To our friend Andrey Pogrebkov on the occasion of his 80th birthday
Introduction
The story of the noncommutative versions of the fundamental theorem of matrix algebra — the Cayley-Hamilton theorem — dates back to the early 1970s with the observation that the generators of the universal enveloping algebra U ( g ) U(g) U ( g ) [6 , 7 ] , subsequently received an abstract-algebraic interpretation [8 ] and were generalized to the case of Lie superalgebras (for a survey and list of references, see [9 ] ).
With the discovery of quantum groups [10 , 11 , 5 ] , the list of associative algebras generated by mutually non-commuting matrix components has expanded significantly. Examples of characteristic identities have been derived for quantized universal enveloping algebras [12 , 13 ] , for their dual algebras of quantized functions on Lie groups (so-called RTT-algebras) [14 , 15 , 16 ] , and for Reflection Equation (RE-) algebras [17 , 18 ] .1
In [21 ] , the Cayley-Hamilton theorem was proved in a unified manner for a family of RE-algebras defined by G L ( n ) GL(n) G L ( n ) [21 ] also yield structural results for RTT-algebras [22 , 23 , 24 ] . Subsequently, the Cayley-Hamilton theorem was generalized to a broader class of G L ( n ) GL(n) G L ( n ) [25 , 26 ] . To date, this line of research has also led to the construction of characteristic identities for families of QM-algebras of type G L ( m ∣ n ) GL(m|n) G L ( m ∣ n ) [27 , 28 ] 2 S p ( 2 n ) Sp(2n) S p ( 2 n ) O ( n ) O(n) O ( n ) [1 , 2 , 3 , 4 ] .
Many other studies of characteristic identities and related issues in the context of quantum group theory have been limited to the canonical (Drinfeld-Jimbo’s) quantization of universal enveloping algebras and their dual RTT-algebras (see, e.g., [12 , 17 , 15 , 16 , 31 , 32 , 13 , 33 ] ). In these cases, known explicit relations for the algebra’s generators can be used to construct a linearly ordered basis for the space of polynomials in these generators, which can then be applied in further reasoning and calculations. This approach is well-suited for the case of general linear quantum (super)groups. However, in orthogonal quantum matrix algebras, the set of relations turns out to be redundant: in addition to ordering relations, it includes supplementary orthogonality conditions (see [5 ] ). In such a situation, constructing a linear basis for a quantum matrix algebra and using it for calculations becomes problematic.
In this note, we consider the simplest non-trivial example of the orthogonal O ( 3 ) O(3) O ( 3 ) F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v q q q v v v q q q v v v 3
We establish a suitable linear order on the set of the algebra generators and explicitly write down the ordering and orthogonality relations for these generators. We then investigate the impact of the orthogonality conditions on the construction of a linear basis in F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v [3 , 4 ] . Finally, we consider F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v v = 1 v=1 v = 1 g g g e 3 e_3 e 3 § 3 ) – called, respectively, the contraction and the 3-elementary sum become central. A quotient algebra F q ( O ( 3 ) ) : = F q , v = 1 / ⟨ q 2 ( n − 1 ) g − 1 ⟩ \mathcal{F}_q(O(3)) := \mathcal{F}_{q,v=1}/\langle q^{2(n-1)}g-1\rangle F q ( O ( 3 )) := F q , v = 1 / ⟨ q 2 ( n − 1 ) g − 1 ⟩ O ( 3 ) O(3) O ( 3 )
Summarizing our study of the algebra F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v
Multiparametric orthogonal R-matrices in dimension 3
In this preparatory Section, we have gathered the necessary information about a specific orthogonal O ( 3 ) O(3) O ( 3 ) R R R R R R O ( n ) O(n) O ( n ) n n n
The standard orthogonal R-matrix R ∘ ∈ Aut ( V ⊗ 2 ) R^\circ\in\operatorname{Aut}(V^{\otimes2}) R ∘ ∈ Aut ( V ⊗ 2 ) V V V [5 ] :
R ∘ = ( q ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ( 1 − 1 q ) ⋅ − l q ⋅ 1 q ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − l q ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ⋅ 1 ⋅ ⋅ ⋅ 1 q ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ q ) , λ : = q − q − 1 . R^\circ=
\begin{pmatrix}
q & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& l & \cdot& 1 & \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& l(1-\frac{1}{q}) & \cdot& -\frac{l}{\sqrt{q}} & \cdot& \frac{1}{q} & \cdot& \cdot\\
\cdot& 1 & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& -\frac{l}{\sqrt{q}} & \cdot& 1 & \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& \cdot& \cdot& \cdot& l & \cdot& 1 & \cdot\\
\cdot& \cdot& \frac{1}{q} & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& \cdot& \cdot& \cdot& 1 & \cdot& \cdot& \cdot\\
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& q
\end{pmatrix},
\qquad\lambda:= q-q^{-1}. R ∘ = q ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ( 1 − q 1 ) ⋅ − q l ⋅ q 1 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − q l ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ⋅ 1 ⋅ ⋅ ⋅ q 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ q , λ := q − q − 1 . Here, dots mark the positions of the zero components of the R-matrix. In this note, we will use a ‘twisted’ version of this R-matrix equipped with two additional non-zero complex parameters u u u v v v R ∘ R^\circ R ∘
R = ( q ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ⋅ v ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ( 1 − 1 q ) ⋅ − l u q ⋅ v 2 q ⋅ ⋅ ⋅ 1 v ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − l u v q ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ⋅ v ⋅ ⋅ ⋅ 1 q v 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1 v ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ q ) . (1) R =
\begin{pmatrix}
q & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& l & \cdot& v & \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& l(1-\frac{1}{q}) & \cdot& -\frac{lu}{\sqrt{q}} & \cdot& \frac{v^2}{q} & \cdot& \cdot\\
\cdot& \frac{1}{v} & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& -\frac{l}{u v\sqrt{q}} & \cdot& 1 & \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& \cdot& \cdot& \cdot& l & \cdot& v & \cdot\\
\cdot& \cdot& \frac{1}{qv^2} & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\
\cdot& \cdot& \cdot& \cdot& \cdot& \frac{1}{v} & \cdot& \cdot& \cdot\\
\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& q
\end{pmatrix}. \tag*{(1)} R = q ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ⋅ v 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ( 1 − q 1 ) ⋅ − uv q l ⋅ q v 2 1 ⋅ ⋅ ⋅ v ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ − q l u ⋅ 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ l ⋅ v 1 ⋅ ⋅ ⋅ q v 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ v ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ q . (1) The parameter u u u V V V
u = q . (2) u=\sqrt{q}. \tag*{(2)} u = q . (2) On the contrary, the parameter v v v [34 , 35 ] ). Unlike the purely gauge parameter u u u v v v (1) . We shall keep this parameter arbitrary until the last Section of the paper.
The R-matrix (1) satisfies the braid relation, also known as the constant Yang-Baxter equation
R 1 R 2 R 1 = R 2 R 1 R 2 . (3) R_{1}R_{2}R_{1}=R_{2}R_{1}R_{2}. \tag*{(3)} R 1 R 2 R 1 = R 2 R 1 R 2 . (3) Here, we have introduced the now-standard matrix notations R 1 = R ⊗ Id V R_{1}=R\otimes\operatorname{Id}_{V} R 1 = R ⊗ Id V R 2 = Id V ⊗ R R_{2}=\operatorname{Id}_{V}\otimes R R 2 = Id V ⊗ R Id V \operatorname{Id}_{V} Id V V V V X ∈ End ( V ⊗ 2 ) X\in\operatorname{End}(V^{\otimes2}) X ∈ End ( V ⊗ 2 )
X i : = Id V ⊗ ( i − 1 ) ⊗ X ⊗ Id V ⊗ ( n − i − 1 ) ∈ End ( V ⊗ n ) , 1 ≤ i ≤ n − 1 , X_{i}:=\operatorname{Id}_{V}^{\otimes(i-1)}\otimes X\otimes\operatorname{Id}_{V}^{\otimes(n-i-1)}\in\operatorname{End}(V^{\otimes n}),\qquad1\leq i\leq n-1, X i := Id V ⊗ ( i − 1 ) ⊗ X ⊗ Id V ⊗ ( n − i − 1 ) ∈ End ( V ⊗ n ) , 1 ≤ i ≤ n − 1 , acting on the spaces V ⊗ n V^{\otimes n} V ⊗ n n ≥ 2 n\geq2 n ≥ 2
For generic q q q (1) has 3 distinct eigenvalues: q q q − q − 1 -q^{-1} − q − 1 q − 2 q^{-2} q − 2 [36 , 37 ] . We will need several formulas following from this fact. Details of their derivation, as well as a general description of Birman-Murakami-Wenzl algebras and of their R-matrix representations can be found in [1 , 3 ] and the references cited therein.
Before proceeding, we make a remark regarding notation: throughout the text, the subscript “R” in the notation X R X_{R} X R X X X R R R
K R : = q 2 λ ( q Id V ⊗ 2 − R ) ( q − 1 Id V ⊗ 2 + R ) , ( K R ) 2 = η K R , η : = q + 1 + q − 1 , K_{R}:=\frac{q^{2}}{\lambda}(q\operatorname{Id}_{V^{\otimes2}}-R)(q^{-1}\operatorname{Id}_{V^{\otimes2}}+R),\qquad(K_{R})^{2}=\eta K_{R},\qquad\eta:=q+1+q^{-1}, K R := λ q 2 ( q Id V ⊗ 2 − R ) ( q − 1 Id V ⊗ 2 + R ) , ( K R ) 2 = η K R , η := q + 1 + q − 1 , onto the subspace of eigenvectors of R R R q − 2 q^{-2} q − 2 [35 ] ). The explicit expression for K K K
K R = ( ⋅ ⋅ 1 ∣ ⋅ 1 ⋅ ∣ q v 2 ⋅ ⋅ ) t ( ⋅ ⋅ q − 1 ∣ ⋅ 1 ⋅ ∣ v 2 ⋅ ⋅ ) . (4) K_{R}=(\mathord{\cdot}\ \mathord{\cdot}\ 1\mid\mathord{\cdot}\ 1\mathord{\cdot}\mid\frac{q}{v^{2}}\mathord{\cdot}\ \mathord{\cdot})^{t}(\mathord{\cdot}\ \mathord{\cdot}\ q^{-1}\mid\mathord{\cdot}\ 1\mathord{\cdot}\mid v^{2}\mathord{\cdot}\ \mathord{\cdot}). \tag*{(4)} K R = ( ⋅ ⋅ 1 ∣ ⋅ 1 ⋅ ∣ v 2 q ⋅ ⋅ ) t ( ⋅ ⋅ q − 1 ∣ ⋅ 1 ⋅ ∣ v 2 ⋅ ⋅ ) . (4) Secondly, in Birman–Murakami–Wenzl algebras, one can introduce a remarkable sequence of idempotents known as q q q [1 ] , Sections 2.3 and 2.4). Their images in R-matrix representations (see [1 ] , Section 3.1) have proven to be important technical tools in the study of QM-algebras. These matrix images we will, with an abuse of notation, also call q q q A R ( i ) A_{R}^{(i)} A R ( i ) i = 1 , 2 , … i=1,2,\ldots i = 1 , 2 , … i i i V V V A R ( i ) ∈ End ( V ⊗ i ) A_{R}^{(i)}\in\operatorname{End}(V^{\otimes i}) A R ( i ) ∈ End ( V ⊗ i )
For R R R O ( n ) O(n) O ( n ) n n n q q q q q q [3 ] , Sections 3.1, 3.2). Below we present formulas for the nonzero q q q O ( 3 ) O(3) O ( 3 ) (1) .
A R ( 1 ) = Id V , A R ( 2 ) = q 3 ( 1 + q ) ( 1 + q 2 ) ( q Id V ⊗ 2 − R ) ( q − 2 Id V ⊗ 2 − R ) , (5) A_{R}^{(1)}=\operatorname{Id}_{V},\qquad A_{R}^{(2)}=\frac{q^{3}}{(1+q)(1+q^{2})}(q\operatorname{Id}_{V^{\otimes2}}-R)(q^{-2}\operatorname{Id}_{V^{\otimes2}}-R), \tag*{(5)} A R ( 1 ) = Id V , A R ( 2 ) = ( 1 + q ) ( 1 + q 2 ) q 3 ( q Id V ⊗ 2 − R ) ( q − 2 Id V ⊗ 2 − R ) , (5) ( A R ( 3 ) ) i 1 i 2 i 3 j 1 j 2 j 3 = ϵ R j 1 j 2 j 3 ⊗ ϵ R i 1 i 2 i 3 ( 1 + q 2 ) ( 1 + q + q 2 ) , i j k 123 132 213 222 231 312 321 ∣ ϵ R ⟩ i j k 1 − q u − q u q ( q − 1 ) v q u 3 q u 3 − q 2 v 4 ⟨ ϵ R ∣ i j k 1 − q v − q v 1 − q v q v 3 q v 3 − q 2 v 4 (6) \left(A_{R}^{(3)}\right)^{j_{1}j_{2}j_{3}}_{i_{1}i_{2}i_{3}}=\frac{\epsilon_{R}^{j_{1}j_{2}j_{3}}\otimes\epsilon_{R\,i_{1}i_{2}i_{3}}}{(1+q^{2})(1+q+q^{2})},
\qquad
\begin{array}{c|ccccccc}
i\,j\,k & 123 & 132 & 213 & 222 & 231 & 312 & 321\\
\hline
\lvert\epsilon_{R}\rangle^{ijk} & 1 & -qu & -qu & q(q-1)v & qu^{3} & qu^{3} & -q^{2}v^{4}\\
\langle\epsilon_{R}\rvert_{ijk} & 1 & -\frac{q}{v} & -\frac{q}{v} & \frac{1-q}{v} & \frac{q}{v^{3}} & \frac{q}{v^{3}} & -\frac{q^{2}}{v^{4}}
\end{array} \tag*{(6)} ( A R ( 3 ) ) i 1 i 2 i 3 j 1 j 2 j 3 = ( 1 + q 2 ) ( 1 + q + q 2 ) ϵ R j 1 j 2 j 3 ⊗ ϵ R i 1 i 2 i 3 , i j k ∣ ϵ R ⟩ ij k ⟨ ϵ R ∣ ij k 123 1 1 132 − q u − v q 213 − q u − v q 222 q ( q − 1 ) v v 1 − q 231 q u 3 v 3 q 312 q u 3 v 3 q 321 − q 2 v 4 − v 4 q 2 (6) Here, we write the rank-1 matrix A R ( 3 ) A_{R}^{(3)} A R ( 3 ) 4
Furthermore, there exist two invertible matrices C R C_{R} C R D R ∈ Aut ( V ) D_{R}\in\operatorname{Aut}(V) D R ∈ Aut ( V )
Tr ( 1 ) ( C R ) 1 R 1 = ( Id V ) 2 , Tr ( 2 ) ( D R ) 2 R 1 = ( Id V ) 1 . \operatorname{Tr}_{(1)}(C_{R})_{1}R_{1}=(\operatorname{Id}_{V})_{2},\qquad\operatorname{Tr}_{(2)}(D_{R})_{2}R_{1}=(\operatorname{Id}_{V})_{1}. Tr ( 1 ) ( C R ) 1 R 1 = ( Id V ) 2 , Tr ( 2 ) ( D R ) 2 R 1 = ( Id V ) 1 . Here the symbol Tr ( i ) \operatorname{Tr}_{(i)} Tr ( i ) i i i V V V C R C_R C R D R D_R D R (1) are given by:
D R = diag ( q − 3 , q − 2 , q − 1 ) , C R = diag ( q − 1 , q − 2 , q − 3 ) . (7) D_R=\operatorname{diag}\left(q^{-3},q^{-2},q^{-1}\right),\qquad C_R=\operatorname{diag}\left(q^{-1},q^{-2},q^{-3}\right). \tag*{(7)} D R = diag ( q − 3 , q − 2 , q − 1 ) , C R = diag ( q − 1 , q − 2 , q − 3 ) . (7) These matrices allow for the introduction of the concepts of left and right R-traces, see, e.g., [38 ] . In the following, we will use the right version of the R-trace:
Tr R X : = Tr ( D R X ) , X ∈ End ( V ) ⊗ W , (8) \operatorname{Tr}_{R}X:=\operatorname{Tr}(D_RX),\quad X\in\operatorname{End}(V)\otimes W, \tag*{(8)} Tr R X := Tr ( D R X ) , X ∈ End ( V ) ⊗ W , (8) where W W W X X X
Finally, there are two more invertible matrices O R O_R O R G R ∈ Aut ( V ) G_R\in\operatorname{Aut}(V) G R ∈ Aut ( V ) (1) are given below.
G R : = Tr ( 2 ) P 1 ( K R ) 1 = diag ( v − 2 , 1 , v 2 ) , (9) G_R:=\operatorname{Tr}_{(2)}P_1(K_R)_1=\operatorname{diag}\left(v^{-2},1,v^2\right), \tag*{(9)} G R := Tr ( 2 ) P 1 ( K R ) 1 = diag ( v − 2 , 1 , v 2 ) , (9) O R : = q ( q 2 + 1 + q − 2 ) q 2 − q + 1 Tr ( 2 , 3 ) ( P 1 P 2 A R ( 3 ) ) = diag ( v − 3 , 1 , v 3 ) . (10) O_R:=\frac{q\left(q^2+1+q^{-2}\right)}{q^2-q+1}\operatorname{Tr}_{(2,3)}\left(P_1P_2A_R^{(3)}\right)=\operatorname{diag}\left(v^{-3},1,v^3\right). \tag*{(10)} O R := q 2 − q + 1 q ( q 2 + 1 + q − 2 ) Tr ( 2 , 3 ) ( P 1 P 2 A R ( 3 ) ) = diag ( v − 3 , 1 , v 3 ) . (10) Here P ∈ Aut ( V ⊗ 2 ) P\in\operatorname{Aut}(V^{\otimes2}) P ∈ Aut ( V ⊗ 2 ) P ( x ⊗ y ) = y ⊗ x ∀ x , y ∈ V P(x\otimes y)=y\otimes x\ \forall x,y\in V P ( x ⊗ y ) = y ⊗ x ∀ x , y ∈ V (9) and (10) are variants of the general definitions of the matrices G R G_R G R O R O_R O R [3 ] , adapted to our specific case of the RTT-algebra and calculated for the R-matrix (1) . Note that G R G_R G R O R O_R O R ( G R ) 3 = ( O R ) 2 (G_R)^3=(O_R)^2 ( G R ) 3 = ( O R ) 2 [3 ] .
General results on the structure of orthogonal RTT-algebras
In this Section, we adapt general structural results concerning orthogonal type O ( n ) O(n) O ( n ) [3 , 4 ] to the case of the standard RTT-algebra of type O ( 3 ) O(3) O ( 3 )
Recall that the RTT-algebra F ( R ) \mathcal{F}(R) F ( R ) R ∈ Aut ( V ⊗ 2 ) R\in\operatorname{Aut}(V^{\otimes2}) R ∈ Aut ( V ⊗ 2 ) 5 W = C ⟨ T b a ⟩ a , b = 1 dim V W=\mathbb{C}\langle T^a_b\rangle_{a,b=1}^{\dim V} W = C ⟨ T b a ⟩ a , b = 1 d i m V
R T 1 T 2 = T 1 T 2 R . (11) RT_1T_2=T_1T_2R. \tag*{(11)} R T 1 T 2 = T 1 T 2 R . (11) Here T = ∥ T b a ∥ a , b = 1 dim V T=\left\|T^a_b\right\|_{a,b=1}^{\dim V} T = ∥ T b a ∥ a , b = 1 d i m V T 1 = T ⊗ Id V T_1=T\otimes\operatorname{Id}_V T 1 = T ⊗ Id V T 2 = Id V ⊗ T T_2=\operatorname{Id}_V\otimes T T 2 = Id V ⊗ T M M M
Each QM-algebra contains a distinguished abelian subalgebra—the so-called characteristic subalgebra . In the case of the RTT-algebra F ( R ) \mathcal{F}(R) F ( R ) O ( n ) O(n) O ( n ) 6
e 0 : = 1 , e i : = Tr R ( 1 , 2 , … , i ) ( T 1 T 2 … T i A R ( i ) ) , i = 1 , 2 , … , n , (12) e_0:=1,\qquad e_i:=\operatorname{Tr}_{R(1,2,\ldots,i)}\left(T_1T_2\ldots T_iA_R^{(i)}\right),\quad i=1,2,\ldots,n, \tag*{(12)} e 0 := 1 , e i := Tr R ( 1 , 2 , … , i ) ( T 1 T 2 … T i A R ( i ) ) , i = 1 , 2 , … , n , (12) called the elementary sums , and one additional element
g : = λ q n ( q n − 1 ) ( q n − 1 + q ) Tr R ( 1 , 2 ) ( T 1 T 2 K R ) , (13) g:=\frac{\lambda q^n}{(q^n-1)(q^{n-1}+q)}\operatorname{Tr}_{R(1,2)}\left(T_1T_2K_R\right), \tag*{(13)} g := ( q n − 1 ) ( q n − 1 + q ) λ q n Tr R ( 1 , 2 ) ( T 1 T 2 K R ) , (13) called the contraction . The last non-zero elementary sum e n e_n e n q q q determinant of the quantum matrix T T T [3 ] , Section 4.4):
det q T = q n ( n − 1 ) e n . (14) \det\nolimits_qT=q^{n(n-1)}e_n. \tag*{(14)} det q T = q n ( n − 1 ) e n . (14) The generators of the characteristic subalgebra are not algebraically independent. They satisfy the following system of reciprocal relations (see [3 ] , Section 4.3)
g i e n − i = e n e i , i = 1 , 2 , … , n . (15) g^{i}e_{n-i}=e_{n}e_{i},\qquad i=1,2,\ldots,n. \tag*{(15)} g i e n − i = e n e i , i = 1 , 2 , … , n . (15) Two of these elements – g g g e n e_{n} e n K R K_{R} K R A R ( n ) A_{R}^{(n)} A R ( n ) [3 ] , Propositions 4.6 and 4.12):
T g = g ( G R − 1 T G R ) , T e n = e n ( O R − 1 T O R ) . (16) Tg=g\left(G_{R}^{-1}TG_{R}\right),\qquad Te_{n}=e_{n}\left(O_{R}^{-1}TO_{R}\right). \tag*{(16)} T g = g ( G R − 1 T G R ) , T e n = e n ( O R − 1 T O R ) . (16) Elements of the characteristic subalgebra appear as coefficients of the powers of the quantum matrix T T T F ( R ) \mathcal{F}(R) F ( R ) T T T 7
Consider a map ϕ R \phi_{R} ϕ R X = ∥ X b a ∥ a , b = 1 dim V X=\lVert X^{a}_{b}\rVert_{a,b=1}^{\dim V} X = ∥ X b a ∥ a , b = 1 d i m V
ϕ R ( X ) 1 : = Tr R ( 2 ) ( X 2 R ) , (17) \phi_{R}(X)_{1}:=\operatorname{Tr}_{R(2)}\left(X_{2}R\right), \tag*{(17)} ϕ R ( X ) 1 := Tr R ( 2 ) ( X 2 R ) , (17) An analogue of the quantum matrix k k k power T ‾ k \overline{T}^{k} T k k k k
T ‾ 0 : = Id V , T ‾ k : = T ⋅ ϕ R ( T ‾ k − 1 ) . (18) \overline{T}^{0}:=\operatorname{Id}_{V},\qquad\overline{T}^{k}:=T\cdot\phi_{R}\left(\overline{T}^{k-1}\right). \tag*{(18)} T 0 := Id V , T k := T ⋅ ϕ R ( T k − 1 ) . (18) Here ⋅ \cdot ⋅
Now we are ready to formulate the analogue of the Cayley-Hamilton theorem for the orthogonal O ( n ) O(n) O ( n ) F ( R ) \mathcal{F}(R) F ( R ) O ( 2 ℓ − 1 ) O(2\ell-1) O ( 2 ℓ − 1 ) ℓ = 2 , 3 , … \ell=2,3,\ldots ℓ = 2 , 3 , … O ( 2 ℓ ) O(2\ell) O ( 2 ℓ ) [4 ] , Section 4).
Theorem 3.1 The quantum matrix T T T F ( R ) \mathcal{F}(R) F ( R ) O ( n ) O(n) O ( n ) n = 2 ℓ − 1 n=2\ell-1 n = 2 ℓ − 1
∑ j = 0 ℓ − 1 ( − q ) j ∑ r = 0 ℓ − j − 1 { T ‾ n − j − 2 r ( g ℓ − 1 e j − e n e j − 1 ) − T ‾ n − j − 2 r − 1 ( e n e j − g ℓ e j − 1 ) } ( q 2 g ) 2 r = 0 , (19) \sum_{j=0}^{\ell-1}(-q)^{j}\sum_{r=0}^{\ell-j-1}\left\{\overline{T}^{n-j-2r}\left(g^{\ell-1}e_{j}-e_{n}e_{j-1}\right)-\overline{T}^{n-j-2r-1}\left(e_{n}e_{j}-g^{\ell}e_{j-1}\right)\right\}(q^{2}g)^{2r}=0, \tag*{(19)} j = 0 ∑ ℓ − 1 ( − q ) j r = 0 ∑ ℓ − j − 1 { T n − j − 2 r ( g ℓ − 1 e j − e n e j − 1 ) − T n − j − 2 r − 1 ( e n e j − g ℓ e j − 1 ) } ( q 2 g ) 2 r = 0 , (19) where we set $e_{-1}=0.
We present the characteristic identity in unfactorized form: formula (19) is an expansion of equality (4.61) from [4 ] . Factorizing this equality requires the explicit use of the element g − 1 g^{-1} g − 1 [4 ] , Theorem 4.5 and Section 5). On the other hand, the unfactorized version (19) better serves the purpose of this work, that is, verifying the characteristic identity using the specific example of the RTT algebra F ( R ) \mathcal{F}(R) F ( R )
Standard O ( 3 ) O(3) O ( 3 )
Ordering of the generators and linear basis
In this Section we shall write in components the set of quadratic relations in the RTT-algebra F ( R ) \mathcal{F}(R) F ( R ) R R R (1) , (2) . In the following, we will explicitly indicate the parameters q q q v v v R R R F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v T T T
T = ( x 1 a + c + a − x 2 b + c − b − x 3 ) T=\begin{pmatrix}
x_{1} & a_{+} & c_{+}\\
a_{-} & x_{2} & b_{+}\\
c_{-} & b_{-} & x_{3}
\end{pmatrix} T = x 1 a − c − a + x 2 b − c + b + x 3 Analyzing quadratic relations (11) we choose suitable ordering for the generators:
( a + , a − , x 1 ) < ( c + , c − , x 2 ) < ( x 3 , b + , b − ) . (20) (a_{+},a_{-},x_{1}) < (c_{+},c_{-},x_{2}) < (x_{3},b_{+},b_{-}). \tag*{(20)} ( a + , a − , x 1 ) < ( c + , c − , x 2 ) < ( x 3 , b + , b − ) . (20) The relations inside the elements in each brackets are very easy, so, the orderings inside each bracket is of no importance; we assume them anti-alphabetic inside each bracket, for example.
This ordering allows us to express unordered quadratic monomials of generators as linear combinations of ordered ones with coefficients that are polynomials (not rational functions) in variables q ± 1 q^{\pm1} q ± 1 v ± 1 v^{\pm1} v ± 1
x 1 a + = q v 2 a + x 1 , x 2 c + = 1 v 2 c + x 2 , b + x 3 = q v x 3 b + , x 1 a − = q v a − x 1 , x 2 c − = v 2 c − x 2 , b − x 3 = q v x 3 b − . (21) \begin{aligned}
x_{1}a_{+} &= \frac{q}{v^{2}}a_{+}x_{1}, & x_{2}c_{+} &= \frac{1}{v^{2}}c_{+}x_{2}, & b_{+}x_{3} &= qv x_{3}b_{+},\\
x_{1}a_{-} &= qv\,a_{-}x_{1}, & x_{2}c_{-} &= v^{2}c_{-}x_{2}, & b_{-}x_{3} &= \frac{q}{v}x_{3}b_{-}. \tag*{(21)}
\end{aligned} x 1 a + x 1 a − = v 2 q a + x 1 , = q v a − x 1 , x 2 c + x 2 c − = v 2 1 c + x 2 , = v 2 c − x 2 , b + x 3 b − x 3 = q v x 3 b + , = v q x 3 b − . (21) a − a + = 1 v 2 a + a − , c − c + = 1 v 4 c + c − , b − b + = 1 v 2 b + b − . (22) a_{-}a_{+}=\frac{1}{v^{2}}a_{+}a_{-},\qquad c_{-}c_{+}=\frac{1}{v^{4}}c_{+}c_{-},\qquad b_{-}b_{+}=\frac{1}{v^{2}}b_{+}b_{-}. \tag*{(22)} a − a + = v 2 1 a + a − , c − c + = v 4 1 c + c − , b − b + = v 2 1 b + b − . (22) We don’t specify the order within the triplets. We usually use them in the order they are given in (4.1.1), but sometimes we rearrange them to apply other (orthogonality or diamond) relations.
Ordering relations between c ± c_{\pm} c ±
c + x 1 = v 2 q 2 x 1 c + , x 3 c + = 1 q 2 v 2 c + x 3 . (23) c_{+}x_{1}=\frac{v^{2}}{q^{2}}x_{1}c_{+},\qquad x_{3}c_{+}=\frac{1}{q^{2}v^{2}}c_{+}x_{3}. \tag*{(23)} c + x 1 = q 2 v 2 x 1 c + , x 3 c + = q 2 v 2 1 c + x 3 . (23) c − x 1 = 1 q 2 v 2 x 1 c − , x 3 c − = v 2 q 2 c − x 3 . (24) c_{-}x_{1}=\frac{1}{q^{2}v^{2}}x_{1}c_{-},\qquad x_{3}c_{-}=\frac{v^{2}}{q^{2}}c_{-}x_{3}. \tag*{(24)} c − x 1 = q 2 v 2 1 x 1 c − , x 3 c − = q 2 v 2 c − x 3 . (24) c + a + = v q a + c + , b + c + = 1 q v c + b + . (25) c_{+}a_{+}=\frac{v}{q}a_{+}c_{+},\qquad b_{+}c_{+}=\frac{1}{qv}c_{+}b_{+}. \tag*{(25)} c + a + = q v a + c + , b + c + = q v 1 c + b + . (25) c − a + = 1 q v 3 a + c − , b + c − = v 3 q c − b + . (26) c_{-}a_{+}=\frac{1}{qv^{3}}a_{+}c_{-},\qquad b_{+}c_{-}=\frac{v^{3}}{q}c_{-}b_{+}. \tag*{(26)} c − a + = q v 3 1 a + c − , b + c − = q v 3 c − b + . (26) c + a − = v 3 q a − c + , b − c + = 1 q v 3 c + b − . (27) c_{+}a_{-}=\frac{v^{3}}{q}a_{-}c_{+},\qquad b_{-}c_{+}=\frac{1}{qv^{3}}c_{+}b_{-}. \tag*{(27)} c + a − = q v 3 a − c + , b − c + = q v 3 1 c + b − . (27) c − a − = 1 q v a − c − , b − c − = v q c − b − . (28) c_{-}a_{-}=\frac{1}{qv}a_{-}c_{-},\qquad b_{-}c_{-}=\frac{v}{q}c_{-}b_{-}. \tag*{(28)} c − a − = q v 1 a − c − , b − c − = q v c − b − . (28) Remaining ordering relations are
x 2 a + = 1 − λ v a + x 2 − λ v x 1 b + , x 2 a − = v ( 1 − λ ) a − x 2 − λ v q x 1 b − . (29) x_{2}a_{+}=\frac{1-\lambda}{v}a_{+}x_{2}-\frac{\lambda}{v}x_{1}b_{+},\qquad x_{2}a_{-}=v(1-\lambda)a_{-}x_{2}-\frac{\lambda v}{q}x_{1}b_{-}. \tag*{(29)} x 2 a + = v 1 − λ a + x 2 − v λ x 1 b + , x 2 a − = v ( 1 − λ ) a − x 2 − q λ v x 1 b − . (29) b + x 2 = v ( 1 − λ ) x 2 b + − λ v q a + x 3 , b − x 2 = 1 − λ v x 2 b − − λ v a − x 3 . (30) b_{+}x_{2}=v(1-\lambda)x_{2}b_{+}-\frac{\lambda v}{q}a_{+}x_{3},\qquad b_{-}x_{2}=\frac{1-\lambda}{v}x_{2}b_{-}-\frac{\lambda}{v}a_{-}x_{3}. \tag*{(30)} b + x 2 = v ( 1 − λ ) x 2 b + − q λ v a + x 3 , b − x 2 = v 1 − λ x 2 b − − v λ a − x 3 . (30) b + x 1 = v q ( 1 + λ ) x 1 b + + λ v q a + x 2 , b − x 1 = 1 + λ q v x 1 b − + λ v a − x 2 . (31) b_{+}x_{1}=\frac{v}{q}(1+\lambda)x_{1}b_{+}+\frac{\lambda v}{q}a_{+}x_{2},\qquad b_{-}x_{1}=\frac{1+\lambda}{qv}x_{1}b_{-}+\frac{\lambda}{v}a_{-}x_{2}. \tag*{(31)} b + x 1 = q v ( 1 + λ ) x 1 b + + q λ v a + x 2 , b − x 1 = q v 1 + λ x 1 b − + v λ a − x 2 . (31) x 3 a + = 1 + λ q v a + x 3 + λ v x 2 b + , x 3 a − = v q ( 1 + λ ) a − x 3 + λ v q x 2 b − . (32) x_{3}a_{+}=\frac{1+\lambda}{qv}a_{+}x_{3}+\frac{\lambda}{v}x_{2}b_{+},\qquad x_{3}a_{-}=\frac{v}{q}(1+\lambda)a_{-}x_{3}+\frac{\lambda v}{q}x_{2}b_{-}. \tag*{(32)} x 3 a + = q v 1 + λ a + x 3 + v λ x 2 b + , x 3 a − = q v ( 1 + λ ) a − x 3 + q λ v x 2 b − . (32) x 2 x 1 = x 1 x 2 − λ v a + a − , x 3 x 2 = x 2 x 3 − λ v b + b − . (33) x_{2}x_{1}=x_{1}x_{2}-\frac{\lambda}{v}a_{+}a_{-},\qquad x_{3}x_{2}=x_{2}x_{3}-\frac{\lambda}{v}b_{+}b_{-}. \tag*{(33)} x 2 x 1 = x 1 x 2 − v λ a + a − , x 3 x 2 = x 2 x 3 − v λ b + b − . (33) x 3 x 1 = x 1 x 3 + λ ( q − 1 ) ( x 2 2 − x 1 x 3 ) − λ ( q − 3 + q − 1 ) a + b − . (34) x_{3}x_{1}=x_{1}x_{3}+\lambda(q-1)(x_{2}^{2}-x_{1}x_{3})-\lambda(q-3+q^{-1})a_{+}b_{-}. \tag*{(34)} x 3 x 1 = x 1 x 3 + λ ( q − 1 ) ( x 2 2 − x 1 x 3 ) − λ ( q − 3 + q − 1 ) a + b − . (34) b + a + = a + b + − λ v c + x 2 , b − a − = a − b − − λ v c − x 2 . (35) b_{+}a_{+}=a_{+}b_{+}-\frac{\lambda}{v}c_{+}x_{2},\qquad b_{-}a_{-}=a_{-}b_{-}-\lambda v c_{-}x_{2}. \tag*{(35)} b + a + = a + b + − v λ c + x 2 , b − a − = a − b − − λ v c − x 2 . (35) b + a − = v 2 ( q + λ ( 1 − q ) ) a − b + + λ v 2 ( x 2 2 − x 1 x 3 ) . (36) b_{+}a_{-}=v^{2}(q+\lambda(1-q))a_{-}b_{+}+\lambda v^{2}(x_{2}^{2}-x_{1}x_{3}). \tag*{(36)} b + a − = v 2 ( q + λ ( 1 − q )) a − b + + λ v 2 ( x 2 2 − x 1 x 3 ) . (36) b − a + = 1 v 2 ( q + λ ( 1 − q ) ) a + b − + λ q v 2 ( x 2 2 − x 1 x 3 ) . (37) b_{-}a_{+}=\frac{1}{v^{2}}(q+\lambda(1-q))a_{+}b_{-}+\frac{\lambda q}{v^{2}}(x_{2}^{2}-x_{1}x_{3}). \tag*{(37)} b − a + = v 2 1 ( q + λ ( 1 − q )) a + b − + v 2 λ q ( x 2 2 − x 1 x 3 ) . (37) In total, we have 36 = 9 ⋅ 8 / 2 = ( 5 + 1 ) ⋅ 3 + 3 ⋅ ( 5 + 1 ) 36 = 9 \cdot8/2 = (5 + 1) \cdot3 + 3 \cdot(5 + 1) 36 = 9 ⋅ 8/2 = ( 5 + 1 ) ⋅ 3 + 3 ⋅ ( 5 + 1 ) (1) constitutes a decomposition of the space V ⊗ V V \otimes V V ⊗ V R R R q − 2 q^{-2} q − 2 − q − 1 -q^{-1} − q − 1 q q q V ⊗ 2 V^{\otimes2} V ⊗ 2 V ⊗ 2 V^{\otimes2} V ⊗ 2
In addition, we have 10 = 5 ⋅ 1 + 1 ⋅ 5 10 = 5 \cdot1 + 1 \cdot5 10 = 5 ⋅ 1 + 1 ⋅ 5 orthogonality relations . These relations are contained within invariant subspaces of End ( V ⊗ V ) \operatorname{End}(V \otimes V) End ( V ⊗ V )
a + 2 = − 1 + q q x 1 c + , a − 2 = − 1 + q q 2 x 1 c − . (38) a_{+}^{2}=-\frac{1+q}{q}x_{1}c_{+}, \qquad a_{-}^{2}=-\frac{1+q}{q^{2}}x_{1}c_{-}. \tag*{(38)} a + 2 = − q 1 + q x 1 c + , a − 2 = − q 2 1 + q x 1 c − . (38) b + 2 = − 1 + q q 2 c + x 3 , b − 2 = − 1 + q q c − x 3 . (39) b_{+}^{2}=-\frac{1+q}{q^{2}}c_{+}x_{3}, \qquad b_{-}^{2}=-\frac{1+q}{q}c_{-}x_{3}. \tag*{(39)} b + 2 = − q 2 1 + q c + x 3 , b − 2 = − q 1 + q c − x 3 . (39) a − c + = − 1 v x 1 b − − 1 v a + x 2 , a + c − = − v x 1 b − − q v a − x 2 . (40) a_{-}c_{+}=-\frac{1}{v}x_{1}b_{-}-\frac{1}{v}a_{+}x_{2}, \qquad a_{+}c_{-}=-v x_{1}b_{-}-qv a_{-}x_{2}. \tag*{(40)} a − c + = − v 1 x 1 b − − v 1 a + x 2 , a + c − = − v x 1 b − − q v a − x 2 . (40) c − b + = − 1 v x 2 b − − 1 v a − x 3 , c + b − = − q v x 2 b + − v a + x 3 . (41) c_{-}b_{+}=-\frac{1}{v}x_{2}b_{-}-\frac{1}{v}a_{-}x_{3}, \qquad c_{+}b_{-}=-qv x_{2}b_{+}-v a_{+}x_{3}. \tag*{(41)} c − b + = − v 1 x 2 b − − v 1 a − x 3 , c + b − = − q v x 2 b + − v a + x 3 . (41) a − b + = 1 q a + b − , c + c − = ( 2 − q ) v 2 a + b − + q v 2 ( x 2 2 − x 1 x 3 ) . (42) a_{-}b_{+}=\frac{1}{q}a_{+}b_{-}, \qquad c_{+}c_{-}=(2-q)v^{2}a_{+}b_{-}+qv^{2}(x_{2}^{2}-x_{1}x_{3}). \tag*{(42)} a − b + = q 1 a + b − , c + c − = ( 2 − q ) v 2 a + b − + q v 2 ( x 2 2 − x 1 x 3 ) . (42) Remark 4.1 Note that for the algebra of coordinate functions on the classical orthogonal group O ( 3 ) O(3) O ( 3 ) T t Ω T = Ω T^{t}\Omega T=\Omega T t Ω T = Ω 3 × 3 3 \times3 3 × 3 T t T^{t} T t T T T Ω \Omega Ω Ω = ( 0 0 1 0 1 0 1 0 0 ) \Omega= \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} Ω = 0 0 1 0 1 0 1 0 0 T t Ω T = Ω T^{t}\Omega T=\Omega T t Ω T = Ω T Ω T t = Ω T\Omega T^{t}=\Omega T Ω T t = Ω T T T Ω \Omega Ω
In our setting, the orthogonality relations in the classical case (i.e., when R R R T t Ω T = g Ω T^{t}\Omega T=g\Omega T t Ω T = g Ω T Ω T t = g Ω T\Omega T^{t}=g\Omega T Ω T t = g Ω g g g (13) . These two matrix relations are independent, provided that g g g T T T g g g
The same phenomenon also for symplectic QM-algebras as well; see [4 ] , the discussion following equation (4.37).
Remark 4.2 It is worth noting a remarkable consistency between the ordering relations and the orthogonality relations, in the following sense. Let us apply the diamond lemma [39 , 40 ] to the ordering relations. Unlike in the G L GL G L R R R 0 = 0 0=0 0 = 0
We notice that the ordering and the orthogonality relations are invariant under an involutive F q , v \mathcal{F}_{q,v} F q , v
a ± ↔ b ∓ , c + ↔ c − , x 1 ↔ x 3 , x 2 ↔ x 2 . (43) a_{\pm} \leftrightarrow b_{\mp}, \qquad c_{+} \leftrightarrow c_{-}, \qquad x_{1} \leftrightarrow x_{3}, \qquad x_{2} \leftrightarrow x_{2}. \tag*{(43)} a ± ↔ b ∓ , c + ↔ c − , x 1 ↔ x 3 , x 2 ↔ x 2 . (43) It corresponds to the reflection of the quantum matrix T T T x 2 x_{2} x 2
Moreover, there exists an involutive homomorphism on the family of algebras F ( R ) q , v ↔ F ( R ) q , v − 1 \mathcal{F}(R)_{q,v} \leftrightarrow\mathcal{F}(R)_{q,v^{-1}} F ( R ) q , v ↔ F ( R ) q , v − 1
a + ↔ a − , b + ↔ q − 1 b − , c + ↔ q − 1 c − , x i ↔ x i , i = 1 , 2 , 3. (44) a_{+} \leftrightarrow a_{-}, \qquad b_{+} \leftrightarrow q^{-1}b_{-}, \qquad c_{+} \leftrightarrow q^{-1}c_{-}, \qquad x_{i} \leftrightarrow x_{i},\ i=1,2,3. \tag*{(44)} a + ↔ a − , b + ↔ q − 1 b − , c + ↔ q − 1 c − , x i ↔ x i , i = 1 , 2 , 3. (44) This homomorphism corresponds to the transposition of the quantum matrix T T T
Finally, there is a two-parameter family of scaling automorphisms of the algebra F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v
a ± ↦ t 1 ± 1 t 2 a ± , b ± ↦ t 1 ± 1 t 2 − 1 b ± , c ± ↦ t 1 ± 2 c ± , x 1 ↦ t 2 2 x 1 , x 2 ↦ x 2 , x 3 ↦ t 2 − 2 x 3 . (45) a_{\pm} \mapsto t_{1}^{\pm1}t_{2}a_{\pm}, \qquad b_{\pm} \mapsto t_{1}^{\pm1}t_{2}^{-1}b_{\pm}, \qquad c_{\pm} \mapsto t_{1}^{\pm2}c_{\pm}, \qquad x_{1} \mapsto t_{2}^{2}x_{1}, \qquad x_{2} \mapsto x_{2}, \qquad x_{3} \mapsto t_{2}^{-2}x_{3}. \tag*{(45)} a ± ↦ t 1 ± 1 t 2 a ± , b ± ↦ t 1 ± 1 t 2 − 1 b ± , c ± ↦ t 1 ± 2 c ± , x 1 ↦ t 2 2 x 1 , x 2 ↦ x 2 , x 3 ↦ t 2 − 2 x 3 . (45) The scaling parameters t 1 t_1 t 1 t 2 t_2 t 2 T T T Z 2 \mathbb{Z}^2 Z 2 F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v
grad a ± = ( ± 1 , 1 ) , grad b ± = ( ± 1 , − 1 ) , grad c ± = ( ± 2 , 0 ) , (46) \operatorname{grad} a_{\pm}=(\pm1,1),\qquad\operatorname{grad} b_{\pm}=(\pm1,-1),\qquad\operatorname{grad} c_{\pm}=(\pm2,0), \tag*{(46)} grad a ± = ( ± 1 , 1 ) , grad b ± = ( ± 1 , − 1 ) , grad c ± = ( ± 2 , 0 ) , (46) grad x 1 = ( 0 , 2 ) , grad x 2 = ( 0 , 0 ) , grad x 3 = ( 0 , − 2 ) . (47) \operatorname{grad} x_1=(0,2),\qquad\operatorname{grad} x_2=(0,0),\qquad\operatorname{grad} x_3=(0,-2). \tag*{(47)} grad x 1 = ( 0 , 2 ) , grad x 2 = ( 0 , 0 ) , grad x 3 = ( 0 , − 2 ) . (47) This bi-grading is also explained by the commutation relations of ρ : = c + c − \rho:=c_+c_- ρ := c + c −
ρ a + = v − 2 q − 2 a + ρ , ρ a − = v 2 q − 2 ρ , ρ x 1 = q − 4 x 1 ρ , ρ c + = v − 4 c + ρ , ρ c − = v 4 c − ρ , ρ x 2 = x 2 ρ , \rho a_+=v^{-2}q^{-2}a_+\rho,\quad\rho a_-=v^2q^{-2}\rho,\quad\rho x_1=q^{-4}x_1\rho,\quad\rho c_+=v^{-4}c_+\rho,\quad\rho c_-=v^4c_-\rho,\quad\rho x_2=x_2\rho, ρ a + = v − 2 q − 2 a + ρ , ρ a − = v 2 q − 2 ρ , ρ x 1 = q − 4 x 1 ρ , ρ c + = v − 4 c + ρ , ρ c − = v 4 c − ρ , ρ x 2 = x 2 ρ , ρ x 3 = q 4 x 3 ρ , ρ b + = v − 2 q 2 b + ρ , ρ b − = v 2 q 2 b − ρ . \rho x_3=q^4x_3\rho,\quad\rho b_+=v^{-2}q^2b_+\rho,\quad\rho b_-=v^2q^2b_-\rho. ρ x 3 = q 4 x 3 ρ , ρ b + = v − 2 q 2 b + ρ , ρ b − = v 2 q 2 b − ρ . Let us consider the set of cubic polynomials in generators of F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v
a − x i b + = 1 q v 2 a + x i b − , i = 1 , 2 , 3 , (48) a_-x_i b_+=\frac{1}{qv^2}a_+x_i b_-,\qquad i=1,2,3, \tag*{(48)} a − x i b + = q v 2 1 a + x i b − , i = 1 , 2 , 3 , (48) a + a − x 2 = ( 1 − λ ) a + x 1 b − + v ( 1 + q ) q x 1 ( x 2 2 − x 1 x 3 ) , (49) a_+a_-x_2=(1-\lambda)a_+x_1b_-+\frac{v(1+q)}{q}x_1(x_2^2-x_1x_3), \tag*{(49)} a + a − x 2 = ( 1 − λ ) a + x 1 b − + q v ( 1 + q ) x 1 ( x 2 2 − x 1 x 3 ) , (49) a + a − b − = v ( 1 + q ) q ( x 1 x 2 b − + q v a − x 1 x 3 ) , (50) a_+a_-b_-=\frac{v(1+q)}{q}\left(x_1x_2b_-+qv\,a_-x_1x_3\right), \tag*{(50)} a + a − b − = q v ( 1 + q ) ( x 1 x 2 b − + q v a − x 1 x 3 ) , (50) a + b + b − = v ( 1 + q ) q ( a + x 2 x 3 + q v x 1 x 3 b + ) , (51) a_+b_+b_-=\frac{v(1+q)}{q}\left(a_+x_2x_3+qv\,x_1x_3b_+\right), \tag*{(51)} a + b + b − = q v ( 1 + q ) ( a + x 2 x 3 + q v x 1 x 3 b + ) , (51) x 2 b + b − = ( 1 − λ ) a + x 3 b − + v ( 1 + q ) q ( x 2 2 − x 1 x 3 ) x 3 , (52) x_2b_+b_-=(1-\lambda)a_+x_3b_-+\frac{v(1+q)}{q}(x_2^2-x_1x_3)x_3, \tag*{(52)} x 2 b + b − = ( 1 − λ ) a + x 3 b − + q v ( 1 + q ) ( x 2 2 − x 1 x 3 ) x 3 , (52) x 1 b + b − = a + a − x 3 . (53) x_1b_+b_-=a_+a_-x_3. \tag*{(53)} x 1 b + b − = a + a − x 3 . (53) We call these cubic relations the diamond relations . Let us comment on their derivation.
Consider three reducible cubic monomials x 1 a − b + x_1a_-b_+ x 1 a − b + a − b + x 3 a_-b_+x_3 a − b + x 3 x 2 a − b + x_2a_-b_+ x 2 a − b + a − b + a_-b_+ a − b +
Relations (4.1.31)–(4.1.35) are obtained in the same way, that is, by comparing results of the two different reductions of the monomials b − x 1 a + b_-x_1a_+ b − x 1 a + a − a − b + a_-a_-b_+ a − a − b + a − b + b + a_-b_+b_+ a − b + b + x 3 b + a − x_3b_+a_- x 3 b + a − x 3 x 2 x 1 x_3x_2x_1 x 3 x 2 x 1
From now on we call irreducible the monomial that can not be transformed by an application of the ordering relations (4.1.5)–(4.1.19), the orthogonality relations (4.1.20)–(4.1.24), the diamond relations (4.1.30)–(4.1.35) and their consequent equalities (4.1.43) even after reshuffling its terms in the triples ( a + , a − , x 1 ) (a_+,a_-,x_1) ( a + , a − , x 1 ) ( c + , c − , x 2 ) (c_+,c_-,x_2) ( c + , c − , x 2 ) ( x 3 , b + , b − ) (x_3,b_+,b_-) ( x 3 , b + , b − )
The complete set of the irreducible monomials in F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v
x 1 a c ± b x 2 c x 3 d x_1^a c_{\pm}^b x_2^c x_3^d x 1 a c ± b x 2 c x 3 d
a ± x 1 a c ± b x 2 c x 3 d a_{\pm}x_1^a c_{\pm}^b x_2^c x_3^d a ± x 1 a c ± b x 2 c x 3 d
x 1 a c ± b x 2 c x 3 d b ± x_1^a c_{\pm}^b x_2^c x_3^d b_{\pm} x 1 a c ± b x 2 c x 3 d b ±
a ± x 1 a c ± b x 2 c x 3 d b ± a_{\pm}x_1^a c_{\pm}^b x_2^c x_3^d b_{\pm} a ± x 1 a c ± b x 2 c x 3 d b ± ∀ a , b , c , d ≥ 0 \forall a,b,c,d\geq0 ∀ a , b , c , d ≥ 0
a + x 1 a x 2 b x 3 c b − ∼ a − x 1 a x 2 b x 3 c b + a_+x_1^a x_2^b x_3^c b_-\sim a_-x_1^a x_2^b x_3^c b_+ a + x 1 a x 2 b x 3 c b − ∼ a − x 1 a x 2 b x 3 c b + ∀ a , b , c ≥ 0 \forall a,b,c\geq0 ∀ a , b , c ≥ 0
6a. a + a − x 1 a x 3 b + 1 ∼ x 1 a + 1 x 3 b b + b − a_+a_-x_1^a x_3^{b+1}\sim x_1^{a+1}x_3^b b_+b_- a + a − x 1 a x 3 b + 1 ∼ x 1 a + 1 x 3 b b + b − ∀ a , b ≥ 0 \forall a,b\geq0 ∀ a , b ≥ 0
6b. a + a − x 1 a , x 3 a b + b − a_+a_-x_1^a,\ x_3^a b_+b_- a + a − x 1 a , x 3 a b + b − ∀ a ≥ 0 \forall a\geq0 ∀ a ≥ 0
Here the symbol ∼ \sim ∼ a + a − a_+a_- a + a − x 1 a x_1^a x 1 a
Lemma 4.3 Let Z ( n ) Z^{(n)} Z ( n ) n n n c ± c_{\pm} c ± x i x_i x i i = 1 , 2 , 3 i = 1,2,3 i = 1 , 2 , 3
a − Z ( n ) b + = 1 q v 2 n a + Z ( n ) b − , a − Z ( n ) a + = 1 v 2 ( n + 1 ) a + Z ( n ) a − . (54) a_{-}Z^{(n)}b_{+} = \frac{1}{qv^{2n}}a_{+}Z^{(n)}b_{-}, \qquad a_{-}Z^{(n)}a_{+} = \frac{1}{v^{2(n+1)}}a_{+}Z^{(n)}a_{-}. \tag*{(54)} a − Z ( n ) b + = q v 2 n 1 a + Z ( n ) b − , a − Z ( n ) a + = v 2 ( n + 1 ) 1 a + Z ( n ) a − . (54) Proof. The proof is by induction on n n n Z ( n ) Z^{(n)} Z ( n ) x 2 x_2 x 2
We note that the set of monomials (4.1.36)–(4.1.42) is ∼ \sim ∼
Theorem 4.4 Using relations (4.1.2)–(4.1.24) and (4.1.30)–(4.1.35) a finite number of times, every monomial in generators a ± a_{\pm} a ± b ± b_{\pm} b ± c ± c_{\pm} c ± x i x_i x i i = 1 , 2 , 3 i = 1,2,3 i = 1 , 2 , 3 F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v
Before we prove the theorem, let’s derive ordering relations between certain combinations of generators of F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v x i x_i x i
Lemma 4.5 Permutation relations for x 1 n x_1^n x 1 n n > 0 n > 0 n > 0
x 2 x 1 n = x 1 n x 2 + 1 − q 2 n v q a + a − x 1 n − 1 , (55) x_2x_1^n = x_1^nx_2 + \frac{1-q^{2n}}{vq}a_{+}a_{-}x_1^{n-1}, \tag*{(55)} x 2 x 1 n = x 1 n x 2 + v q 1 − q 2 n a + a − x 1 n − 1 , (55) b + x 1 n = v n ( q 2 n + q 2 n − 1 − 1 ) q 3 n − 1 x 1 n b + + v ( q 2 n − 1 ) q 2 n a + x 1 n − 1 x 2 , (56) b_{+}x_1^n = \frac{v^n(q^{2n}+q^{2n-1}-1)}{q^{3n-1}}x_1^nb_{+} + \frac{v(q^{2n}-1)}{q^{2n}}a_{+}x_1^{n-1}x_2, \tag*{(56)} b + x 1 n = q 3 n − 1 v n ( q 2 n + q 2 n − 1 − 1 ) x 1 n b + + q 2 n v ( q 2 n − 1 ) a + x 1 n − 1 x 2 , (56) x 3 x 1 n = x 1 n x 3 + ( 1 − q 2 n ) ( 1 − q 2 n − 1 ) q 4 n − 3 x 1 n − 1 ( x 2 2 − x 1 x 3 ) + ( 1 − q 2 n ) [ ( 2 − q ) ( 1 − q 2 n − 1 ) − q 2 n − 2 ] v n − 1 q 3 n − 1 a + x 1 n − 1 b − , (57) \begin{aligned}
x_3x_1^n ={}& x_1^nx_3 + \frac{(1-q^{2n})(1-q^{2n-1})}{q^{4n-3}}x_1^{n-1}(x_2^2-x_1x_3) \\
&+ \frac{(1-q^{2n})\left[(2-q)(1-q^{2n-1})-q^{2n-2}\right]}{v^{n-1}q^{3n-1}}a_{+}x_1^{n-1}b_{-}, \tag*{(57)}
\end{aligned} x 3 x 1 n = x 1 n x 3 + q 4 n − 3 ( 1 − q 2 n ) ( 1 − q 2 n − 1 ) x 1 n − 1 ( x 2 2 − x 1 x 3 ) + v n − 1 q 3 n − 1 ( 1 − q 2 n ) [ ( 2 − q ) ( 1 − q 2 n − 1 ) − q 2 n − 2 ] a + x 1 n − 1 b − , (57) ( x 2 2 − x 1 x 3 ) x 1 n = 1 + q ( 1 + q ) ( 2 − q ) ( 1 − q 2 n ) q 4 n x 1 n ( x 2 2 − x 1 x 3 ) + [ 1 + q ( 1 + q ) ( 2 − q ) ] ( 2 − q ) ( 1 − q 2 n ) v n q 3 n + 1 a + x 1 n b − . (58) \begin{aligned}
(x_2^2-x_1x_3)x_1^n ={}& \frac{1+q(1+q)(2-q)(1-q^{2n})}{q^{4n}}x_1^n(x_2^2-x_1x_3) \\
&+ \frac{[1+q(1+q)(2-q)](2-q)(1-q^{2n})}{v^nq^{3n+1}}a_{+}x_1^nb_{-}. \tag*{(58)}
\end{aligned} ( x 2 2 − x 1 x 3 ) x 1 n = q 4 n 1 + q ( 1 + q ) ( 2 − q ) ( 1 − q 2 n ) x 1 n ( x 2 2 − x 1 x 3 ) + v n q 3 n + 1 [ 1 + q ( 1 + q ) ( 2 − q )] ( 2 − q ) ( 1 − q 2 n ) a + x 1 n b − . (58) Proof. Relations (4.1.44)–(4.1.47) are proved by induction on n n n
The proof of relation (4.1.44) is a direct check, the induction base is eq. (4.1.15).
Let us rewrite relation (4.1.45) in a concise form
b + x 1 n = v n X n x 1 n b + + v Y n a + x 1 n − 1 x 2 , (59) b_{+}x_1^n = v^nX_nx_1^nb_{+} + vY_na_{+}x_1^{n-1}x_2, \tag*{(59)} b + x 1 n = v n X n x 1 n b + + v Y n a + x 1 n − 1 x 2 , (59) where X n X_n X n Y n Y_n Y n q ± 1 q^{\pm1} q ± 1 X 0 = 1 X_0=1 X 0 = 1 Y 0 = 0 Y_0=0 Y 0 = 0 X 1 = 1 + λ q X_1=\frac{1+\lambda}{q} X 1 = q 1 + λ Y 1 = λ q Y_1=\frac{\lambda}{q} Y 1 = q λ
The induction step n → n + 1 n \to n+1 n → n + 1 b + x 1 n + 1 = ( b + x 1 n ) x 1 b_{+}x_1^{n+1}=(b_{+}x_1^n)x_1 b + x 1 n + 1 = ( b + x 1 n ) x 1 X n X_n X n Y n Y_n Y n
X n + 1 = q − 1 ( 1 + λ ) X n − λ q − n − 1 ( 1 + q ) Y n , Y n + 1 = λ q n − 1 X n + ( q − 2 − λ ) Y n . \begin{aligned}
X_{n+1} &= q^{-1}(1+\lambda)X_n-\lambda q^{-n-1}(1+q)Y_n,\\
Y_{n+1} &= \lambda q^{n-1}X_n+(q^{-2}-\lambda)Y_n.
\end{aligned} X n + 1 Y n + 1 = q − 1 ( 1 + λ ) X n − λ q − n − 1 ( 1 + q ) Y n , = λ q n − 1 X n + ( q − 2 − λ ) Y n . It is now easy to verify that the coefficients in (4.1.45) satisfy these relations.
Now, let us rewrite relations (4.1.46), (4.1.47) in a concise form
x 3 x 1 n = x 1 n x 3 + A n x 1 n − 1 ( x 2 2 − x 1 x 3 ) + v 1 − n B n a + x 1 n − 1 b − , (60) x_3x_1^n = x_1^nx_3+A_nx_1^{n-1}(x_2^2-x_1x_3)+v^{1-n}B_na_{+}x_1^{n-1}b_{-}, \tag*{(60)} x 3 x 1 n = x 1 n x 3 + A n x 1 n − 1 ( x 2 2 − x 1 x 3 ) + v 1 − n B n a + x 1 n − 1 b − , (60) ( x 2 2 − x 1 x 3 ) x 1 n = C n x 1 n ( x 2 2 − x 1 x 3 ) + v − n D n a + x 1 n b − . (61) (x_2^2-x_1x_3)x_1^n = C_nx_1^n(x_2^2-x_1x_3)+v^{-n}D_na_{+}x_1^nb_{-}. \tag*{(61)} ( x 2 2 − x 1 x 3 ) x 1 n = C n x 1 n ( x 2 2 − x 1 x 3 ) + v − n D n a + x 1 n b − . (61) where A n A_n A n B n B_n B n C n C_n C n D n D_n D n q ± 1 q^{\pm1} q ± 1 A 0 = B 0 = D 0 = 0 A_0=B_0=D_0=0 A 0 = B 0 = D 0 = 0 C 0 = 1 C_0=1 C 0 = 1
A 1 = λ ( q − 1 ) , B 1 = − λ ( q − 3 + q − 1 ) . A_1=\lambda(q-1),\qquad B_1=-\lambda(q-3+q^{-1}). A 1 = λ ( q − 1 ) , B 1 = − λ ( q − 3 + q − 1 ) . Calculation of the coefficients C 1 C_1 C 1 D 1 D_1 D 1 ( x 2 2 − x 1 x 3 ) x 1 (x_2^2-x_1x_3)x_1 ( x 2 2 − x 1 x 3 ) x 1
C 1 = q − 4 [ 1 + q ( 1 + q ) ( 2 − q ) ( 1 − q 2 ) ] , D 1 = q − 4 [ 1 + q ( 1 + q ) ( 2 − q ) ] ( 2 − q ) ( 1 − q 2 ) . C_1=q^{-4}[1+q(1+q)(2-q)(1-q^2)],\qquad D_1=q^{-4}[1+q(1+q)(2-q)](2-q)(1-q^2). C 1 = q − 4 [ 1 + q ( 1 + q ) ( 2 − q ) ( 1 − q 2 )] , D 1 = q − 4 [ 1 + q ( 1 + q ) ( 2 − q )] ( 2 − q ) ( 1 − q 2 ) . which agrees with the general formula (4.1.47). Thus, we have verified the induction base for proving the relations (4.1.46), (4.1.47)`.
In the proof of (4.1.46) the induction step n → n + 1 n\to n+1 n → n + 1 x 3 x 1 n + 1 = ( x 3 x 1 n ) x 1 x_3x_1^{n+1}=(x_3x_1^n)x_1 x 3 x 1 n + 1 = ( x 3 x 1 n ) x 1 for $n=1$ (this relation was verified above) and {xref}(4.1.13), and then using relations {xref}(4.1.2) several times and {xref}(4.1.31)` we obtain the following recurrence relations for the coefficients A n A_n A n B n B_n B n
A n + 1 = A 1 + C 1 A n + λ q − n ( 1 + q ) B n , B n + 1 = q n B 1 + q n − 1 D 1 A n + ( q + λ q − λ 2 ) B n . \begin{aligned}
A_{n+1}&=A_1+C_1A_n+\lambda q^{-n}(1+q)B_n,\\
B_{n+1}&=q^nB_1+q^{n-1}D_1A_n+\left(q+\frac{\lambda}{q}-\lambda^2\right)B_n.
\end{aligned} A n + 1 B n + 1 = A 1 + C 1 A n + λ q − n ( 1 + q ) B n , = q n B 1 + q n − 1 D 1 A n + ( q + q λ − λ 2 ) B n . Coefficients in formula (4.1.46) verify these relations.
The induction step n → n + 1 n\to n+1 n → n + 1 ( ( x 1 2 − x 1 x 3 ) x 1 n ) x 1 ((x_1^2-x_1x_3)x_1^n)x_1 (( x 1 2 − x 1 x 3 ) x 1 n ) x 1 C n C_n C n D n D_n D n
C n + 1 = C 1 C n + λ q − n − 1 ( 1 + q ) D n , D n + 1 = q n D 1 C n + ( q + λ q − λ 2 ) D n . \begin{aligned}
C_{n+1}&=C_1C_n+\lambda q^{-n-1}(1+q)D_n,\\
D_{n+1}&=q^nD_1C_n+\left(q+\frac{\lambda}{q}-\lambda^2\right)D_n.
\end{aligned} C n + 1 D n + 1 = C 1 C n + λ q − n − 1 ( 1 + q ) D n , = q n D 1 C n + ( q + q λ − λ 2 ) D n . Coefficients in formula (4.1.47) verify these relations.
Applying (anti)homomorphic maps (4.1.25), (4.1.26) to relations {xref}(4.1.44)–(4.1.47)`, one can obtain permutation relations of x 1 n x_1^n x 1 n b − b_- b − a ± a_\pm a ± x 3 n x_3^n x 3 n x 1 , 2 x_{1,2} x 1 , 2 a ± a_\pm a ± b ± b_\pm b ±
Corollary 4.6. Some of the permutation relations for x 3 n x_3^n x 3 n n > 0 n>0 n > 0
x 3 n x 2 = x 2 x 3 n + 1 − q 2 n v q 2 n − 1 b + b − x 3 n − 1 . (62) x_3^n x_2=x_2x_3^n+\frac{1-q^{2n}}{v q^{2n-1}}b_+b_-x_3^{n-1}. \tag*{(62)} x 3 n x 2 = x 2 x 3 n + v q 2 n − 1 1 − q 2 n b + b − x 3 n − 1 . (62) x 3 n a + = q 2 n + q 2 n − 1 − 1 v n q 3 n − 1 a + x 3 n + q 2 n − 1 v q 2 n − 1 x 2 x 3 n − 1 b + . (63) x_3^n a_+=\frac{q^{2n}+q^{2n-1}-1}{v^nq^{3n-1}}a_+x_3^n+\frac{q^{2n}-1}{v q^{2n-1}}x_2x_3^{n-1}b_+. \tag*{(63)} x 3 n a + = v n q 3 n − 1 q 2 n + q 2 n − 1 − 1 a + x 3 n + v q 2 n − 1 q 2 n − 1 x 2 x 3 n − 1 b + . (63) The permutation relations of the generators with x 2 n x_2^n x 2 n v v v λ = q − q − 1 \lambda=q-q^{-1} λ = q − q − 1 q ± 1 q^{\pm1} q ± 1 v ± 1 v^{\pm1} v ± 1
Lemma 4.7. Permutation relations for x 2 n x_2^n x 2 n n > 1 n>1 n > 1
b + x 2 n = v n ∑ i ≥ 0 λ i Q n , i x 1 i x 2 n − 2 i x 3 i b + + v ∑ i ≥ 0 λ i + 1 T n , i a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 , (64) b_+x_2^n=v^n\sum_{i\geq0}\lambda^iQ_{n,i}x_1^ix_2^{n-2i}x_3^ib_++v\sum_{i\geq0}\lambda^{i+1}T_{n,i}a_+x_1^ix_2^{n-2i-1}x_3^{i+1}, \tag*{(64)} b + x 2 n = v n i ≥ 0 ∑ λ i Q n , i x 1 i x 2 n − 2 i x 3 i b + + v i ≥ 0 ∑ λ i + 1 T n , i a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 , (64) b − x 2 n = v − n ∑ i ≥ 0 λ i Q n , i x 1 i x 2 n − 2 i x 3 i b − + v − 1 ∑ i ≥ 0 q λ i + 1 T n , i a − x 1 i x 2 n − 2 i − 1 x 3 i + 1 , (65) b_-x_2^n=v^{-n}\sum_{i\geq0}\lambda^iQ_{n,i}x_1^ix_2^{n-2i}x_3^ib_-+v^{-1}\sum_{i\geq0}q\lambda^{i+1}T_{n,i}a_-x_1^ix_2^{n-2i-1}x_3^{i+1}, \tag*{(65)} b − x 2 n = v − n i ≥ 0 ∑ λ i Q n , i x 1 i x 2 n − 2 i x 3 i b − + v − 1 i ≥ 0 ∑ q λ i + 1 T n , i a − x 1 i x 2 n − 2 i − 1 x 3 i + 1 , (65) x 2 n a + = v − n ∑ i ≥ 0 λ i Q n , i a + x 1 i x 2 n − 2 i x 3 i + v − 1 ∑ i ≥ 0 q λ i + 1 T n , i x 1 i + 1 x 2 n − 2 i − 1 x 3 i b + , (66) x_2^na_+=v^{-n}\sum_{i\geq0}\lambda^iQ_{n,i}a_+x_1^ix_2^{n-2i}x_3^i+v^{-1}\sum_{i\geq0}q\lambda^{i+1}T_{n,i}x_1^{i+1}x_2^{n-2i-1}x_3^ib_+, \tag*{(66)} x 2 n a + = v − n i ≥ 0 ∑ λ i Q n , i a + x 1 i x 2 n − 2 i x 3 i + v − 1 i ≥ 0 ∑ q λ i + 1 T n , i x 1 i + 1 x 2 n − 2 i − 1 x 3 i b + , (66) x 2 n a − = v n ∑ i ≥ 0 λ i Q n , i a − x 1 i x 2 n − 2 i x 3 i + v ∑ i ≥ 0 λ i + 1 T n , i x 1 i + 1 x 2 n − 2 i − 1 x 3 i b − (67) x_2^n a_- = v^n \sum_{i\geq0} \lambda^i Q_{n,i} a_- x_1^i x_2^{n-2i} x_3^i + v \sum_{i\geq0} \lambda^{i+1} T_{n,i} x_1^{i+1} x_2^{n-2i-1} x_3^i b_- \tag*{(67)} x 2 n a − = v n i ≥ 0 ∑ λ i Q n , i a − x 1 i x 2 n − 2 i x 3 i + v i ≥ 0 ∑ λ i + 1 T n , i x 1 i + 1 x 2 n − 2 i − 1 x 3 i b − (67) x 3 x 2 n = ∑ i ≥ 0 λ i S n , i x 1 i x 2 n − 2 i x 3 i + 1 + v 1 − n ∑ i ≥ 0 λ i + 1 U n , i a + x 1 i x 2 n − 2 i − 2 x 3 i + 1 b − + ϵ n v − 1 λ n + 1 2 W n x 1 n − 3 2 a + a − x 3 n + 1 2 (68) x_3 x_2^n = \sum_{i\geq0} \lambda^i S_{n,i} x_1^i x_2^{n-2i} x_3^{i+1} + v^{1-n} \sum_{i\geq0} \lambda^{i+1} U_{n,i} a_+ x_1^i x_2^{n-2i-2} x_3^{i+1} b_- + \epsilon_n v^{-1} \lambda^{\frac{n+1}{2}} W_n x_1^{\frac{n-3}{2}} a_+ a_- x_3^{\frac{n+1}{2}} \tag*{(68)} x 3 x 2 n = i ≥ 0 ∑ λ i S n , i x 1 i x 2 n − 2 i x 3 i + 1 + v 1 − n i ≥ 0 ∑ λ i + 1 U n , i a + x 1 i x 2 n − 2 i − 2 x 3 i + 1 b − + ϵ n v − 1 λ 2 n + 1 W n x 1 2 n − 3 a + a − x 3 2 n + 1 (68) x 2 n x 1 = ∑ i ≥ 0 λ i S n , i x 1 i + 1 x 2 n − 2 i x 3 i + v 1 − n ∑ i ≥ 0 λ i + 1 U n , i x 1 i + 1 x 2 n − 2 i − 2 x 3 i b − + ϵ n v − 1 λ n + 1 2 W n x 1 n − 1 2 a + a − x 3 n − 1 2 (69) x_2^n x_1 = \sum_{i\geq0} \lambda^i S_{n,i} x_1^{i+1} x_2^{n-2i} x_3^i + v^{1-n} \sum_{i\geq0} \lambda^{i+1} U_{n,i} x_1^{i+1} x_2^{n-2i-2} x_3^i b_- + \epsilon_n v^{-1} \lambda^{\frac{n+1}{2}} W_n x_1^{\frac{n-1}{2}} a_+ a_- x_3^{\frac{n-1}{2}} \tag*{(69)} x 2 n x 1 = i ≥ 0 ∑ λ i S n , i x 1 i + 1 x 2 n − 2 i x 3 i + v 1 − n i ≥ 0 ∑ λ i + 1 U n , i x 1 i + 1 x 2 n − 2 i − 2 x 3 i b − + ϵ n v − 1 λ 2 n + 1 W n x 1 2 n − 1 a + a − x 3 2 n − 1 (69) Here Q n , i Q_{n,i} Q n , i T n , i T_{n,i} T n , i S n , i S_{n,i} S n , i U n , i U_{n,i} U n , i W n W_n W n q ± 1 q^{\pm1} q ± 1 ϵ n \epsilon_n ϵ n n n n λ \lambda λ
Q n , 0 = ( 1 − λ ) n , S n , 0 = ( q − 2 − λ ) n ∀ n > 1. (70) Q_{n,0} = (1-\lambda)^n, \qquad S_{n,0} = (q^{-2}-\lambda)^n \ \forall n>1. \tag*{(70)} Q n , 0 = ( 1 − λ ) n , S n , 0 = ( q − 2 − λ ) n ∀ n > 1. (70) Remark 4.8 The calculation of the numerical coefficients in formulas (4.1.53)–(4.1.57) for n ≤ 6 n\leq6 n ≤ 6 Q n , i Q_{n,i} Q n , i T n , i T_{n,i} T n , i λ i \lambda^i λ i S n , i S_{n,i} S n , i ( 1 + q ) (1+q) ( 1 + q )
Before proceeding to the proof of the lemma 4.7 we will deduce its corollary.
Corollary 4.9 Assuming validity of the relations (4.1.55) for x 2 m a + x_2^m a_+ x 2 m a + x 2 m x 1 x_2^m x_1 x 2 m x 1 m = 1 , … , n − 1 m=1,\ldots,n-1 m = 1 , … , n − 1 x 2 n b + b − x_2^n b_+b_- x 2 n b + b − n > 0 n>0 n > 0
x 2 n b + b − = v q − 1 ( 1 + q ) x 2 n + 1 x 3 + v ∑ i ≥ 0 λ i M n , i x 1 i + 1 x 2 n − 2 i − 1 x 3 i + 2 x_2^n b_+b_- = vq^{-1}(1+q)x_2^{n+1}x_3 + v\sum_{i\geq0}\lambda^i M_{n,i}x_1^{i+1}x_2^{n-2i-1}x_3^{i+2} x 2 n b + b − = v q − 1 ( 1 + q ) x 2 n + 1 x 3 + v i ≥ 0 ∑ λ i M n , i x 1 i + 1 x 2 n − 2 i − 1 x 3 i + 2 + v 1 − n ∑ i ≥ 0 λ i J n , i a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 b − + ϵ n − 1 λ n 2 L n x 1 n 2 a + a − x 3 n 2 − 1 . (71) {}+v^{1-n}\sum_{i\geq0}\lambda^i J_{n,i}a_+x_1^i x_2^{n-2i-1}x_3^{i+1}b_- + \epsilon_{n-1}\lambda^{\frac n2}L_nx_1^{\frac n2}a_+a_-x_3^{\frac n2-1}. \tag*{(71)} + v 1 − n i ≥ 0 ∑ λ i J n , i a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 b − + ϵ n − 1 λ 2 n L n x 1 2 n a + a − x 3 2 n − 1 . (71) Here M n , i M_{n,i} M n , i J n , i J_{n,i} J n , i L n L_n L n q ± 1 q^{\pm1} q ± 1 Q m , i Q_{m,i} Q m , i T m , i T_{m,i} T m , i S m , i S_{m,i} S m , i U m , i U_{m,i} U m , i W m W_m W m m = 1 , 2 , … , n − 1 m=1,2,\ldots,n-1 m = 1 , 2 , … , n − 1 J n , 0 J_{n,0} J n , 0
J n , 0 = ( 1 − λ ) Q n − 1 , 0 = ( 1 − λ ) n . (72) J_{n,0}=(1-\lambda)Q_{n-1,0}=(1-\lambda)^n. \tag*{(72)} J n , 0 = ( 1 − λ ) Q n − 1 , 0 = ( 1 − λ ) n . (72) Proof. Let us calculate x 2 n b + b − x_2^n b_+b_- x 2 n b + b −
x 2 n b + b − = x 2 n − 1 ( x 2 b + b − ) = v 1 + q q x 2 n + 1 x 3 + ( 1 − λ ) ( x 2 n − 1 a + ) x 3 b − − v 1 + q q ( x 2 n − 1 x 1 ) x 3 2 x_2^n b_+b_- = x_2^{n-1}(x_2b_+b_-) = v\frac{1+q}{q}x_2^{n+1}x_3 +(1-\lambda)(x_2^{n-1}a_+)x_3b_- - v\frac{1+q}{q}(x_2^{n-1}x_1)x_3^2 x 2 n b + b − = x 2 n − 1 ( x 2 b + b − ) = v q 1 + q x 2 n + 1 x 3 + ( 1 − λ ) ( x 2 n − 1 a + ) x 3 b − − v q 1 + q ( x 2 n − 1 x 1 ) x 3 2 Here we used the diamond relation (4.1.34) at the beginning of the transformation. The last two terms in the right hand side of this calculation can be transformed further with the use of relations (4.1.55) and (4.1.58). Using eqs. (4.1.2), (4.1.3), (4.1.35) and collecting similar terms we finally obtain
x 2 n b + b − = v 1 + q q x 2 n + 1 x 3 + ( 1 − λ ) ∑ i ≥ 0 λ i + 1 q − 2 i T n − 1 , i x 1 i + 1 x 2 n − 2 ( i + 1 ) b + b − x 3 i + 1 − v 1 + q q ∑ i ≥ 0 λ i S n − 1 , i x 1 i + 1 x 2 n − 2 i − 1 x 3 i + 2 + v 1 − n ∑ i ≥ 0 λ i ( ( 1 − λ ) Q n − 1 , i − q ( 1 + q ) U n − 1 , i − 1 ) a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 b − − ϵ n − 1 1 + q q λ n 2 W n − 1 x 1 n 2 b + b − x 3 n 2 . \begin{aligned}
x_2^n b_+b_- ={}& v\frac{1+q}{q}x_2^{n+1}x_3 +(1-\lambda)\sum_{i\geq0}\lambda^{i+1}q^{-2i}T_{n-1,i}x_1^{i+1}x_2^{n-2(i+1)}b_+b_-x_3^{i+1}\\
&-v\frac{1+q}{q}\sum_{i\geq0}\lambda^iS_{n-1,i}x_1^{i+1}x_2^{n-2i-1}x_3^{i+2}\\
&+v^{1-n}\sum_{i\geq0}\lambda^i\left((1-\lambda)Q_{n-1,i}-q(1+q)U_{n-1,i-1}\right)a_+x_1^i x_2^{n-2i-1}x_3^{i+1}b_-\\
&-\epsilon_{n-1}\frac{1+q}{q}\lambda^{\frac n2}W_{n-1}x_1^{\frac n2}b_+b_-x_3^{\frac n2}.
\end{aligned} x 2 n b + b − = v q 1 + q x 2 n + 1 x 3 + ( 1 − λ ) i ≥ 0 ∑ λ i + 1 q − 2 i T n − 1 , i x 1 i + 1 x 2 n − 2 ( i + 1 ) b + b − x 3 i + 1 − v q 1 + q i ≥ 0 ∑ λ i S n − 1 , i x 1 i + 1 x 2 n − 2 i − 1 x 3 i + 2 + v 1 − n i ≥ 0 ∑ λ i ( ( 1 − λ ) Q n − 1 , i − q ( 1 + q ) U n − 1 , i − 1 ) a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 b − − ϵ n − 1 q 1 + q λ 2 n W n − 1 x 1 2 n b + b − x 3 2 n . In this expression the only reducible term is the rightmost one in the first line. This term contains the underlined combination x 2 n − 2 ( i + 1 ) b + b − x_2^{n-2(i+1)}b_+b_- x 2 n − 2 ( i + 1 ) b + b − x 2 n b + b − x_2^n b_+b_- x 2 n b + b − x 2 x_2 x 2 n − 2 ( i + 1 ) < n n-2(i+1)<n n − 2 ( i + 1 ) < n n n n
Proof of Lemma 4.7. Among six relations (4.1.53)–(4.1.58), only two, say (4.1.53) and (4.1.57), are independent. The other four follow from this pair by applying (anti)homomorphisms (4.1.25) and (4.1.26).
We will start by considering the relation (4.1.53). The proof goes by induction on n n n n → n + 1 n \to n+1 n → n + 1
b + x 2 n + 1 = ( b + x 2 n ) x 2 = l.c. { v n λ i x 1 i x 2 n − 2 i x 3 i b + x 2 , v λ i + 1 a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 x 2 } i ≥ 0 . (73) b_{+}x_{2}^{n+1}=(b_{+}x_{2}^{n})x_{2}=\operatorname{l.c.}\{v^{n}\lambda^{i}x_{1}^{i}x_{2}^{n-2i}x_{3}^{i}b_{+}x_{2},\ v\lambda^{i+1}a_{+}x_{1}^{i}x_{2}^{n-2i-1}x_{3}^{i+1}x_{2}\}_{i\geq0}. \tag*{(73)} b + x 2 n + 1 = ( b + x 2 n ) x 2 = l.c. { v n λ i x 1 i x 2 n − 2 i x 3 i b + x 2 , v λ i + 1 a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 x 2 } i ≥ 0 . (73) Here we apply the induction assumption and use an abbreviated notation: l.c. { … } \operatorname{l.c.}\{\ldots\} l.c. { … } q ± 1 q^{\pm1} q ± 1
Consider transformation of the typical monomial from the first subset in the braces in (4.1.62). Applying relations (4.1.12) and then, (4.1.51) and (4.1.52), we obtain
v n λ i x 1 i x 2 n − 2 i x 3 i b + x 2 = l.c. { v n + 1 λ i x 1 i x 2 n + 1 − 2 i x 3 i b + , v n λ i + 1 x 1 i x 2 n − 2 i x 3 i − 1 b + b − b + , v^{n}\lambda^{i}x_{1}^{i}x_{2}^{n-2i}x_{3}^{i}b_{+}x_{2}=\operatorname{l.c.}\{v^{n+1}\lambda^{i}x_{1}^{i}x_{2}^{n+1-2i}x_{3}^{i}b_{+},\ v^{n}\lambda^{i+1}x_{1}^{i}x_{2}^{n-2i}x_{3}^{i-1}b_{+}b_{-}b_{+}, v n λ i x 1 i x 2 n − 2 i x 3 i b + x 2 = l.c. { v n + 1 λ i x 1 i x 2 n + 1 − 2 i x 3 i b + , v n λ i + 1 x 1 i x 2 n − 2 i x 3 i − 1 b + b − b + , v n + 1 − i λ i + 1 x 1 i x 2 n − 2 i a + x 3 i + 1 , v n λ i + 2 x 1 i x 2 n − 2 i + 1 x 3 i − 1 b + x 3 } . (74) v^{n+1-i}\lambda^{i+1}x_{1}^{i}x_{2}^{n-2i}a_{+}x_{3}^{i+1},\ v^{n}\lambda^{i+2}x_{1}^{i}x_{2}^{n-2i+1}x_{3}^{i-1}b_{+}x_{3}\}. \tag*{(74)} v n + 1 − i λ i + 1 x 1 i x 2 n − 2 i a + x 3 i + 1 , v n λ i + 2 x 1 i x 2 n − 2 i + 1 x 3 i − 1 b + x 3 } . (74) Here, the first term in braces is irreducible. It corresponds to the left term in the right hand side of the formula (4.1.53) we are testing. Upon permuting b + b_{+} b + x 3 x_{3} x 3
Next, we transform the third term in the braces in (4.1.63).
v n + 1 − i λ i + 1 x 1 i ( x 2 n − 2 i a + ) x 3 i + 1 = l.c. { v i + 1 λ i + j + 1 x 1 i ( a + x 1 j x 2 n − 2 ( i + j ) x 3 j ) x 3 i + 1 , v^{n+1-i}\lambda^{i+1}x_{1}^{i}(x_{2}^{n-2i}a_{+})x_{3}^{i+1}=\operatorname{l.c.}\{v^{i+1}\lambda^{i+j+1}x_{1}^{i}(a_{+}x_{1}^{j}x_{2}^{n-2(i+j)}x_{3}^{j})x_{3}^{i+1}, v n + 1 − i λ i + 1 x 1 i ( x 2 n − 2 i a + ) x 3 i + 1 = l.c. { v i + 1 λ i + j + 1 x 1 i ( a + x 1 j x 2 n − 2 ( i + j ) x 3 j ) x 3 i + 1 , v n − i λ i + j + 2 x 1 i ( x 3 j + 1 x 2 n − 2 ( i + j ) − 1 x 3 j b + ) x 3 i + 1 } j ≥ 0 = l.c. { v λ i + j + 1 a + x 1 i + j x 2 n − 2 ( i + j ) x 3 i + j + 1 , v n + 1 λ i + j + 2 x 1 i + j + 1 x 2 n + 1 − 2 ( i + j + 1 ) x 3 i + j + 1 b + } j ≥ 0 . (75) \begin{aligned}
v^{n-i}\lambda^{i+j+2}x_{1}^{i}(x_{3}^{j+1}x_{2}^{n-2(i+j)-1}x_{3}^{j}b_{+})x_{3}^{i+1}\}_{j\geq0} \\
&=\operatorname{l.c.}\{v\lambda^{i+j+1}a_{+}x_{1}^{i+j}x_{2}^{n-2(i+j)}x_{3}^{i+j+1},\ v^{n+1}\lambda^{i+j+2}x_{1}^{i+j+1}x_{2}^{n+1-2(i+j+1)}x_{3}^{i+j+1}b_{+}\}_{j\geq0}. \tag*{(75)}
\end{aligned} v n − i λ i + j + 2 x 1 i ( x 3 j + 1 x 2 n − 2 ( i + j ) − 1 x 3 j b + ) x 3 i + 1 } j ≥ 0 = l.c. { v λ i + j + 1 a + x 1 i + j x 2 n − 2 ( i + j ) x 3 i + j + 1 , v n + 1 λ i + j + 2 x 1 i + j + 1 x 2 n + 1 − 2 ( i + j + 1 ) x 3 i + j + 1 b + } j ≥ 0 . (75) Here, we substituted formula (4.1.55) for x 2 n − 2 i a + x_{2}^{n-2i}a_{+} x 2 n − 2 i a + a + , x 1 a_{+},x_{1} a + , x 1 b + , x 3 b_{+},x_{3} b + , x 3
Finally, we shall bring the second term in the braces in (4.1.63) to the irreducible form. To this end we transform its last underlined in (4.1.63) part:
x 3 i − 1 b + b − b + ∼ v − i − 1 b + 2 b − x 3 i − 1 ∼ v − i c + b − x 3 i = l.c. { v x 2 x 3 i b + , v 1 − i a + x 3 i + 1 } . (76) x_{3}^{i-1}b_{+}b_{-}b_{+}\sim v^{-i-1}b_{+}^{2}b_{-}x_{3}^{i-1}\sim v^{-i}c_{+}b_{-}x_{3}^{i}=\operatorname{l.c.}\{v x_{2}x_{3}^{i}b_{+},\ v^{1-i}a_{+}x_{3}^{i+1}\}. \tag*{(76)} x 3 i − 1 b + b − b + ∼ v − i − 1 b + 2 b − x 3 i − 1 ∼ v − i c + b − x 3 i = l.c. { v x 2 x 3 i b + , v 1 − i a + x 3 i + 1 } . (76) Here symbol “∼ \sim ∼ q ± 1 q^{\pm1} q ± 1 x 3 , b ± x_{3}, b_{\pm} x 3 , b ± x 3 x_{3} x 3 b + b_{+} b +
Thus, we confirmed that the first subset in braces in (4.1.62) reduces to a linear combination of the irreducible terms which are present in the formula (4.1.53).
It remains to transform monomials from the second subset in the braces in (4.1.62). We apply relation (4.1.51)
v λ i + 1 a + x 1 i x 2 n − 2 i − 1 ( x 3 i + 1 x 2 ) = l.c. { v λ i + 1 a + x 1 i x 2 n − 2 i x 3 i + 1 , λ i + 2 a + x 1 i x 2 n − 2 i − 1 b + b − x 3 i } . (77) v\lambda^{i+1}a_{+}x_{1}^{i}x_{2}^{n-2i-1}(x_{3}^{i+1}x_{2})=\operatorname{l.c.}\{v\lambda^{i+1}a_{+}x_{1}^{i}x_{2}^{n-2i}x_{3}^{i+1},\ \lambda^{i+2}a_{+}x_{1}^{i}x_{2}^{n-2i-1}b_{+}b_{-}x_{3}^{i}\}. \tag*{(77)} v λ i + 1 a + x 1 i x 2 n − 2 i − 1 ( x 3 i + 1 x 2 ) = l.c. { v λ i + 1 a + x 1 i x 2 n − 2 i x 3 i + 1 , λ i + 2 a + x 1 i x 2 n − 2 i − 1 b + b − x 3 i } . (77) The first term in the r.h.s. of (4.1.66) is irreducible. It has a form of the second term in (4.1.53). The second term in the r.h.s. of (4.1.66) is reducible. Applying relations (4.1.60) we continue its transformation
λ i + 2 a + x 1 i ( x 2 n − 2 i − 1 b + b − ) x 3 i = l.c. { v λ i + 2 a + x 1 i x 2 n − 2 i x 3 i + 1 , v λ i + j + 2 a + x 1 i + j + 1 x 2 n − 2 ( i + j + 1 ) x 3 i + j + 2 , \lambda^{i+2}a_{+}x_{1}^{i}(x_{2}^{n-2i-1}b_{+}b_{-})x_{3}^{i}=\operatorname{l.c.}\{v\lambda^{i+2}a_{+}x_{1}^{i}x_{2}^{n-2i}x_{3}^{i+1},\ v\lambda^{i+j+2}a_{+}x_{1}^{i+j+1}x_{2}^{n-2(i+j+1)}x_{3}^{i+j+2}, λ i + 2 a + x 1 i ( x 2 n − 2 i − 1 b + b − ) x 3 i = l.c. { v λ i + 2 a + x 1 i x 2 n − 2 i x 3 i + 1 , v λ i + j + 2 a + x 1 i + j + 1 x 2 n − 2 ( i + j + 1 ) x 3 i + j + 2 , v 2 ( i + 1 ) − n λ i + j + 2 a + x 1 i a + x 1 j x 2 n − 2 ( i + j + 1 ) x 3 j + 1 b − x 3 i , ε n λ n + 3 2 a + x 1 n − 3 2 a + a − x 3 n + 1 2 } j ≥ 0 . (78) v^{2(i+1)-n}\lambda^{i+j+2}a_{+}x_{1}^{i}a_{+}x_{1}^{j}x_{2}^{n-2(i+j+1)}x_{3}^{j+1}b_{-}x_{3}^{i},\ \varepsilon_{n}\lambda^{\frac{n+3}{2}}a_{+}x_{1}^{\frac{n-3}{2}}a_{+}a_{-}x_{3}^{\frac{n+1}{2}}\}_{j\geq0}. \tag*{(78)} v 2 ( i + 1 ) − n λ i + j + 2 a + x 1 i a + x 1 j x 2 n − 2 ( i + j + 1 ) x 3 j + 1 b − x 3 i , ε n λ 2 n + 3 a + x 1 2 n − 3 a + a − x 3 2 n + 1 } j ≥ 0 . (78) The first two terms in the r.h.s. of (4.1.67) are irreducible and have a form of the second term in (4.1.53). Continuing transformation of the last term in the r.h.s. of (4.1.67) we obtain
ε n λ n + 3 2 a + x 1 n − 3 2 a + a − x 3 n + 1 2 = ε n λ n + 3 2 v n − 3 2 x 1 n − 1 2 c + a − x 3 n + 1 2 = l.c. { ε n v n + 1 λ n + 3 2 x 1 n + 1 2 x 3 n + 1 2 b + , ε n v λ n + 3 2 a + x 1 n + 1 2 x 2 x 3 n − 1 2 } . (79) \begin{aligned}
\varepsilon_{n}\lambda^{\frac{n+3}{2}}a_{+}x_{1}^{\frac{n-3}{2}}a_{+}a_{-}x_{3}^{\frac{n+1}{2}}
=\varepsilon_{n}\lambda^{\frac{n+3}{2}}v^{\frac{n-3}{2}}x_{1}^{\frac{n-1}{2}}c_{+}a_{-}x_{3}^{\frac{n+1}{2}} \\
&=\operatorname{l.c.}\{\varepsilon_{n}v^{n+1}\lambda^{\frac{n+3}{2}}x_{1}^{\frac{n+1}{2}}x_{3}^{\frac{n+1}{2}}b_{+},\ \varepsilon_{n}v\lambda^{\frac{n+3}{2}}a_{+}x_{1}^{\frac{n+1}{2}}x_{2}x_{3}^{\frac{n-1}{2}}\}. \tag*{(79)}
\end{aligned} ε n λ 2 n + 3 a + x 1 2 n − 3 a + a − x 3 2 n + 1 = ε n λ 2 n + 3 v 2 n − 3 x 1 2 n − 1 c + a − x 3 2 n + 1 = l.c. { ε n v n + 1 λ 2 n + 3 x 1 2 n + 1 x 3 2 n + 1 b + , ε n v λ 2 n + 3 a + x 1 2 n + 1 x 2 x 3 2 n − 1 } . (79) In the first step of this transformation we permuted a + a_+ a + x 1 x_1 x 1 c + c_+ c + a − a_- a − b + b_+ b + x 1 x_1 x 1 x 1 x_1 x 1 a + a_+ a +
It remains to consider the third term in the r.h.s. of (4.1.67). The transformation goes as follows
v 2 ( i + 1 ) − n λ i + j + 2 a + x 1 i a + x 1 j x 2 n − 2 ( i + j + 1 ) x 3 j + 1 b − x 3 i = v n − k − 1 λ k + 2 x 1 k + 1 x 2 n − 2 ( k + 1 ) c + b − x 3 k + 1 ∣ k = i + j = l.c. { v n + 1 λ k + 2 x 1 k + 1 x 2 n − 2 ( k + 1 ) x 3 k + 1 b − , v n + 1 − ( k + 1 ) λ k + 2 x 1 k + 1 x 2 n − 2 ( k + 1 ) a + x 3 k + 2 } ∣ k = i + j . (80) \begin{aligned}
v^{2(i+1)-n}\lambda^{i+j+2}a_+x_1^ia_+x_1^jx_2^{n-2(i+j+1)}x_3^{j+1}b_-x_3^i
= v^{n-k-1}\lambda^{k+2}x_1^{k+1}x_2^{n-2(k+1)}c_+b_-x_3^{k+1}\big|_{k=i+j} \\
&= \operatorname{l.c.}\{v^{n+1}\lambda^{k+2}x_1^{k+1}x_2^{n-2(k+1)}x_3^{k+1}b_-,v^{n+1-(k+1)}\lambda^{k+2}x_1^{k+1}x_2^{n-2(k+1)}a_+x_3^{k+2}\}\big|_{k=i+j}. \tag*{(80)}
\end{aligned} v 2 ( i + 1 ) − n λ i + j + 2 a + x 1 i a + x 1 j x 2 n − 2 ( i + j + 1 ) x 3 j + 1 b − x 3 i = v n − k − 1 λ k + 2 x 1 k + 1 x 2 n − 2 ( k + 1 ) c + b − x 3 k + 1 k = i + j = l.c. { v n + 1 λ k + 2 x 1 k + 1 x 2 n − 2 ( k + 1 ) x 3 k + 1 b − , v n + 1 − ( k + 1 ) λ k + 2 x 1 k + 1 x 2 n − 2 ( k + 1 ) a + x 3 k + 2 } k = i + j . (80) Here in the first line we moved b − b_- b − x 3 x_3 x 3 a + a_+ a + x 1 x_1 x 1 c + c_+ c + x 2 x_2 x 2 b + b_+ b + x 3 x_3 x 3
The proof of the induction step n → n + 1 n \to n+1 n → n + 1 n → n + 1 n \to n+1 n → n + 1
x 3 x 2 n + 1 = ( x 3 x 2 n ) x 2 = l.c. { λ i x 1 i x 2 n − 2 i x 3 i + 1 x 2 , v i + 2 − n λ i + 1 a + x 1 i x 2 n − 2 i − 2 b − x 3 i + 1 x 2 , ε n v − 1 n + 1 2 x 1 n − 3 2 a + a − x 3 n + 1 2 x 2 } i ≥ 0 . (81) x_3x_2^{n+1}=(x_3x_2^n)x_2=\operatorname{l.c.}\{\lambda^ix_1^ix_2^{n-2i}x_3^{i+1}x_2,v^{i+2-n}\lambda^{i+1}a_+x_1^ix_2^{n-2i-2}b_-x_3^{i+1}x_2,\varepsilon_nv^{-1}\frac{n+1}{2}x_1^{\frac{n-3}{2}}a_+a_-x_3^{\frac{n+1}{2}}x_2\}_{i\geq0}. \tag*{(81)} x 3 x 2 n + 1 = ( x 3 x 2 n ) x 2 = l.c. { λ i x 1 i x 2 n − 2 i x 3 i + 1 x 2 , v i + 2 − n λ i + 1 a + x 1 i x 2 n − 2 i − 2 b − x 3 i + 1 x 2 , ε n v − 1 2 n + 1 x 1 2 n − 3 a + a − x 3 2 n + 1 x 2 } i ≥ 0 . (81) Here we applied the relation (4.1.57) (our induction assumption) and rearranged b − b_- b − x 3 x_3 x 3
The first term in the r.h.s. of (4.1.70) can be transformed using the relations (4.1.51) and (4.1.60). Its irreducible expression contains all terms that are present in the r.h.s. of (4.1.57).
The last term in the r.h.s. of (4.1.70) can be brought to an irreducible form with the help of orthogonality relations (4.1.20), (4.1.21), (4.1.24) and the diamond relation (4.1.31). Its irreducible expression contains first and second terms from the r.h.s. of (4.1.57).
We will take a closer look at the most time-consuming calculation of the second term in the r.h.s. of (4.1.70). To transform this term we use relation (4.1.51) and then, permute generators b − b_- b − x 2 x_2 x 2
v i + 2 − n λ i + 1 a + x 1 i x 2 n − 2 i − 2 b − x 3 i + 1 x 2 = l.c. { v − n λ i + 1 a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 b − , v i + 1 − n λ i + 2 a + x 1 i x 2 n − 2 i − 2 a − x 3 i + 2 , v 1 − n λ i + 2 a + x 1 i x 2 n − 2 i − 2 x 3 i b − b + b − } . (82) v^{i+2-n}\lambda^{i+1}a_+x_1^ix_2^{n-2i-2}b_-x_3^{i+1}x_2
=\operatorname{l.c.}\{v^{-n}\lambda^{i+1}a_+x_1^ix_2^{n-2i-1}x_3^{i+1}b_-,
v^{i+1-n}\lambda^{i+2}a_+x_1^ix_2^{n-2i-2}a_-x_3^{i+2},
v^{1-n}\lambda^{i+2}a_+x_1^ix_2^{n-2i-2}x_3^ib_-b_+b_-\}. \tag*{(82)} v i + 2 − n λ i + 1 a + x 1 i x 2 n − 2 i − 2 b − x 3 i + 1 x 2 = l.c. { v − n λ i + 1 a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 b − , v i + 1 − n λ i + 2 a + x 1 i x 2 n − 2 i − 2 a − x 3 i + 2 , v 1 − n λ i + 2 a + x 1 i x 2 n − 2 i − 2 x 3 i b − b + b − } . (82) The first term in the r.h.s of (4.1.73) is irreducible. It has a form of the second term in (4.1.57).
To bring the second term in the r.h.s of (4.1.73) to the irreducible form we apply consecutively the relation (4.1.56) and the image of the relation (4.1.60) under the antihomomorphism (4.1.25) which gives the reduced expressions for the monomials a + a − x 3 n a_+a_-x_3^n a + a − x 3 n x 1 x_1 x 1 a + a_+ a + b − b_- b − x 3 x_3 x 3
The last term in the r.h.s. of (4.1.73) contains the monomial x 3 i b − b + b − x_3^ib_-b_+b_- x 3 i b − b + b −
x 3 i b − b + b − = l.c. { v − 1 x 2 x 3 i + 1 b − , v i a − x 3 i + 2 } . (83) x_3^ib_-b_+b_-=\operatorname{l.c.}\{v^{-1}x_2x_3^{i+1}b_-,v^ia_-x_3^{i+2}\}. \tag*{(83)} x 3 i b − b + b − = l.c. { v − 1 x 2 x 3 i + 1 b − , v i a − x 3 i + 2 } . (83) Therefore for the last term in the r.h.s. of (4.1.73) we have
v 1 − n λ i + 2 a + x 1 i x 2 n − 2 i − 2 x 3 i b − b + b − = l.c. { v − n λ i + 2 a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 b − , v i + 1 − n λ i + 2 a + x 1 i x 2 n − 2 i − 2 a − x 3 i + 2 } . (84) v^{1-n}\lambda^{i+2}a_+x_1^ix_2^{n-2i-2}x_3^ib_-b_+b_-
=\operatorname{l.c.}\{v^{-n}\lambda^{i+2}a_+x_1^ix_2^{n-2i-1}x_3^{i+1}b_-,
v^{i+1-n}\lambda^{i+2}a_+x_1^ix_2^{n-2i-2}a_-x_3^{i+2}\}. \tag*{(84)} v 1 − n λ i + 2 a + x 1 i x 2 n − 2 i − 2 x 3 i b − b + b − = l.c. { v − n λ i + 2 a + x 1 i x 2 n − 2 i − 1 x 3 i + 1 b − , v i + 1 − n λ i + 2 a + x 1 i x 2 n − 2 i − 2 a − x 3 i + 2 } . (84) Thus, we expressed the last term on the r.h.s. of the relation (4.1.73) as a linear combination of the first two terms from there. Hence, its reduced expression contains terms from the r.h.s. of (4.1.57) only. The proof of the induction step for the relation (4.1.57) is finished.
Proof of Theorem 4.4. Verifying the statement of the theorem is a straightforward although lengthy calculation. It goes by induction on the degree of the monomials in the algebra generators.
The base of the induction is the set of second-degree monomials. Indeed, the 9 generators of the algebra F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v 9 2 = 81 9^2=81 9 2 = 81 35 = 9 2 − 36 − 10 35=9^2-36-10 35 = 9 2 − 36 − 10
The inductive step consists of checking that monomials of degree n + 1 n+1 n + 1 n n n n + 1 n+1 n + 1 x i x_i x i
Let us place all irreducible cubic monomials on a square lattice: the coordinate of a monomial on the lattice coincides with its Z 2 \mathbb{Z}^2 Z 2
Figure 1.
Here vertices of the lattice and their bi-gradings are colored blue.
The figure above suggests a conjecture that when placed on the Z 2 \mathbb{Z}^2 Z 2 n n n d n d_n d n n n n F q , v \mathcal{F}_{q,v} F q , v
d n = ( n + 1 ) + 8 n ( n + 1 ) ( n + 2 ) 6 = ( 2 n + 3 3 ) , n ≥ 2. (85) d_n=(n+1)+8\frac{n(n+1)(n+2)}{6}=\binom{2n+3}{3},\qquad n\ge2. \tag*{(85)} d n = ( n + 1 ) + 8 6 n ( n + 1 ) ( n + 2 ) = ( 3 2 n + 3 ) , n ≥ 2. (85) This formula does not work only for n = 1 n=1 n = 1 d 1 = 9 = ( 5 3 ) − 1 d_1=9=\binom{5}{3}-1 d 1 = 9 = ( 3 5 ) − 1 F q , v ( R ) \mathcal{F}_{q,v}(R) F q , v ( R )
P ( t ) = 1 − t + ∑ n ≥ 1 ( 2 n + 3 3 ) t n = 8 t ( 1 − t ) 4 + 1 ( 1 − t ) 2 − t . (86) P(t)=1-t+\sum_{n\ge1}\binom{2n+3}{3}t^n=\frac{8t}{(1-t)^4}+\frac{1}{(1-t)^2}-t. \tag*{(86)} P ( t ) = 1 − t + n ≥ 1 ∑ ( 3 2 n + 3 ) t n = ( 1 − t ) 4 8 t + ( 1 − t ) 2 1 − t . (86) Let us summarize the results obtained.
Proposition 4.10 Any polynomial of degree n ≥ 0 n\ge0 n ≥ 0 F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v d n = ( 2 n + 3 3 ) d_n=\binom{2n+3}{3} d n = ( 3 2 n + 3 ) Z 2 \mathbb{Z}^2 Z 2 Z 2 \mathbb{Z}^2 Z 2
grad = ( 0 , 2 p ) , p ≥ 0 : x 1 i + p x 2 n − 2 i − p x 3 i , 0 ≤ i ≤ ⌊ n − p 2 ⌋ , \operatorname{grad}=(0,2p),\ p\ge0:\quad x_1^{i+p}x_2^{n-2i-p}x_3^i,\qquad0\le i\le\left\lfloor\frac{n-p}{2}\right\rfloor, grad = ( 0 , 2 p ) , p ≥ 0 : x 1 i + p x 2 n − 2 i − p x 3 i , 0 ≤ i ≤ ⌊ 2 n − p ⌋ , a + x 1 i + p x 2 n − 2 i − p − 2 b − x 3 i , 0 ≤ i ≤ ⌊ n − p − 2 2 ⌋ , (87) a_{+}x_{1}^{i+p}x_{2}^{n-2i-p-2}b_{-}x_{3}^{i}, \qquad0\le i\le\left\lfloor\frac{n-p-2}{2}\right\rfloor, \tag*{(87)} a + x 1 i + p x 2 n − 2 i − p − 2 b − x 3 i , 0 ≤ i ≤ ⌊ 2 n − p − 2 ⌋ , (87) a + a − x 1 n + p − 3 2 x 3 n − p − 1 2 , if ( n − p ) is odd , a_{+}a_{-}x_{1}^{\frac{n+p-3}{2}}x_{3}^{\frac{n-p-1}{2}}, \qquad\text{if }(n-p)\text{ is odd}, a + a − x 1 2 n + p − 3 x 3 2 n − p − 1 , if ( n − p ) is odd , grad = ( 2 k , 2 p ) , k > 0 , p ≥ 0 : x 1 i + p c + k x 2 n − 2 i − p − k x 3 i , 0 ≤ i ≤ ⌊ n − p − k 2 ⌋ , a + x 1 i + p c + k − 1 x 2 n − 2 i − p − k − 1 b + x 3 i , 0 ≤ i ≤ ⌊ n − p − k − 1 2 ⌋ , (88) \begin{aligned}
\operatorname{grad}=(2k,2p),\ k>0,\ p\ge0:\quad&
x_{1}^{i+p}c_{+}^{k}x_{2}^{n-2i-p-k}x_{3}^{i},\qquad0\le i\le\left\lfloor\frac{n-p-k}{2}\right\rfloor,\\
&a_{+}x_{1}^{i+p}c_{+}^{k-1}x_{2}^{n-2i-p-k-1}b_{+}x_{3}^{i},\qquad0\le i\le\left\lfloor\frac{n-p-k-1}{2}\right\rfloor, \tag*{(88)}
\end{aligned} grad = ( 2 k , 2 p ) , k > 0 , p ≥ 0 : x 1 i + p c + k x 2 n − 2 i − p − k x 3 i , 0 ≤ i ≤ ⌊ 2 n − p − k ⌋ , a + x 1 i + p c + k − 1 x 2 n − 2 i − p − k − 1 b + x 3 i , 0 ≤ i ≤ ⌊ 2 n − p − k − 1 ⌋ , (88) grad = ( 2 k + 1 , 2 p + 1 ) , k ≥ 0 , p ≥ 0 : a + x 1 i + p c + k x 2 n − 2 i − p − k − 1 x 3 i , 0 ≤ i ≤ ⌊ n − p − k − 1 2 ⌋ , x 1 i + p + 1 c + k x 2 n − 2 i − p − k − 2 b + x 3 i , 0 ≤ i ≤ ⌊ n − p − k − 2 2 ⌋ . (89) \begin{aligned}
\operatorname{grad}=(2k+1,2p+1),\ k\ge0,\ p\ge0:\quad&
a_{+}x_{1}^{i+p}c_{+}^{k}x_{2}^{n-2i-p-k-1}x_{3}^{i},\qquad0\le i\le\left\lfloor\frac{n-p-k-1}{2}\right\rfloor,\\
&x_{1}^{i+p+1}c_{+}^{k}x_{2}^{n-2i-p-k-2}b_{+}x_{3}^{i},\qquad0\le i\le\left\lfloor\frac{n-p-k-2}{2}\right\rfloor. \tag*{(89)}
\end{aligned} grad = ( 2 k + 1 , 2 p + 1 ) , k ≥ 0 , p ≥ 0 : a + x 1 i + p c + k x 2 n − 2 i − p − k − 1 x 3 i , 0 ≤ i ≤ ⌊ 2 n − p − k − 1 ⌋ , x 1 i + p + 1 c + k x 2 n − 2 i − p − k − 2 b + x 3 i , 0 ≤ i ≤ ⌊ 2 n − p − k − 2 ⌋ . (89) The remaining monomials contain negative components in their Z 2 \mathbb{Z}^{2} Z 2
Proof. The expressions (4.1.76)–(4.1.78) for irreducible Z 2 \mathbb{Z}^{2} Z 2 n n n Z 2 \mathbb{Z}^{2} Z 2 n n n ( n − p − k + 1 ) (n-p-k+1) ( n − p − k + 1 ) ( 2 k , 2 p ) (2k,2p) ( 2 k , 2 p ) ( n − p − k ) (n-p-k) ( n − p − k ) ( 2 k + 1 , 2 p + 1 ) (2k+1,2p+1) ( 2 k + 1 , 2 p + 1 ) n n n
Transition to the quotient algebra F q ( O ( 3 ) ) \mathcal{F}_{q}(O(3)) F q ( O ( 3 )) O ( 3 ) O(3) O ( 3 )
Keeping in mind that the contraction g g g F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v g g g x 2 2 x_{2}^{2} x 2 2
x 2 2 = q 2 g + λ − 2 q a + b − + λ q x 1 x 3 . x_{2}^{2}=q^{2}g+\frac{\lambda-2}{q}a_{+}b_{-}+\frac{\lambda}{q}x_{1}x_{3}. x 2 2 = q 2 g + q λ − 2 a + b − + q λ x 1 x 3 . With this substitution the ordering relations (4.1.16), (4.1.18), (4.1.19), the orthogonality relation (4.1.24) and the diamond relations (4.1.31), (4.1.34) take a form
x 3 x 1 = 1 + λ q 2 x 1 x 3 + λ q 2 a + b − + λ q 2 ( q − 1 ) g . (90) x_{3}x_{1}=\frac{1+\lambda}{q^{2}}x_{1}x_{3}+\frac{\lambda}{q^{2}}a_{+}b_{-}+\lambda q^{2}(q-1)g. \tag*{(90)} x 3 x 1 = q 2 1 + λ x 1 x 3 + q 2 λ a + b − + λ q 2 ( q − 1 ) g . (90) b + a − = v 4 q b − a + = v 2 ( 1 − λ q 2 a + b − − λ q 2 x 1 x 3 + λ q 2 g ) . (91) b_{+}a_{-}=\frac{v^{4}}{q}b_{-}a_{+}=v^{2}\left(\frac{1-\lambda}{q^{2}}a_{+}b_{-}-\frac{\lambda}{q^{2}}x_{1}x_{3}+\lambda q^{2}g\right). \tag*{(91)} b + a − = q v 4 b − a + = v 2 ( q 2 1 − λ a + b − − q 2 λ x 1 x 3 + λ q 2 g ) . (91) c + c − = v 2 ( − q − 1 a + b − − q − 1 x 1 x 3 + q 3 g ) . (92) c_{+}c_{-}=v^{2}\left(-q^{-1}a_{+}b_{-}-q^{-1}x_{1}x_{3}+q^{3}g\right). \tag*{(92)} c + c − = v 2 ( − q − 1 a + b − − q − 1 x 1 x 3 + q 3 g ) . (92) a + a − x 2 = − 1 + 2 q q 2 a + x 1 b − + v 1 + q q 3 x 1 ( q 4 g − x 1 x 3 ) . (93) a_{+}a_{-}x_{2}=-\frac{1+2q}{q^{2}}a_{+}x_{1}b_{-}+v\frac{1+q}{q^{3}}x_{1}(q^{4}g-x_{1}x_{3}). \tag*{(93)} a + a − x 2 = − q 2 1 + 2 q a + x 1 b − + v q 3 1 + q x 1 ( q 4 g − x 1 x 3 ) . (93) x 2 b + b − = − 1 + 2 q q 2 a + x 3 b − + v 1 + q q 3 ( q 4 g − x 1 x 3 ) x 3 . (94) x_{2}b_{+}b_{-}=-\frac{1+2q}{q^{2}}a_{+}x_{3}b_{-}+v\frac{1+q}{q^{3}}(q^{4}g-x_{1}x_{3})x_{3}. \tag*{(94)} x 2 b + b − = − q 2 1 + 2 q a + x 3 b − + v q 3 1 + q ( q 4 g − x 1 x 3 ) x 3 . (94) Then, all even powers of x 2 x_{2} x 2 g g g
Proposition 4.11
x 2 2 n = ∑ i ≥ 0 F n , i g n − i x 1 i x 3 i + v − 2 n ∑ i ≥ 0 G n , i a + g n − i − 1 x 1 i x 3 i b − , ∀ n > 0. (95) x_{2}^{2n}=\sum_{i\ge0}F_{n,i}g^{n-i}x_{1}^{i}x_{3}^{i}+v^{-2n}\sum_{i\ge0}G_{n,i}a_{+}g^{n-i-1}x_{1}^{i}x_{3}^{i}b_{-},\qquad\forall n>0. \tag*{(95)} x 2 2 n = i ≥ 0 ∑ F n , i g n − i x 1 i x 3 i + v − 2 n i ≥ 0 ∑ G n , i a + g n − i − 1 x 1 i x 3 i b − , ∀ n > 0. (95) where F n , i F_{n,i} F n , i G n , i G_{n,i} G n , i q ± 1 q^{\pm1} q ± 1
The proof of this statement follows the same pattern as the proofs of lemmas 4.5 and 4.7; therefore, we shall provide here only two essential remarks regarding it.
1). Formula (4.2.6) is proved by induction on n n n ( a + b − ) n (a_{+}b_{-})^{n} ( a + b − ) n q q q
2). It is easy to find F n , 0 = q 2 n F_{n,0}=q^{2n} F n , 0 = q 2 n F n , i F_{n,i} F n , i G n , i G_{n,i} G n , i λ \lambda λ
We conclude that in the list (4.1.36)–(4.1.42) of irreducible monomials, the factors x 2 2 k x_2^{2k} x 2 2 k x 2 2 k + 1 x_2^{2k+1} x 2 2 k + 1 g k g^k g k g k x 2 g^k x_2 g k x 2
Now, let us assume that the contraction g g g ϵ 3 \epsilon_3 ϵ 3 (9) , (10) , and (16) , we conclude, that this is possible if and only if v = 1 v=1 v = 1 F q , v = 1 ( R ) \mathcal{F}_{q,v=1}(R) F q , v = 1 ( R ) q 2 ( n − 1 ) g = 1 q^{2(n-1)}g=1 q 2 ( n − 1 ) g = 1 q 2 ( n − 1 ) g q^{2(n-1)}g q 2 ( n − 1 ) g F q ( O ( 3 ) ) \mathcal{F}_q(O(3)) F q ( O ( 3 )) O ( 3 ) O(3) O ( 3 ) [5 ] ). To conclude this section, we compute the Hilbert–Poincaré series for this algebra.
The list of irreducible monomials in F q ( O ( 3 ) ) \mathcal{F}_q(O(3)) F q ( O ( 3 )) x 2 x_2 x 2
*) 1 + 2 2 + 3 2 + ⋯ + ( n − 1 ) 2 = n ( n − 1 ) ( 2 n − 1 ) 6 1+2^2+3^2+\cdots+(n-1)^2=\frac{n(n-1)(2n-1)}{6} 1 + 2 2 + 3 2 + ⋯ + ( n − 1 ) 2 = 6 n ( n − 1 ) ( 2 n − 1 ) n n n x 2 n , x 2 n − 1 , … , x 2 2 x_2^n,x_2^{n-1},\ldots,x_2^2 x 2 n , x 2 n − 1 , … , x 2 2
**) 4 4 4 1 + 3 + 6 + ⋯ + ( n − 1 ) ( n − 2 ) 2 = n ( n − 1 ) ( n − 2 ) 6 1+3+6+\cdots+\frac{(n-1)(n-2)}{2}=\frac{n(n-1)(n-2)}{6} 1 + 3 + 6 + ⋯ + 2 ( n − 1 ) ( n − 2 ) = 6 n ( n − 1 ) ( n − 2 ) n n n X x 2 n − 1 , X x 2 n − 2 , … , X x 2 2 Xx_2^{n-1},Xx_2^{n-2},\ldots,Xx_2^2 X x 2 n − 1 , X x 2 n − 2 , … , X x 2 2 X X X a + a_+ a + a − a_- a − b + b_+ b + b − b_- b −
***) 2 2 2 1 + 3 + ⋯ + ( n − 2 ) ( n − 3 ) 2 = ( n − 1 ) ( n − 2 ) ( n − 3 ) 6 1+3+\cdots+\frac{(n-2)(n-3)}{2}=\frac{(n-1)(n-2)(n-3)}{6} 1 + 3 + ⋯ + 2 ( n − 2 ) ( n − 3 ) = 6 ( n − 1 ) ( n − 2 ) ( n − 3 ) n n n X x 2 n − 2 Y , … , X x 2 2 Y Xx_2^{n-2}Y,\ldots,Xx_2^2Y X x 2 n − 2 Y , … , X x 2 2 Y { X , Y } \{X,Y\} { X , Y } { a + , b + } \{a_+,b_+\} { a + , b + } { a − , b − } \{a_-,b_-\} { a − , b − }
****) 1 + 2 + 3 + ⋯ + ( n − 3 ) = ( n − 2 ) ( n − 3 ) 2 1+2+3+\cdots+(n-3)=\frac{(n-2)(n-3)}{2} 1 + 2 + 3 + ⋯ + ( n − 3 ) = 2 ( n − 2 ) ( n − 3 ) n n n a + x 2 n − 2 b − , … , a + x 2 2 b − a_+x_2^{n-2}b_-,\ldots,a_+x_2^2b_- a + x 2 n − 2 b − , … , a + x 2 2 b −
Altogether we exclude from the list of irreducible monomials
n ( n − 1 ) ( 2 n − 1 ) 6 + 4 × n ( n − 1 ) ( n − 2 ) 6 + 2 × ( n − 1 ) ( n − 2 ) ( n − 3 ) 6 + ( n − 2 ) ( n − 3 ) 2 = 8 ( n 3 ) + 1 \frac{n(n-1)(2n-1)}{6}+4\times\frac{n(n-1)(n-2)}{6}+2\times\frac{(n-1)(n-2)(n-3)}{6}+\frac{(n-2)(n-3)}{2}=8\binom{n}{3}+1 6 n ( n − 1 ) ( 2 n − 1 ) + 4 × 6 n ( n − 1 ) ( n − 2 ) + 2 × 6 ( n − 1 ) ( n − 2 ) ( n − 3 ) + 2 ( n − 2 ) ( n − 3 ) = 8 ( 3 n ) + 1 elements containing x 2 x_2 x 2
d ~ 0 = 1 , d ~ n = ( 2 n + 3 3 ) − 8 ( n 3 ) − 1 = n ( 8 n + 1 ) ∀ n > 0 \widetilde d_0=1,\qquad\widetilde d_n=\binom{2n+3}{3}-8\binom{n}{3}-1=n(8n+1)\quad\forall n>0 d 0 = 1 , d n = ( 3 2 n + 3 ) − 8 ( 3 n ) − 1 = n ( 8 n + 1 ) ∀ n > 0 elements. This is the number of the irreducible degree n n n O q ( 3 ) O_q(3) O q ( 3 ) F q ( O ( 3 ) ) \mathcal{F}_q(O(3)) F q ( O ( 3 ))
P ~ ( t ) = 1 + ∑ n ≥ 1 n ( 8 n + 1 ) t n = 1 + t ( 7 t + 9 ) ( 1 − t ) 3 . (96) \widetilde P(t)=1+\sum_{n\geq1}n(8n+1)t^n=1+\frac{t(7t+9)}{(1-t)^3}. \tag*{(96)} P ( t ) = 1 + n ≥ 1 ∑ n ( 8 n + 1 ) t n = 1 + ( 1 − t ) 3 t ( 7 t + 9 ) . (96) Verification of the general theory results
In this subsection, we verify the results presented in § 3 regarding the general structure of QM-algebras, using the RTT-algebra F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v § 4.1 . As we have seen, the calculations prove to be quite cumbersome; the Mathematica symbolic computation software was used to carry them out.
The expansions of the generators of the characteristic subalgebra of the algebra F ( R ) q , v \mathcal{F}(R)_{q,v} F ( R ) q , v
g : = q q 2 + q + 1 Tr R ( 1 , 2 ) ( T 1 T 2 K R ) = 2 − λ q 3 a + b − − λ q 3 x 1 x 3 + 1 q 2 x 2 2 (97) g:=\frac{q}{q^2+q+1}\operatorname{Tr}_{R(1,2)}(T_1T_2K_R)=\frac{2-\lambda}{q^3}a_+b_- -\frac{\lambda}{q^3}x_1x_3+\frac{1}{q^2}x_2^2 \tag*{(97)} g := q 2 + q + 1 q Tr R ( 1 , 2 ) ( T 1 T 2 K R ) = q 3 2 − λ a + b − − q 3 λ x 1 x 3 + q 2 1 x 2 2 (97) ϵ 1 : = Tr R T = q − 3 x 1 + q − 2 x 2 + q − 1 x 3 (98) \epsilon_1:=\operatorname{Tr}_R T=q^{-3}x_1+q^{-2}x_2+q^{-1}x_3 \tag*{(98)} ϵ 1 := Tr R T = q − 3 x 1 + q − 2 x 2 + q − 1 x 3 (98) ϵ 2 : = Tr R ( 1 , 2 ) ( T 1 T 2 A R ( 2 ) ) = 1 q 5 v ( v x 1 x 2 − q a + a − ) + 1 q 4 ( ( 1 + q ) x 1 x 3 − q x 2 2 + ( λ − 1 ) a + b − ) + 1 q 3 v ( v x 2 x 3 − q b + b − ) (99) \epsilon_2:=\operatorname{Tr}_{R(1,2)}(T_1T_2A_R^{(2)})=\frac{1}{q^5v}(vx_1x_2-qa_+a_-)
+\frac{1}{q^4}\left((1+q)x_1x_3-qx_2^2+(\lambda-1)a_+b_-\right)+\frac{1}{q^3v}(vx_2x_3-qb_+b_-) \tag*{(99)} ϵ 2 := Tr R ( 1 , 2 ) ( T 1 T 2 A R ( 2 ) ) = q 5 v 1 ( v x 1 x 2 − q a + a − ) + q 4 1 ( ( 1 + q ) x 1 x 3 − q x 2 2 + ( λ − 1 ) a + b − ) + q 3 v 1 ( v x 2 x 3 − q b + b − ) (99) ϵ 3 : = Tr R ( 1 , 2 , 3 ) ( T 1 T 2 T 3 A R ( 3 ) ) = 1 + q 3 q 6 x 1 x 2 x 3 − 1 q 3 x 2 3 − 2 q 6 v ( 1 + q + λ q 2 ( q − 1 ) ) a + a − x 3 − 1 q 6 v ( 1 + q 2 + q 4 − 3 λ q 2 ) a + x 2 b − (100) \epsilon_3:=\operatorname{Tr}_{R(1,2,3)}(T_1T_2T_3A_R^{(3)})=\frac{1+q^3}{q^6}x_1x_2x_3-\frac{1}{q^3}x_2^3-\frac{2}{q^6v}\left(1+q+\lambda q^2(q-1)\right)a_+a_-x_3-\frac{1}{q^6v}\left(1+q^2+q^4-3\lambda q^2\right)a_+x_2b_- \tag*{(100)} ϵ 3 := Tr R ( 1 , 2 , 3 ) ( T 1 T 2 T 3 A R ( 3 ) ) = q 6 1 + q 3 x 1 x 2 x 3 − q 3 1 x 2 3 − q 6 v 2 ( 1 + q + λ q 2 ( q − 1 ) ) a + a − x 3 − q 6 v 1 ( 1 + q 2 + q 4 − 3 λ q 2 ) a + x 2 b − (100) Here elements g g g e 3 e_3 e 3 Z 2 \mathbb{Z}^2 Z 2 ( 0 , 0 ) (0,0) ( 0 , 0 ) e 1 e_1 e 1 e 2 e_2 e 2 ( 0 , 2 ) (0,2) ( 0 , 2 ) ( 0 , 0 ) (0,0) ( 0 , 0 ) ( 0 , − 2 ) (0,-2) ( 0 , − 2 )
First, we have checked explicitly commutation relations (16) . Note that these computations involve the ordering and the orthogonality relations, and all the diamond relations too.
Relations (16) guarantee the commutativity of g g g e 3 e_3 e 3 e 1 e_1 e 1 e 2 e_2 e 2 e 1 e_1 e 1 e 2 e_2 e 2 Z 2 \mathbb{Z}^2 Z 2 ( 0 , 4 ) (0,4) ( 0 , 4 ) ( 0 , 2 ) (0,2) ( 0 , 2 ) … \ldots … ( 0 , − 4 ) (0,-4) ( 0 , − 4 ) [ e 1 , e 2 ] [e_1,e_2] [ e 1 , e 2 ]
Next, we checked the reciprocal relations
g e 2 = e 1 e 3 , g 2 e 1 = e 2 e 3 , g 3 = e 3 2 . (101) ge_2=e_1e_3,\qquad g^2e_1=e_2e_3,\qquad g^3=e_3^2. \tag*{(101)} g e 2 = e 1 e 3 , g 2 e 1 = e 2 e 3 , g 3 = e 3 2 . (101) These calculations are quite labor-intensive, but they can be automated. We performed them using the Wolfram Mathematica software equipped with the NCAlgebra package, designed for operations with noncommutative associative algebras. To illustrate the complexity of the calculations, we present the irreducible expressions for g 2 g^2 g 2 g 3 g^3 g 3
g 2 = 1 q 4 x 4 2 + ( 1 + q ) ( 3 + 9 q + 5 q 2 − 12 q 3 − 8 q 4 + 11 q 5 + 5 q 6 − 6 q 7 + q 8 ) q 10 x 1 x 2 2 x 3 − ( 1 + q ) 2 ( 3 + 6 q − 2 q 2 − 9 q 3 + 12 q 5 − 7 q 6 + q 7 ) q 10 x 1 2 x 3 2 + ( 1 + 2 q − q 2 ) ( 1 + 2 q − 2 q 3 + q 4 ) q 8 v 2 a + x 2 2 b − + ( 1 + q ) ( 6 + 15 q − 3 q 2 − 27 q 3 + 2 q 4 + 26 q 5 − q 6 − 17 q 7 + 8 q 8 − q 9 ) q 9 v 2 a + x 1 x 3 b − . (102) \begin{aligned}
g^2={}&\frac{1}{q^4}x_4^2+\frac{(1+q)(3+9q+5q^2-12q^3-8q^4+11q^5+5q^6-6q^7+q^8)}{q^{10}}x_1x_2^2x_3\\
&-\frac{(1+q)^2(3+6q-2q^2-9q^3+12q^5-7q^6+q^7)}{q^{10}}x_1^2x_3^2\\
&+\frac{(1+2q-q^2)(1+2q-2q^3+q^4)}{q^8v^2}a_+x_2^2b_-\\
&+\frac{(1+q)(6+15q-3q^2-27q^3+2q^4+26q^5-q^6-17q^7+8q^8-q^9)}{q^9v^2}a_+x_1x_3b_- . \tag*{(102)}
\end{aligned} g 2 = q 4 1 x 4 2 + q 10 ( 1 + q ) ( 3 + 9 q + 5 q 2 − 12 q 3 − 8 q 4 + 11 q 5 + 5 q 6 − 6 q 7 + q 8 ) x 1 x 2 2 x 3 − q 10 ( 1 + q ) 2 ( 3 + 6 q − 2 q 2 − 9 q 3 + 12 q 5 − 7 q 6 + q 7 ) x 1 2 x 3 2 + q 8 v 2 ( 1 + 2 q − q 2 ) ( 1 + 2 q − 2 q 3 + q 4 ) a + x 2 2 b − + q 9 v 2 ( 1 + q ) ( 6 + 15 q − 3 q 2 − 27 q 3 + 2 q 4 + 26 q 5 − q 6 − 17 q 7 + 8 q 8 − q 9 ) a + x 1 x 3 b − . (102) g 3 = 1 q 6 x 6 2 + ( 1 + q ) ( 5 + 30 q + 61 q 2 − 162 q 4 − 125 q 5 + 210 q 6 + 245 q 7 − 177 q 8 − 228 q 9 + 132 q 10 + 110 q 11 − 82 q 12 − 14 q 13 + 27 q 14 − 9 q 15 + q 16 ) q 16 x 1 x 4 2 x 3 + ( 1 + q ) 2 q 21 ( 40 + 240 q + 405 q 2 − 444 q 3 − 2150 q 4 − 943 q 5 + 4418 q 6 + 4695 q 7 − 5112 q 8 − 8571 q 9 + 3935 q 10 + 9597 q 11 − 2640 q 12 − 7377 q 13 + 2130 q 14 + 3863 q 15 − 1692 q 16 − 1144 q 17 + 877 q 18 + 31 q 19 − 210 q 20 + 86 q 21 − 15 q 22 + q 23 ) x 1 2 x 2 2 x 3 2 − ( 1 + q ) 3 q 21 ( 40 + 200 q + 205 q 2 − 649 q 3 − 1501 q 4 + 563 q 5 + 3880 q 6 + 851 q 7 − 5999 q 8 − 2699 q 9 + 6637 q 10 + 3166 q 11 − 5766 q 12 − 1829 q 13 + 3950 q 14 + 54 q 15 − 1777 q 16 + 582 q 17 + 332 q 18 − 311 q 19 + 102 q 20 − 16 q 21 + q 22 ) x 1 3 x 3 3 + ( 1 + 2 q − q 2 ) ( 1 + 3 q + q 2 − 3 q 3 + q 4 ) ( 1 + q − q 2 − q 3 + q 4 ) q 12 v 4 a + x 2 4 b − + 1 + q q 17 v 4 ( 20 + 125 q + 245 q 2 − 79 q 3 − 918 q 4 − 703 q 5 + 1485 q 6 + 2098 q 7 − 1420 q 8 − 3049 q 9 + 1064 q 10 + 2845 q 11 − 914 q 12 − 1779 q 13 + 841 q 14 + 622 q 15 − 517 q 16 − 17 q 17 + 145 q 18 − 66 q 19 + 13 q 20 − q 21 ) a + x 1 x 2 2 x 3 b − + ( 1 + q ) 2 q 18 v 4 ( 80 + 420 q + 465 q 2 − 1413 q 3 − 3542 q 4 + 1082 q 5 + 9598 q 6 + 2984 q 7 − 15431 q 8 − 9126 q 9 + 17532 q 10 + 12266 q 11 − 15685 q 12 − 9866 q 13 + 11647 q 14 + 4535 q 15 − 6914 q 16 − 400 q 17 + 2738 q 18 − 773 q 19 − 455 q 20 + 382 q 21 − 116 q 22 + 17 q 23 − q 24 ) a + x 1 2 x 3 b − . (103) \begin{aligned}
g^3={}&\frac{1}{q^6}x_6^2+\frac{(1+q)(5+30q+61q^2-162q^4-125q^5+210q^6+245q^7-177q^8-228q^9+132q^{10}+110q^{11}-82q^{12}-14q^{13}+27q^{14}-9q^{15}+q^{16})}{q^{16}}x_1x_4^2x_3\\
&+\frac{(1+q)^2}{q^{21}}(40+240q+405q^2-444q^3-2150q^4-943q^5+4418q^6+4695q^7-5112q^8-8571q^9+3935q^{10}+9597q^{11}-2640q^{12}\\
&\qquad-7377q^{13}+2130q^{14}+3863q^{15}-1692q^{16}-1144q^{17}+877q^{18}+31q^{19}-210q^{20}+86q^{21}-15q^{22}+q^{23})x_1^2x_2^2x_3^2\\
&-\frac{(1+q)^3}{q^{21}}(40+200q+205q^2-649q^3-1501q^4+563q^5+3880q^6+851q^7-5999q^8-2699q^9+6637q^{10}+3166q^{11}\\
&\qquad-5766q^{12}-1829q^{13}+3950q^{14}+54q^{15}-1777q^{16}+582q^{17}+332q^{18}-311q^{19}+102q^{20}-16q^{21}+q^{22})x_1^3x_3^3\\
&+\frac{(1+2q-q^2)(1+3q+q^2-3q^3+q^4)(1+q-q^2-q^3+q^4)}{q^{12}v^4}a_+x_2^4b_-\\
&+\frac{1+q}{q^{17}v^4}(20+125q+245q^2-79q^3-918q^4-703q^5+1485q^6+2098q^7-1420q^8-3049q^9+1064q^{10}+2845q^{11}\\
&\qquad-914q^{12}-1779q^{13}+841q^{14}+622q^{15}-517q^{16}-17q^{17}+145q^{18}-66q^{19}+13q^{20}-q^{21})a_+x_1x_2^2x_3b_-\\
&+\frac{(1+q)^2}{q^{18}v^4}(80+420q+465q^2-1413q^3-3542q^4+1082q^5+9598q^6+2984q^7-15431q^8-9126q^9+17532q^{10}\\
&\qquad+12266q^{11}-15685q^{12}-9866q^{13}+11647q^{14}+4535q^{15}-6914q^{16}-400q^{17}+2738q^{18}-773q^{19}\\
&\qquad-455q^{20}+382q^{21}-116q^{22}+17q^{23}-q^{24})a_+x_1^2x_3b_- . \tag*{(103)}
\end{aligned} g 3 = q 6 1 x 6 2 + q 16 ( 1 + q ) ( 5 + 30 q + 61 q 2 − 162 q 4 − 125 q 5 + 210 q 6 + 245 q 7 − 177 q 8 − 228 q 9 + 132 q 10 + 110 q 11 − 82 q 12 − 14 q 13 + 27 q 14 − 9 q 15 + q 16 ) x 1 x 4 2 x 3 + q 21 ( 1 + q ) 2 ( 40 + 240 q + 405 q 2 − 444 q 3 − 2150 q 4 − 943 q 5 + 4418 q 6 + 4695 q 7 − 5112 q 8 − 8571 q 9 + 3935 q 10 + 9597 q 11 − 2640 q 12 − 7377 q 13 + 2130 q 14 + 3863 q 15 − 1692 q 16 − 1144 q 17 + 877 q 18 + 31 q 19 − 210 q 20 + 86 q 21 − 15 q 22 + q 23 ) x 1 2 x 2 2 x 3 2 − q 21 ( 1 + q ) 3 ( 40 + 200 q + 205 q 2 − 649 q 3 − 1501 q 4 + 563 q 5 + 3880 q 6 + 851 q 7 − 5999 q 8 − 2699 q 9 + 6637 q 10 + 3166 q 11 − 5766 q 12 − 1829 q 13 + 3950 q 14 + 54 q 15 − 1777 q 16 + 582 q 17 + 332 q 18 − 311 q 19 + 102 q 20 − 16 q 21 + q 22 ) x 1 3 x 3 3 + q 12 v 4 ( 1 + 2 q − q 2 ) ( 1 + 3 q + q 2 − 3 q 3 + q 4 ) ( 1 + q − q 2 − q 3 + q 4 ) a + x 2 4 b − + q 17 v 4 1 + q ( 20 + 125 q + 245 q 2 − 79 q 3 − 918 q 4 − 703 q 5 + 1485 q 6 + 2098 q 7 − 1420 q 8 − 3049 q 9 + 1064 q 10 + 2845 q 11 − 914 q 12 − 1779 q 13 + 841 q 14 + 622 q 15 − 517 q 16 − 17 q 17 + 145 q 18 − 66 q 19 + 13 q 20 − q 21 ) a + x 1 x 2 2 x 3 b − + q 18 v 4 ( 1 + q ) 2 ( 80 + 420 q + 465 q 2 − 1413 q 3 − 3542 q 4 + 1082 q 5 + 9598 q 6 + 2984 q 7 − 15431 q 8 − 9126 q 9 + 17532 q 10 + 12266 q 11 − 15685 q 12 − 9866 q 13 + 11647 q 14 + 4535 q 15 − 6914 q 16 − 400 q 17 + 2738 q 18 − 773 q 19 − 455 q 20 + 382 q 21 − 116 q 22 + 17 q 23 − q 24 ) a + x 1 2 x 3 b − . (103) Our final check concerns the characteristic identity (19) for the quantum matrix T T T O ( 3 ) O(3) O ( 3 )
T ‾ 3 g − T ‾ 2 ( q g e 1 + ( 1 − q ) e 3 ) + T ( q g e 2 + q ( q − 1 ) g 2 ) − Id V q 2 g e 3 = 0. (104) \overline{T}^{3}g-\overline{T}^{2}(qge_1+(1-q)e_3)+T(qge_2+q(q-1)g^2)-\operatorname{Id}_Vq^2ge_3=0. \tag*{(104)} T 3 g − T 2 ( q g e 1 + ( 1 − q ) e 3 ) + T ( q g e 2 + q ( q − 1 ) g 2 ) − Id V q 2 g e 3 = 0. (104) To verify this, an explicit formula is required for the linear map ϕ R \phi_R ϕ R (17) involved in the construction of the powers (18) of the quantum matrix T T T
ϕ R ( T ) = ( 1 q 2 x 1 + λ q 2 x 2 + λ ( q − 1 ) q 2 x 3 v q 3 a + − λ q 2 b + + v 2 q 4 c + 1 q 2 v a − − λ q 2 b − 1 q 2 x 2 + λ q 2 x 3 v 2 q 2 b + 1 q 2 v c − 1 q v b − x 3 ) . (105) \phi_R(T)=
\begin{pmatrix}
\frac{1}{q^2}x_1+\frac{\lambda}{q^2}x_2+\frac{\lambda(q-1)}{q^2}x_3 & \frac{v}{q^3}a_+-\frac{\lambda}{q^2}b_++\frac{v^2}{q^4}c_+\\
\frac{1}{q^2v}a_--\frac{\lambda}{q^2}b_- & \frac{1}{q^2}x_2+\frac{\lambda}{q^2}x_3 & \frac{v^2}{q^2}b_+\\
\frac{1}{q^2v}c_- & \frac{1}{qv}b_- & x_3
\end{pmatrix}. \tag*{(105)} ϕ R ( T ) = q 2 1 x 1 + q 2 λ x 2 + q 2 λ ( q − 1 ) x 3 q 2 v 1 a − − q 2 λ b − q 2 v 1 c − q 3 v a + − q 2 λ b + + q 4 v 2 c + q 2 1 x 2 + q 2 λ x 3 q v 1 b − q 2 v 2 b + x 3 . (105) Direct calculations with the use of Wolfram Mathematica confirm the identity (4.3.8).
Multiparametric orthogonal R-matrices
Here we describe multiparametric O ( N ) O(N) O ( N ) R R R [41 , 42 ] (see also [35 ] ), which can be derived from the standard orthogonal R R R [5 ] by the twist procedure [34 ] . We give a parametrization for these R R R
Let E i j E_{ij} E ij N × N N \times N N × N i ′ : = N + 1 − i i^\prime:= N + 1 - i i ′ := N + 1 − i n : = ⌊ N / 2 ⌋ n := \lfloor N/2 \rfloor n := ⌊ N /2 ⌋
ρ i = − ρ i ′ : = N 2 − i ∀ i = 1 , … , n , \rho_i = -\rho_{i^\prime} := \frac{N}{2} - i \quad\forall\ i = 1,\ldots,n, ρ i = − ρ i ′ := 2 N − i ∀ i = 1 , … , n , and additionally ρ n + 1 = 0 \rho_{n+1}=0 ρ n + 1 = 0 N N N N = 2 n + 1 N=2n+1 N = 2 n + 1 O ( N ) O(N) O ( N )
R = ∑ i , j = 1 N u i j E i j ⊗ E j i + ( q − q − 1 ) ∑ i , j = 1 i < j N E i i ⊗ E j j − ( q − q − 1 ) ∑ i , j = 1 i < j ′ N v i j q − ( ρ i + ρ j ) E i j ⊗ E i ′ j ′ (106) R = \sum_{i,j=1}^{N} u_{ij} E_{ij} \otimes E_{ji} + (q-q^{-1}) \sum_{\substack{i,j=1\\i<j}}^{N} E_{ii} \otimes E_{jj} - (q-q^{-1}) \sum_{\substack{i,j=1\\i<j^\prime}}^{N} v_{ij}q^{-(\rho_i+\rho_j)} E_{ij} \otimes E_{i^\prime j^\prime} \tag*{(106)} R = i , j = 1 ∑ N u ij E ij ⊗ E j i + ( q − q − 1 ) i , j = 1 i < j ∑ N E ii ⊗ E j j − ( q − q − 1 ) i , j = 1 i < j ′ ∑ N v ij q − ( ρ i + ρ j ) E ij ⊗ E i ′ j ′ (106) Here parameters u i j ∈ C ∖ { 0 } u_{ij} \in\mathbb{C} \setminus\{0\} u ij ∈ C ∖ { 0 } u i j u_{ij} u ij N N N
N = 2 n : u i i = q ∀ i = 1 , … , N , u j i = u i j − 1 ∀ i , j = 1 , … , n : i ≠ j , u i ′ j = ( u i j x j ) − 1 , u i j ′ = x i u i j − 1 , u i ′ j ′ = u i j x j x i − 1 ∀ i , j = 1 , … , n , (107) \begin{aligned}
N=2n:\quad&u_{ii}=q\ \forall\ i=1,\ldots,N,\qquad u_{ji}=u_{ij}^{-1}\ \forall\ i,j=1,\ldots,n:i\ne j,\\
&u_{i^\prime j}=(u_{ij}x_j)^{-1},\qquad u_{ij^\prime}=x_i u_{ij}^{-1},\qquad u_{i^\prime j^\prime}=u_{ij}x_jx_i^{-1}\ \forall\ i,j=1,\ldots,n, \tag*{(107)}
\end{aligned} N = 2 n : u ii = q ∀ i = 1 , … , N , u j i = u ij − 1 ∀ i , j = 1 , … , n : i = j , u i ′ j = ( u ij x j ) − 1 , u i j ′ = x i u ij − 1 , u i ′ j ′ = u ij x j x i − 1 ∀ i , j = 1 , … , n , (107) N = 2 n + 1 : u i i = q 1 − δ i , n + 1 ∀ i = 1 , … , N , u j i = u i j − 1 ∀ i , j = 1 , … , n : i ≠ j , u i , n + 1 = u n + 1 , i − 1 = u n + 1 , i ′ = u i ′ , n + 1 − 1 = y i ∀ i = 1 , … , n , u i ′ j = ( u i j y j ) − 1 , u i j ′ = y i 2 u i j − 1 , u i ′ j ′ = u i j y j 2 y i − 2 ∀ i , j = 1 , … , n , (108) \begin{aligned}
N=2n+1:\quad&u_{ii}=q^{1-\delta_{i,n+1}}\ \forall\ i=1,\ldots,N,\qquad u_{ji}=u_{ij}^{-1}\ \forall\ i,j=1,\ldots,n:i\ne j,\\
&u_{i,n+1}=u_{n+1,i}^{-1}=u_{n+1,i^\prime}=u_{i^\prime,n+1}^{-1}=y_i\ \forall\ i=1,\ldots,n,\\
&u_{i^\prime j}=(u_{ij}y_j)^{-1},\qquad u_{ij^\prime}=y_i^2u_{ij}^{-1},\qquad u_{i^\prime j^\prime}=u_{ij}y_j^2y_i^{-2}\ \forall\ i,j=1,\ldots,n, \tag*{(108)}
\end{aligned} N = 2 n + 1 : u ii = q 1 − δ i , n + 1 ∀ i = 1 , … , N , u j i = u ij − 1 ∀ i , j = 1 , … , n : i = j , u i , n + 1 = u n + 1 , i − 1 = u n + 1 , i ′ = u i ′ , n + 1 − 1 = y i ∀ i = 1 , … , n , u i ′ j = ( u ij y j ) − 1 , u i j ′ = y i 2 u ij − 1 , u i ′ j ′ = u ij y j 2 y i − 2 ∀ i , j = 1 , … , n , (108) where parameters x i , y i ∈ C ∖ { 0 } x_i,y_i \in\mathbb{C}\setminus\{0\} x i , y i ∈ C ∖ { 0 } i = 1 , … , n i=1,\ldots,n i = 1 , … , n
For parameters v i j v_{ij} v ij
N = 2 n : v i j = z j z i − 1 , v i ′ j = v i j x i − 1 , v i j ′ = v i j x j ∀ i , j = 1 , … , n , (109) \begin{aligned}
N=2n:\quad&v_{ij}=z_jz_i^{-1},\qquad v_{i^\prime j}=v_{ij}x_i^{-1},\qquad v_{ij^\prime}=v_{ij}x_j\ \forall\ i,j=1,\ldots,n, \tag*{(109)}
\end{aligned} N = 2 n : v ij = z j z i − 1 , v i ′ j = v ij x i − 1 , v i j ′ = v ij x j ∀ i , j = 1 , … , n , (109) N = 2 n + 1 : v i j = z j z i − 1 ∀ i , j = 1 , … , n + 1 , v i ′ j = v i j y i − 2 , v i j ′ = v i j y j 2 ∀ i , j = 1 , … , n , (110) \begin{aligned}
N=2n+1:\quad&v_{ij}=z_jz_i^{-1}\ \forall\ i,j=1,\ldots,n+1,\\
&v_{i^\prime j}=v_{ij}y_i^{-2},\qquad v_{ij^\prime}=v_{ij}y_j^2\ \forall\ i,j=1,\ldots,n, \tag*{(110)}
\end{aligned} N = 2 n + 1 : v ij = z j z i − 1 ∀ i , j = 1 , … , n + 1 , v i ′ j = v ij y i − 2 , v i j ′ = v ij y j 2 ∀ i , j = 1 , … , n , (110) where z i ∈ C ∖ { 0 } z_i \in\mathbb{C}\setminus\{0\} z i ∈ C ∖ { 0 } z 1 = 1 z_1=1 z 1 = 1
Altogether the R-matrix (A.1) contains n ( n + 3 ) / 2 − 1 n(n+3)/2-1 n ( n + 3 ) /2 − 1 N = 2 n N=2n N = 2 n n ( n + 3 ) / 2 n(n+3)/2 n ( n + 3 ) /2 N = 2 n + 1 N=2n+1 N = 2 n + 1
Acknowledgments
The authors thank Dimitry Gurevich and Pavel Saponov for their time, fruitful discussions and valuable remarks. The work of the second author (P. P.) was carried out within the project ”Symmetry. Information. Chaos” (HSE-BR-2025-84) of the Basic Research Program at HSE University.
