ L_{\infty} L ∞ L ∞ L_{\infty} L ∞ π \pi π σ \sigma σ N i j L i e n \mathsf{NijLie}_{n} NijLie n π N S = n π \pi N^{\mathsf{S}}=\mathfrak{n}\pi π N S = n π N S σ = σ n N^{\mathsf{S}}\sigma=\sigma\mathfrak{n} N S σ = σ n (31) .
Faithfulness. If Ψ ( Φ , Θ ) = Ψ ( Φ ′ , Θ ′ ) \Psi(\Phi,\Theta)=\Psi(\Phi',\Theta') Ψ ( Φ , Θ ) = Ψ ( Φ ′ , Θ ′ ) V \mathcal{V} V (31) and its analogue for morphisms this recovers every component Φ j \Phi_j Φ j Θ j \Theta_j Θ j ( Φ , Θ ) = ( Φ ′ , Θ ′ ) (\Phi,\Theta)=(\Phi',\Theta') ( Φ , Θ ) = ( Φ ′ , Θ ′ )
Fullness. Let ( φ , ψ ) : Ψ ( V , S ) → Ψ ( V ′ , S ′ ) (\varphi,\psi)\colon\Psi(\mathcal{V},S)\to\Psi(\mathcal{V}',S') ( φ , ψ ) : Ψ ( V , S ) → Ψ ( V ′ , S ′ ) Ext ( g , n ) \operatorname{Ext}(\mathfrak{g},\mathfrak{n}) Ext ( g , n ) φ 1 \varphi_1 φ 1 π \pi π σ \sigma σ φ 1 ( σ x + a ) = σ ′ x + Φ 0 ( a ) \varphi_1(\sigma x+a)=\sigma'x+\Phi_0(a) φ 1 ( σ x + a ) = σ ′ x + Φ 0 ( a ) Φ 0 \Phi_0 Φ 0 φ q \varphi_q φ q ψ q \psi_q ψ q ( σ x 1 , … , σ x q − 1 , a ) (\sigma x_1,\ldots,\sigma x_{q-1},a) ( σ x 1 , … , σ x q − 1 , a ) Φ q − 1 \Phi_{q-1} Φ q − 1 Θ q − 1 \Theta_{q-1} Θ q − 1 V \mathcal{V} V π \pi π σ \sigma σ (25) gives (30) and its Nijenhuis analogue, so ( φ , ψ ) = Ψ ( Φ , Θ ) (\varphi,\psi)=\Psi(\Phi,\Theta) ( φ , ψ ) = Ψ ( Φ , Θ )
Essential surjectivity. Let ( ( g , ℓ ) , N , π , σ ) ((\mathfrak{g},\ell),N,\pi,\sigma) (( g , ℓ ) , N , π , σ ) V \mathcal{V} V σ \sigma σ g 0 = σ ( g ) ⊕ V 0 \mathfrak{g}_0=\sigma(\mathfrak{g})\oplus\mathcal{V}_0 g 0 = σ ( g ) ⊕ V 0 g i = V i \mathfrak{g}_i=\mathcal{V}_i g i = V i i ≥ 1 i\geq1 i ≥ 1 R \mathcal{R} R S \mathcal{S} S (31) backwards, i.e.
R k − 1 ( x 1 , … , x k − 1 ) a = ( − 1 ) k ℓ k ( σ x 1 , … , σ x k − 1 , a ) , R_{k-1}(x_1,\ldots,x_{k-1})a=(-1)^k\ell_k(\sigma x_1,\ldots,\sigma x_{k-1},a), R k − 1 ( x 1 , … , x k − 1 ) a = ( − 1 ) k ℓ k ( σ x 1 , … , σ x k − 1 , a ) , S q − 1 ( x 1 , … , x q − 1 ) a = ( − 1 ) q N q ( σ x 1 , … , σ x q − 1 , a ) , S_{q-1}(x_1,\ldots,x_{q-1})a=(-1)^qN_q(\sigma x_1,\ldots,\sigma x_{q-1},a), S q − 1 ( x 1 , … , x q − 1 ) a = ( − 1 ) q N q ( σ x 1 , … , σ x q − 1 , a ) , together with R 0 = d ∣ V R_0=d|_{\mathcal{V}} R 0 = d ∣ V S 0 = N 1 ∣ V S_0=N_1|_{\mathcal{V}} S 0 = N 1 ∣ V π \pi π V \mathcal{V} V V \mathcal{V} V R \mathcal{R} R (29) , and by construction N S = N N^{S}=N N S = N g ≅ g ⊕ V \mathfrak{g}\cong\mathfrak{g}\oplus\mathcal{V} g ≅ g ⊕ V σ \sigma σ ( V , S ) (\mathcal{V},S) ( V , S ) Rep n ( g , n ) \operatorname{Rep}_{n}(\mathfrak{g},n) Rep n ( g , n ) ( x , a ) ↦ σ ( x ) + a (x,a)\mapsto\sigma(x)+a ( x , a ) ↦ σ ( x ) + a Ψ ( V , S ) ⟶ ≅ ( ( g , ℓ ) , N ) \Psi(\mathcal{V},S)\overset{\cong}{\longrightarrow}((\mathfrak{g},\ell),N) Ψ ( V , S ) ⟶ ≅ (( g , ℓ ) , N ) Ext ( g , n ) \operatorname{Ext}(\mathfrak{g},n) Ext ( g , n ) Theorem 5.1 together with the observation that adjoining a zero term to V \mathcal{V} V g ⋉ V \mathfrak{g}\ltimes\mathcal{V} g ⋉ V
Corollary 7.8. Split abelian extensions of a Nijenhuis Lie algebra ( g , n ) (\mathfrak{g},n) ( g , n ) N i j L i e n \mathit{NijLien} NijLien ( R , S ) (\mathcal{R},S) ( R , S ) Definitions 7.1 and 7.4 modulo the equivalence generated by (30) ; and the classification is stable in n n n m m m n n n m m m
Proof. Combine Theorem 7.7 with Theorem 5.1 .
Concluding remarks
We have defined Nijenhuis operators on L ∞ L_{\infty} L ∞ n n n n n n m m m L ∞ L_{\infty} L ∞
Is the twisted collection always an L ∞ L_{\infty} L ∞ Theorem 4.10 proves this when the linear part n n n Proposition 6.3 proves it for strict objects at every level with no hypothesis. In general the L ∞ L_{\infty} L ∞ ℓ N \ell^{N} ℓ N ker N \ker N ker N ℓ N \ell^{N} ℓ N [29 ] .
Cohomology and deformations. The identities (23) should be the vanishing conditions for a Maurer–Cartan element in a graded Lie algebra built from the Nijenhuis–Richardson bracket on ⨁ k Hom ( S k V , V ) \bigoplus_k\operatorname{Hom}(S^kV,V) ⨁ k Hom ( S k V , V ) N N N L ∞ L_{\infty} L ∞ [11 , 28 ] , in the spirit of the survey [26 ] and of the Maurer–Cartan and cohomological treatments of related operators and derivation-type structures [8 , 6 , 7 ] , and would explain the polynomial degree bound ⌊ m / 2 ⌋ \lfloor m/2\rfloor ⌊ m /2 ⌋
Categorification. Baez and Crans realise 2-term L ∞ L_{\infty} L ∞ [3 ] . The level-n n n n n n L ∞ L_{\infty} L ∞ n n n n − 1 n-1 n − 1 [12 Def. 3.1] and the second-order coherence of [24 §6] . Proving that the resulting category is equivalent to N i j L i e n \mathit{NijLien} NijLien n n n [12 Thm. 4.5] .
Other operators. The scheme of this paper — torsion corrected by the substitutions of the higher components, equivalently the morphism property with respect to an explicitly twisted structure — applies verbatim to Rota–Baxter operators, twisted Rota–Baxter operators and Reynolds operators, and should produce the level-n n n [33 , 13 ] ; see also [6 ] for modified Rota–Baxter operators with derivations. The only input which changes is the definition of the twist (11) and hence of the correction (24) .
Geometry. Nijenhuis forms on L ∞ L_{\infty} L ∞ [2 ] . It would be interesting to compare the notion of the present paper with the Nijenhuis forms of that work; the comparison is not immediate because [2 ] works with a fixed arity while our operator has components in every arity.
Other algebraic structures. The classification and derivation-theoretic techniques developed for associative trialgebras and Hom-trialgebras [21 , 22 ] suggest studying Nijenhuis-type operators, and their torsion, on such structures as well.
Higher crossed modules. Proposition 6.1 identifies strict Lie n n n n n n S N i j L i e n \mathrm{SNijLie}_{n} SNijLie n ( n − 1 ) (n-1) ( n − 1 )
Component identities at low levels
We record the expansion of (23) in components for n = 1 , 2 , 3 n=1,2,3 n = 1 , 2 , 3 x , y , z , t ∈ g 0 x,y,z,t \in\mathfrak{g}_{0} x , y , z , t ∈ g 0 h , k ∈ g 1 h,k \in\mathfrak{g}_{1} h , k ∈ g 1 e ∈ g 2 e \in\mathfrak{g}_{2} e ∈ g 2 ∑ ( 3 ) \sum_{(3)} ∑ ( 3 ) § 2.2 and were produced and cross-checked by the computation of Appendix § 10 .
Level one
The only non-vacuous identity is the arity-two one,
Tors n ( [ , ] ) ( x , y ) = [ n x , n y ] − n ( [ n x , y ] + [ x , n y ] − n [ x , y ] ) = 0. \operatorname{Tors}_{n}([\ ,\ ])(x,y)=[nx,ny]-n\left([nx,y]+[x,ny]-n[x,y]\right)=0. Tors n ([ , ]) ( x , y ) = [ n x , n y ] − n ( [ n x , y ] + [ x , n y ] − n [ x , y ] ) = 0. Level two
Here N = ( n 0 , n 1 , N 2 ) N=(n_{0},n_{1},N_{2}) N = ( n 0 , n 1 , N 2 ) N 2 : g 0 × g 0 → g 1 N_{2}\colon\mathfrak{g}_{0}\times\mathfrak{g}_{0}\to\mathfrak{g}_{1} N 2 : g 0 × g 0 → g 1
m = 1 : n d = d n , (33) m=1:\qquad nd=dn, \tag*{(33)} m = 1 : n d = d n , (33) m = 2 : Tors n ( [ , ] ) ( x , y ) + d N 2 ( x , y ) = 0 , (34) m=2:\qquad\operatorname{Tors}_{n}([\ ,\ ])(x,y)+dN_{2}(x,y)=0, \tag*{(34)} m = 2 : Tors n ([ , ]) ( x , y ) + d N 2 ( x , y ) = 0 , (34) Tors n ( [ , ] ) ( x , h ) + N 2 ( x , d h ) = 0 , (35) \operatorname{Tors}_{n}([\ ,\ ])(x,h)+N_{2}(x,dh)=0, \tag*{(35)} Tors n ([ , ]) ( x , h ) + N 2 ( x , d h ) = 0 , (35) m = 3 : Tors n ( l 3 ) ( x , y , z ) + Λ ( x , y , z ) = 0 , (36) m=3:\qquad\operatorname{Tors}_{n}(l_{3})(x,y,z)+\Lambda(x,y,z)=0, \tag*{(36)} m = 3 : Tors n ( l 3 ) ( x , y , z ) + Λ ( x , y , z ) = 0 , (36) where
Λ ( a , b , c ) : = ∑ ( 3 ) ( [ n a , N 2 ( b , c ) ] − n [ a , N 2 ( b , c ) ] + N 2 ( a , [ b , c ] n ) − n N 2 ( a , [ b , c ] ) ) . \Lambda(a,b,c):=\sum_{(3)}\left([na,N_{2}(b,c)]-n[a,N_{2}(b,c)]+N_{2}(a,[b,c]_{n})-nN_{2}(a,[b,c])\right). Λ ( a , b , c ) := ( 3 ) ∑ ( [ na , N 2 ( b , c )] − n [ a , N 2 ( b , c )] + N 2 ( a , [ b , c ] n ) − n N 2 ( a , [ b , c ]) ) . Identities (A.2)–(A.4) are (5)–(7) of [12 Def. 4.1] . The arity-four identity is vacuous.
Level three
Here N = ( n 0 , n 1 , n 2 , N 2 , N 3 ) N=(n_{0},n_{1},n_{2},N_{2},N_{3}) N = ( n 0 , n 1 , n 2 , N 2 , N 3 ) N 2 N_{2} N 2 g 0 × g 0 \mathfrak{g}_{0}\times\mathfrak{g}_{0} g 0 × g 0 g 0 × g 1 \mathfrak{g}_{0}\times\mathfrak{g}_{1} g 0 × g 1 N 3 : g 0 3 → g 2 N_{3}\colon\mathfrak{g}_{0}^{3}\to\mathfrak{g}_{2} N 3 : g 0 3 → g 2
m = 1 : n d = d n on g 1 and g 2 , (37) m=1:\qquad nd=dn\ \text{on }\mathfrak{g}_{1}\text{ and }\mathfrak{g}_{2}, \tag*{(37)} m = 1 : n d = d n on g 1 and g 2 , (37) m = 2 : Tors n ( [ , ] ) ( x , y ) + d N 2 ( x , y ) = 0 , (38) m=2:\qquad\operatorname{Tors}_{n}([\ ,\ ])(x,y)+dN_{2}(x,y)=0, \tag*{(38)} m = 2 : Tors n ([ , ]) ( x , y ) + d N 2 ( x , y ) = 0 , (38) Tors n ( [ , ] ) ( x , h ) + N 2 ( x , d h ) + d N 2 ( x , h ) = 0 , (39) \operatorname{Tors}_{n}([\ ,\ ])(x,h)+N_{2}(x,dh)+dN_{2}(x,h)=0, \tag*{(39)} Tors n ([ , ]) ( x , h ) + N 2 ( x , d h ) + d N 2 ( x , h ) = 0 , (39) Tors n ( [ , ] ) ( x , e ) + N 2 ( x , d e ) = 0 , (40) \operatorname{Tors}_{n}([\ ,\ ])(x,e)+N_{2}(x,de)=0, \tag*{(40)} Tors n ([ , ]) ( x , e ) + N 2 ( x , d e ) = 0 , (40) Tors n ( [ , ] ) ( h , k ) + N 2 ( d h , k ) − N 2 ( h , d k ) = 0 , (41) \operatorname{Tors}_{n}([\ ,\ ])(h,k)+N_{2}(dh,k)-N_{2}(h,dk)=0, \tag*{(41)} Tors n ([ , ]) ( h , k ) + N 2 ( d h , k ) − N 2 ( h , d k ) = 0 , (41) m = 3 : Tors n ( l 3 ) ( x , y , z ) + Λ ( x , y , z ) + d N 3 ( x , y , z ) = 0 , (42) m=3:\qquad\operatorname{Tors}_{n}(l_{3})(x,y,z)+\Lambda(x,y,z)+dN_{3}(x,y,z)=0, \tag*{(42)} m = 3 : Tors n ( l 3 ) ( x , y , z ) + Λ ( x , y , z ) + d N 3 ( x , y , z ) = 0 , (42) Tors n ( l 3 ) ( x , y , h ) + Λ ( x , y , h ) − N 3 ( x , y , d h ) = 0 , (43) \operatorname{Tors}_{n}(l_{3})(x,y,h)+\Lambda(x,y,h)-N_{3}(x,y,dh)=0, \tag*{(43)} Tors n ( l 3 ) ( x , y , h ) + Λ ( x , y , h ) − N 3 ( x , y , d h ) = 0 , (43) m = 4 : Tors n ( l 4 ) ( x , y , z , t ) + Ξ ( x , y , z , t ) = 0 , (44) m=4:\qquad\operatorname{Tors}_{n}(l_{4})(x,y,z,t)+\Xi(x,y,z,t)=0, \tag*{(44)} m = 4 : Tors n ( l 4 ) ( x , y , z , t ) + Ξ ( x , y , z , t ) = 0 , (44) with Λ \Lambda Λ ( x 1 , x 2 , x 3 , x 4 ) = ( x , y , z , t ) (x_1,x_2,x_3,x_4)=(x,y,z,t) ( x 1 , x 2 , x 3 , x 4 ) = ( x , y , z , t ) ∑ ( 4 ) \sum_{(4)} ∑ ( 4 ) ∑ ( 4 , 2 ) \sum_{(4,2)} ∑ ( 4 , 2 ) i < j i<j i < j ε i j k l \varepsilon_{ijkl} ε ij k l
Ξ = ∑ ( 4 ) ( [ n x i , N 3 ( x ^ i ) ] − n [ x i , N 3 ( x ^ i ) ] ) + ∑ ( 4 ) ( N 2 ( x i , l 3 N ( x ^ i ) ) − n N 2 ( x i , l 3 ( x ^ i ) ) ) + ∑ ( 4 , 2 ) ( l 3 ( n x k , n x l , N 2 ( x i , x j ) ) − n l 3 ( x k , x l , N 2 ( x i , x j ) ) ) + ∑ ( 4 , 2 ) ( N 3 ( x k , x l , [ x i , x j ] n ) − n N 3 ( x k , x l , [ x i , x j ] ) ) + ∑ { i , j } ⊔ { k , l } ε i j k l [ N 2 ( x i , x j ) , N 2 ( x k , x l ) ] . (45) \Xi= \sum_{(4)} \left([n x_i,N_3(\hat{x}_i)]-n[x_i,N_3(\hat{x}_i)]\right)+\sum_{(4)}\left(N_2(x_i,l_3^N(\hat{x}_i))-nN_2(x_i,l_3(\hat{x}_i))\right)
\\
+\sum_{(4,2)}\left(l_3(nx_k,nx_l,N_2(x_i,x_j))-nl_3(x_k,x_l,N_2(x_i,x_j))\right)
\\
+\sum_{(4,2)}\left(N_3(x_k,x_l,[x_i,x_j]_n)-nN_3(x_k,x_l,[x_i,x_j])\right)+\sum_{\{i,j\}\sqcup\{k,l\}}\varepsilon_{ijkl}[N_2(x_i,x_j),N_2(x_k,x_l)]. \tag*{(45)} Ξ = ( 4 ) ∑ ( [ n x i , N 3 ( x ^ i )] − n [ x i , N 3 ( x ^ i )] ) + ( 4 ) ∑ ( N 2 ( x i , l 3 N ( x ^ i )) − n N 2 ( x i , l 3 ( x ^ i )) ) + ( 4 , 2 ) ∑ ( l 3 ( n x k , n x l , N 2 ( x i , x j )) − n l 3 ( x k , x l , N 2 ( x i , x j )) ) + ( 4 , 2 ) ∑ ( N 3 ( x k , x l , [ x i , x j ] n ) − n N 3 ( x k , x l , [ x i , x j ]) ) + { i , j } ⊔ { k , l } ∑ ε ij k l [ N 2 ( x i , x j ) , N 2 ( x k , x l )] . (45) where in the third and fourth sums { k < l } \{k<l\} { k < l } { i < j } \{i<j\} { i < j } l 3 N l_3^N l 3 N (18) . The arity-five identity is vacuous. Identities (A.6)–(A.9), (A.11), (A.12) are (N1)–(N4), (N6), (N7) of [24 Def. 3.6] ; identity (A.10) corrects [24 (N5)] by the term d N 3 dN_3 d N 3 (45) corrects [24 (3.22)] , as explained in Remark 5.5.
Level Four
At level four, the framework incorporates the higher component N 4 : g 0 × 4 → g 3 N_4:\mathfrak{g}_0^{\times4}\to\mathfrak{g}_3 N 4 : g 0 × 4 → g 3 l 5 l_5 l 5 Λ 5 N \Lambda^N_5 Λ 5 N { 1 , … , 5 } \{1,\ldots,5\} { 1 , … , 5 } ( 2 , 2 , 1 ) (2,2,1) ( 2 , 2 , 1 )
l 3 ( n x i , N 2 ( x j , x k ) , N 2 ( x l , x m ) ) − n l 3 ( x i , N 2 ( x j , x k ) , N 2 ( x l , x m ) ) , l_3(nx_i,N_2(x_j,x_k),N_2(x_l,x_m))-nl_3(x_i,N_2(x_j,x_k),N_2(x_l,x_m)), l 3 ( n x i , N 2 ( x j , x k ) , N 2 ( x l , x m )) − n l 3 ( x i , N 2 ( x j , x k ) , N 2 ( x l , x m )) , together with types ( 2 , 1 , 1 , 1 ) (2,1,1,1) ( 2 , 1 , 1 , 1 ) ( 3 , 2 ) (3,2) ( 3 , 2 ) l 2 ( N 2 , N 3 ) l_2(N_2,N_3) l 2 ( N 2 , N 3 ) l 4 ( n ⋅ , n ⋅ , n ⋅ , N 2 ) l_4(n\cdot,n\cdot,n\cdot,N_2) l 4 ( n ⋅ , n ⋅ , n ⋅ , N 2 )
Machine Verification
The consistency of the component identities with [12 , 24 ] was verified using a computer algebra routine implementing the shifted conventions of § 2.2 :
Term Structure: Terms are formalized as directed trees whose leaves carry V V V l k l_k l k − 1 -1 − 1 N q N_q N q 0 0 0 1 ≤ deg ≤ n 1\le\deg\le n 1 ≤ deg ≤ n
Evaluation of Sides: For each arity m m m A m A_m A m { 1 , … , m } \{1,\ldots,m\} { 1 , … , m } B m B_m B m ( k , m − k ) (k,m-k) ( k , m − k ) l k N l_k^N l k N (15) .
Output Generation: Computes the normal forms of A m − B m A_m-B_m A m − B m Tors n ( l m ) \operatorname{Tors}_n(l_m) Tors n ( l m ) Λ m N \Lambda_m^N Λ m N
Computational Outcomes by Truncation Level:
n = 3 n = 3 n = 3 g 0 3 g_{0}^{3} g 0 3 d N 3 ( x , y , z ) dN_{3}(x, y, z) d N 3 ( x , y , z ) n n n
n = 4 n = 4 n = 4 Λ 5 N \Lambda_{5}^{N} Λ 5 N L ∞ L_{\infty} L ∞ [12 , 24 ] .
Acknowledgements
The authors are grateful to the referees for their valuable comments.
