# Reciprocal relations for orthogonal quantum matrices

Oleg Ogievetsky, Pavel Pyatov

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Oleg Ogievetsky, Pavel Pyatov. Reciprocal relations for orthogonal quantum matrices. Journal of Geometry and Physics, 2026, 219, pp.105718. 10.1016/j geomphys.2025.105718. hal-05497677

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Reciprocal Relations for Orthogonal
Quantum Matrices

Oleg Ogievetsky$^{◇†}$ and Pavel Pyatov$^{**}$

◇Aix Marseille Université, Université de Toulon, CNRS,
CPT UMR 7332, 13288, Marseille, France

†I.E.Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospekt
53, 119991 Moscow, Russia

*National Research University "Higher School of Economics", 20 Myasnitskaya street, Moscow 101000, Russia

*Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow region, Russia

ABSTRACT

For the family of the orthogonal quantum matrix algebras we investigate the structure of their characteristic subalgebras — special commutative subalgebras, which for the subfamily of the reflection equation algebras appear to be central. In [OP1] we described three generating sets of the characteristic subalgebras of the symplectic and orthogonal quantum matrix algebras. One of these — the set of the elementary sums — is finite. In the symplectic case the elementary sums are in general algebraically independent. On the contrary, in the orthogonal case the elementary sums turn out to be dependent. We obtain a set of quadratic relations for these generators. We call these relations ‘reciprocal’ because they lie at the heart of the reciprocal (sometimes called palindromic) property of the characteristic polynomial of the orthogonal quantum matrices. Next, we resolve the reciprocal relations for the quantum orthogonal matrix algebra extended by the inverse of the quantum matrix. As an auxiliary result, we derive the commutation relations between the $q$-determinant of the quantum orthogonal matrix and the generators of the quantum matrix algebra, that is, the components of the quantum matrix.
# Contents

1 Introduction 2

2 Preliminaries 4

2.1 BMW algebras 4

2.2 Yang-Baxter representations and R-trace 5

2.3 Compatible Yang-Baxter pairs 6

3 Finite height and orthogonal Yang-Baxter matrices 7

3.1 Finite height 7

3.2 Orthogonal case 7

3.3 Rank-one projectors 8

4 Quantum matrix algebra in the orthogonal case 10

4.1 Orthogonal quantum matrix algebra 10

4.2 Characteristic subalgebra 12

4.3 Reciprocal relations 13

4.4 Group-like elements in the orthogonal QM-algebras 15

4.5 Representations of quantum matrix algebras 19

4.6 Resolution of reciprocal relations 22

A Braided coproduct on quantum matrix algebras 23

## 1 Introduction

A concept of Quantum Matrix (shortly, QM-)algebra unifies a variety of algebras arising in the theory of quantum groups, among them most famous are families of the FRT-algebras [RTF] and the Reflection Equation (RE-)algebras [Ch, KS]. Roughly speaking QM-algebra is a unital associative algebra generated by the matrix components with the defining quadratic relations. These relations are expressed with the help of a Yang-Baxter representation of the braid group (for exact definitions see sections 2.2 and 4.1). Fruitfullness of the concept became apparent when it was realized that for these algebras one can universally define generalizations of the matrix power, the matrix trace and determinant, and, finally, derive a generalization of the fundamental theorem of the matrix algebra — the Cayley-Hamilton theorem [IOP1, IOP2]. In so doing, all the notions and results rely exclusively on the properties of the Yang-Baxter matrices appearing in the definition of the QM-algebra.

So far, one usually considers the QM-algebras based on the representations of either Iwahori-Hecke [K, Iw], or Birman-Murakami-Wenzl algebras [BW, Mr]. In the first case we mean the so called $GL(m|n)$ type Yang-Baxter matrices (including the subseries of the type $GL(n)$); the second case decomposes into the subtypes of symplectic $Sp(2n)$ Yang-Baxter matrices, or orthogonal $O(n)$ Yang-Baxter matrices.<fnref n="1" /> The more general case of orthosymplectic $OSp(m|2n)$ Yang-Baxter matrices is less investigated.

We call the QM-algebras by the types of the $R$-matrices used in their construction. The family of $GL(n)$ type QM-algebras is best studied. The structure results, including the analogue of the Cayley-Hamilton theorem, its generalization, Cayley-Hamilton-Newton theorem [IOP], and the thorough investigation of the special commutative subalgebra of the QM-algebra, called the characteristic subalgebra, were carried out in [IOP1, IOP2]. Many particular examples of the $GL(n)$ type QM-algebras were considered in details (see, e.g., [EOW, NT, PS, GPS1, Zh, JW, OV]). The family of $GL(m|n)$ type QM-algebra was investigated in the general setting in [GPS2, GPS3].

<footnote n="1">¹For examples of these classical series of the $R$-matrices the reader is referred to [RTF, I]</footnote>
The program of investigation of the general orthogonal and symplectic types QM-algebras was initiated in the unpublished work [OP]; our main objective was the generalization of the Cayley-Hamilton theorem for the quantum symplectic and orthogonal groups. Later, in [OP1] the characteristic subalgebra of these QM-algebras was described in terms of generators and relations. Various sets of generators were considered and relations among them were derived. In the paper [OP2] on the base of obtained results we proved an analogue of the Cayley-Hamilton theorem for the symplectic QM-algebras. Our next aim is a generalization of the Cayley-Hamilton theorem for the orthogonal QM-algebras [OP3]. In the orthogonal case the generators of the characteristic subalgebra are not algebraically independent and we need to find additional relations among them. These relations, called reciprocal relations will be necessary for the proper understanding of the fundamental notion of the eigenvalue of the orthogonal quantum matrix. These reciprocal relations are derived in the present work.

Before proceeding with considerations let us make a remark about the subfamily of RE-algebras. They are distinguished among the QM-algebras because their characteristic subalgebras are central (for the proof see [I], proposition 5). Besides, they are closely related to the quantized universal enveloping algebras (see [Sem] and, e.g., [JW], section 1.4). Accordingly, earlier results on characteristic identities for the quantized universal enveloping algebras [GZB, MRS] and for their classical analogs (for a review see [IWG]) agree with the results obtained for the RE-algebras. Structure results on the RE-algebras have found numerous applications. These results include but are not limited to, the quantization of the conjugacy classes [DM, Md, AM1, AM2], investigation of differential geometric constructions over quantum groups [IP, P], description of the diagonal reduction algebras [KhO], and the construction of the generalized Capelli identities [GPtS, JLM, Z].

A word about notation. The main objects in this paper are defined with the help of a matrix solution of the Yang-Baxter equation which is an operator in the tensor square $V \otimes V$ where $V$ is a vector space. Such solution is often called R-matrix; however we shall call it a *Yang-Baxter matrix* to avoid word combinations like “R-matrix R”. Also, we shall call a braid group representation, naturally built on a Yang-Baxter matrix, a *Yang-Baxter representation*.

The main text of the paper is organized as follows.

Section 2 contains necessary information about Birman-Murakami-Wenzl (BMW, for short) algebras and their Yang-Baxter representations.

The BMW algebra $W_n(q, \mu)$ is defined in subsection 2.1. Three sets of mutually orthogonal idempotents in $W_n(q, \mu)$ — the $q$-antisymmetrizers $a^{(n)}$, the $q$-symmetrizers $s^{(n)}$ and the contractors $c^{(n)}$ — are introduced here. Yang-Baxter representations of these elements are used in section 4 in the construction of commutative subalgebras in the QM-algebras.

Yang-Baxter representations of the BMW algebras themselves, and the Yang-Baxter matrix techniques including the notions of the R-trace and of the compatible Yang-Baxter matrix pair are introduced in subsections 2.2 and 2.3. Formulas specific for the BMW type Yang-Baxter matrices are collected here.

Orthogonal ($O(k)$ type) Yang-Baxter matrices are introduced in section 3. This particular family of the BMW type Yang-Baxter matrices is used in the definition of the orthogonal QM-algebras. In subsection 3.3 a series of the rank=1 projectors related to the orthogonal Yang-Baxter matrices are investigated. These projectors serve as a main technical tool in considerations in the next section.

Section 4 begins with a general definition the QM-algebra $M(R, F)$ associated to a compatible pair $\{R, F\}$ of the strict skew invertible Yang-Baxter matrices. Imposing additional conditions on the Yang-Baxter matrix $R$ one specializes to the QM-algebras of the BMW and of the orthogonal types. Note that in our definition the BMW type QM-algebras (and also their orthogonal and symplectic subfamilies) contain a nontrivial quadratic element — the contraction $g$. Traditionally, in consideration of examples of such algebras this element is set proportional to 1. Explicitly, $g = \mu^2$1 in the notations of (4.4) (see, e.g., [RTF], definition 11, or [KSch], section 9.3.1). In our studies we keep the contraction free, not only for sake of generality, but mainly because fixing the value of $g$ makes our derivations less clear.
Next, in subsection 4.2 we define special commutative subalgebra $C(R, F) \subset M(R, F)$ — the characteristic subalgebra — and introduce three sets of its generating elements: the so-called power, elementary and complete sums. In the end of the subsection we define a natural extension of the BMW type QM-algebra containing the inverse of the quantum matrix $M$. Material of this subsection is mainly based on [OP1].

Our main result is contained in subsection 4.3. Here in theorem 4.7 we derive algebraic relations specific for the elementary sums in the orthogonal QM-algebras. These relations are called reciprocal.

The subsection 4.4 is devoted to the investigation of a distinguished element of the orthogonal type QM-algebra — the quantum determinant. We obtain the commutation relations of the quantum determinant with the QM-algebra generators. These relations involve the operator $O \in \text{Aut}(V)$ associated with a compatible pair $\{R, F\}$, where $R$ is of the orthogonal type.

Two representations, $\alpha_+$ and $\alpha_-$, of a general QM-algebra are considered in the subsection 4.5. We calculate images of the contraction and of the quantum determinant in these representations. As discussed in subsection 4.6 these data are useful for a resolution of the reciprocal relations. In the end of the subsection 4.5, using the representation $\alpha_+$, we exhibit some properties of the operator $O$.

Our second main result is presented in the final subsection 4.6. In theorem 4.25 we resolve the reciprocal relations for the extended orthogonal QM-algebras (that is, for the algebras extended by the inverse $g^{-1}$) to the contraction $g$. The solution is sensitive to the parity of $k$ of the orthogonal $O(k)$ type QM-algebra. For even $k$, $k = 2\ell$, one has two different solutions associated with the two ‘components’ of the type $O(2\ell)$ QM-algebra. These components are nothing else but the quantum analogues of the two connected components of the orthogonal matrix group. In the odd case, $k = 2\ell-1$, the solution is unique.

The symbol ▶ denotes ends of remarks as well as ends of proofs of auxiliary lemmas placed inside the proofs of theorems (whose ends are denoted by ■).

# 2 Preliminaries

In this section we recall necessary definitions and formulas concerning the Birman–Murakami–Wenzl algebras and their Yang–Baxter representations. This material was discussed in detail in our preceding works [OP1, OP2].

## 2.1 BMW algebras

The Birman–Murakami–Wenzl (BMW) algebra $W_n(q, \mu)$ [BW, Mr] is a finite dimensional quotient algebra of the group algebra $\mathbb{C}[B_n]$ of the braid group $B_n$. It can be defined in terms of generators $\{\sigma_i, \kappa_i\}_{i=1}^{n-1}$ and relations depending on two complex parameters $q$ and $\mu$:<fnref n="2" />

$$ \sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}, \quad \sigma_i\sigma_j = \sigma_j\sigma_i \quad \forall i,j: |i-j| > 1, \quad (2.1) $$

$$ \kappa_i = \frac{(q1-\sigma_i)(q^{-1}1+\sigma_i)}{\mu(q-q^{-1})}, \quad \sigma_i\kappa_i = \kappa_i\sigma_i = \mu\kappa_i, \quad (2.2) $$

$$ \kappa_i \sigma_{i \pm 1}^{\epsilon} = \kappa_i \kappa_{i \pm 1} \sigma_i^{-\epsilon}, \quad \sigma_{i \pm 1}^{\epsilon} \kappa_i = \sigma_i^{-\epsilon} \kappa_{i \pm 1} \kappa_i, \quad \forall \epsilon \in \{1, -1\}, \quad (2.3) $$

$$ \kappa_i \kappa_{i \pm 1} \kappa_i = \kappa_i, \quad \kappa_i^2 = \eta \kappa_i, \quad \text{where } \eta := \frac{(q-\mu)(q^{-1}+\mu)}{\mu(q-q^{-1})}. \quad (2.4) $$

Here $\epsilon$ and $\pi$ are signs: $\epsilon = \pm 1$ and $\pi = \pm 1$. The first line is the Artin’s presentation of the braid group $B_n$; the rest of relations define the quotient algebra $W_n(q, \mu)$ of the group algebra $\mathbb{C}[B_n]$. The parameters $q$ and $\mu$ are taken in domains<fnref n="3" />

$$ q \in \mathbb{C}\{0, \pm 1\}, \quad \mu \in \mathbb{C}\{0, q, -q^{-1}\}, \quad (2.5) $$

<footnote n="2">²The set of relations (2.1)-(2.4) is more concise than the set given in the definition in section 2.1 of [OP1]. However, it defines the same BMW algebra.</footnote>

<footnote n="3">³For particular values $\mu = \pm q^i$, $i \in \mathbb{Z}$, the limiting cases $q \to \pm 1$ to the Brauer algebra [B] can be consistently defined.</footnote>
so that the elements $\kappa_i$ are well defined and non-nilpotent.

Introduce two sets of the so-called baxterized elements $\sigma_i^{\pm}$ [Jon]:

$$ \sigma_i^{\pm}(x) := 1 + \frac{x-1}{q-q^{-1}} \sigma_i + \frac{\mu(x-1)}{\mu \mp q^{\mp 1} x} \kappa_i, \quad x \in \mathbb{C} \setminus \{\pm q^{\pm 1} \mu\}. \qquad (2.6) $$

Using the baxterized elements one defines two series of the idempotents in $W_n$: $a^{(i)}$ and $s^{(i)}$, $i = 1, \dots, n$, called, respectively, the *q-antisymmetrizers* and the *q-symmetrizers*<fnref n="4" />

$$ a^{(1)} := 1, \qquad s^{(1)} := 1, \qquad (2.7) $$

$$ a^{(i+1)} := \frac{q^i}{(i+1)_q} a^{(i)} \sigma_i^-(q^{-2i}) a^{(i)} = \frac{q^i}{(i+1)_q} a^{(i)\uparrow 1} \sigma_1^-(q^{-2i}) a^{(i)\uparrow 1}, \qquad (2.8) $$

$$ s^{(i+1)} := \frac{q^{-i}}{(i+1)_q} s^{(i)} \sigma_i^+(q^{2i}) s^{(i)} = \frac{q^{-i}}{(i+1)_q} s^{(i)\uparrow 1} \sigma_1^+(q^{2i}) s^{(i)\uparrow 1}. \qquad (2.9) $$

Here $i_q$ are the q-numbers, $i_q := (q^i - q^{-i})/(q - q^{-1})$. Symbol $x^{\uparrow 1}$ in these formulas stands for the image of the element $x \in W_n$ (or, more generally, $x \in B_n$) under the monomorphism $W_n \hookrightarrow W_{n+1}$ (respectively, $B_n \hookrightarrow B_{n+1}$) defined on the generators as $(\sigma_i)^{\uparrow 1} = \sigma_{i+1} \quad \forall i = 1, \dots, k-1$. Further on symbol $x^{\uparrow p}$ will denote p-times iteration of this map: $x^{\uparrow p} := (x^{\uparrow (p-1)})^{\uparrow 1}$.

To avoid singularities in the definition of $a^{(i)}$ (respectively, $s^{(i)}$), $i = 2, \dots, n$, we impose additional restrictions on the parameters of $W_n(q, \mu)$:

$$ q^{2j} \neq 1, \quad \mu \neq -q^{-2j+3} \quad (\text{resp., } \mu \neq q^{2j-3}) \quad \forall j = 2, \dots, n. \qquad (2.10) $$

In [OP1] yet another series of the idempotents was introduced:

$$ c^{(0)} := 1, \qquad c^{(2)} := \eta^{-1} \kappa_1, \qquad c^{(2i+2)} := c^{(2i)\uparrow 1} \kappa_1 \kappa_{2i+1} c^{(2i)\uparrow 1} \quad \forall i \ge 1. \qquad (2.11) $$

The element $c^{(2i)}$ is called the 2i-th order contractor. Alternatively, the element $c^{(2i)}$ can be defined by

$$ c^{(2i)} = \eta^{-1} c^{(2i-2)\uparrow 1} (\kappa_{2i-1} \kappa_{2i-2} \dots \kappa_{i+1}) (\kappa_1 \kappa_2 \dots \kappa_i) \quad \forall i \ge 1. \qquad (2.12) $$

Equivalence of the two definitions follows from the relations (see [OP1])

$$ c^{(2n)} c^{(2i)\uparrow n-i} = c^{(2i)\uparrow n-i} c^{(2n)} = c^{(2n)} \quad \forall i = 1, 2, \dots, n. \qquad (2.13) $$

One also has

$$ c^{(2n)} \sigma_i = c^{(2n)} \sigma_{2n-i}, \qquad \sigma_i c^{(2n)} = \sigma_{2n-i} c^{(2n)} \quad \forall i = 1, 2, \dots, n-1. \qquad (2.14) $$

## 2.2 Yang-Baxter representations and R-trace

We remind basic notations and definitions on the Yang-Baxter representations of the BMW algebras. A detailed exposition can be found in [OP1], section 3.

Let $V$ be a finite dimensional $\mathbb{C}$-linear space, $\dim V = N$; $I := \text{Id}_V$. An operator $R \in \text{Aut}(V^{\otimes 2})$ is called an *Yang-Baxter matrix* (respectively, an *Yang-Baxter matrix of a BMW type*) if the map defined on the set of the braid group (respectively, BMW algebra) generators

$$ \sigma_i \mapsto R_i := I^{\otimes (i-1)} \otimes R \otimes I^{\otimes (n-i-1)} \in \text{Aut}(V^{\otimes n}), \quad 1 \le i \le n-1, \qquad (2.15) $$

extends to a homomorphism

$$ \rho_R : B_n \quad (\text{respectively, } W_n) \to \text{End}(V^{\otimes n}). \qquad (2.16) $$

<footnote n="4">⁴Coincidence of the two expressions for q-(anti)symmetrizers in (2.8),(2.9) and their further properties are explained in [OP1], proposition 2.2.</footnote>
The homomorphism $\rho_R$ is called the *Yang-Baxter representation* of the group $\mathcal{B}_n$ (resp., the algebra $W_n$). Image of the braid relation (2.1) in the representation $\rho_R$

$$R_1 R_2 R_1 = R_2 R_1 R_2, \qquad (2.17)$$

is traditionally called the *braid relation*. This matrix equality is sufficient for the definition of the Yang-Baxter matrix $R$ (that is, it is enough to take $n = 3$ in (2.16)). If $R$ is the Yang-Baxter matrix, then so is $R^{-1}$.

An operator $R \in \text{End}(V^{\otimes 2})$ is called *skew invertible* if there exists an operator $\Psi_R \in \text{End}(V^{\otimes 2})$ such that

$$\text{Tr}_{(2)} R_1 (\Psi_R)_2 = \text{Tr}_{(2)} (\Psi_R)_1 R_2 = P. \qquad (2.18)$$

Here $(\Psi_R)_i$, $i = 1, 2$, are treated as elements of $\text{End}(V^{\otimes 3})$, like in (2.15), $(\Psi_R)_1 := \Psi_R \otimes I$, $(\Psi_R)_2 := I \otimes \Psi_R$. The symbol $\text{Tr}_{(i)}$ stands for the trace over an $i$-th component in the tensor product of the spaces $V$. The symbol $P \in \text{Aut}(V^{\otimes 2})$ is reserved for the permutation operator: $P(u \otimes v) = v \otimes u \quad \forall u, v \in V$. In the following, for an operator $Z$ in the tensor power of the space $V$, $\text{Tr}_{(i_1, i_2, \dots, i_L)}$ stands for the trace over the components $i_1, i_2, \dots, i_L$ in the tensor product.

With the skew invertible Yang-Baxter matrix $R$ one associates a pair of matrices

$$(C_R)_2 := \text{Tr}_{(1)}(\Psi_R)_1 \in \text{End}(V) \quad \text{and} \quad (D_R)_1 := \text{Tr}_{(2)}(\Psi_R)_1 \in \text{End}(V). \qquad (2.19)$$

The meaning of the indices 1 and 2 in the symbols $(D_R)_1$ and $(C_R)_2$ is as follows. For an operator $Z \in \text{End}(V)$ the symbol $Z_i$ stands for the operator which acts as $Z$ in the $i$-th component in the tensor power of the space $V$ and as the identity in all other components. In the sequel we shall often omit the brackets in the symbols $(D_R)_1$ and $(C_R)_2$ and write simply $D_{R1}$ and $C_{R2}$.

The characteristic properties of operators $C$ and $D$ are, respectively,

$$\text{Tr}_{(1)} C_{R1} R_1 = I_2 \quad \text{and} \quad \text{Tr}_{(2)} D_{R2} R_1 = I_1. \qquad (2.20)$$

As shown in, e.g., [O], invertibility of (any of) the matrices $C_R$, $D_R$ is equivalent to the skew invertibility of the Yang-Baxter matrix $R^{-1}$ and moreover

$$C_R D_{R^{-1}} = D_R C_{R^{-1}} = I. \qquad (2.21)$$

Skew invertible Yang-Baxter matrix whose inverse matrix is also skew invertible is called *strict skew invertible*. For a Yang-Baxter matrix $R$ of the BMW type, one has [IOP3]

$$C_R D_R = \mu^2 I \qquad (2.22)$$

and thus, for the BMW type Yang-Baxter matrices skew invertibility implies strict skew invertibility.

With the matrix $D_R$ we associate a linear map on the space of $N \times N$ matrices whose entries belong to a $\mathbb{C}$-linear space $W$

$$\text{Tr}_R : \text{End}(V) \otimes W \to W, \quad \text{Tr}_R(X) = \sum_{i,j=1}^{N} (D_R)^{ij} X_j^i, \quad X \in \text{End}(V) \otimes W,$$

This map is called the *R-trace*. As for $\text{Tr}_{(i)}$, the symbol $\text{Tr}_{R(i)}$ stands for the R-trace over an $i$-th component in the tensor product of the spaces $V$. Similarly, for the symbol $\text{Tr}_{R(i_1, i_2, \dots, i_L)}$.

Let us introduce special notation for the images of the elements $\kappa_i \in W_n$ in the Yang-Baxter representation

$$K_i := \rho_R(\kappa_i).$$

By definition, the matrix $K = \rho_R(\kappa_1) \in \text{End}(V^{\otimes 2})$ is degenerate. As shown in [IOP3], rank of the operator $K$ corresponding to the skew invertible BMW type Yang-Baxter matrix equals 1:

$$\text{rk} K = 1. \qquad (2.23)$$
The following set of relations for a skew invertible BMW type Yang–Baxter matrix R and for related matrices $D_R$, $K$, $\rho_R(a^{(i)})$ will be used in the main sections of the text. The proofs can be found in [OP1], sections 3.2, 3.4, and [OP2], section 2.4.

$$R_1 D_{R1} D_{R2} = D_{R1} D_{R2} R_1, \qquad (2.24)$$

$$K_1 D_{R1} D_{R2} = D_{R1} D_{R2} K_1 = \mu^2 K_1 \qquad (2.25)$$

$$\mathrm{Tr}_{R(2)} K_1 = \mu I_1, \qquad (2.26)$$

$$\mathrm{Tr}_R I = \frac{(q-\mu)(q^{-1}+\mu)}{(q-q^{-1})}, \qquad (2.27)$$

$$\mathrm{Tr}_{R(i)} \rho_R(a^{(i)}) = \delta_i(q, \mu) \rho_R(a^{(i-1)}), \qquad (2.28)$$

$$\text{where } \delta_i(q, \mu) := -q^{i-1} \frac{(\mu+q^{1-2i})(\mu^2-q^{4-2i})}{(\mu+q^{3-2i})(q^i-q^{-i})}.$$

## 2.3 Compatible Yang–Baxter pairs

An ordered pair {$R, F$} of two Yang–Baxter matrices R and F is called a compatible Yang–Baxter pair if the following conditions

$$R_1 F_2 F_1 = F_2 F_1 R_2, \qquad R_2 F_1 F_2 = F_1 F_2 R_1, \qquad (2.29)$$

are satisfied. The equalities (2.29) are sometimes called *twist relations*. Clearly, {$R, P$} and {$R, R$} are compatible Yang–Baxter pairs.

A compatible Yang–Baxter pair {$R, F$} gives rise to a new Yang–Baxter matrix

$$R_F := F^{-1} R F, \qquad (2.30)$$

called the *twisted* Yang–Baxter matrix. It is known that the Yang–Baxter pair {$R_F, F$} is again compatible. If R is skew invertible and F is strict skew invertible, then $R_F$ is skew invertible; if additionally R is strict skew invertible, then $R_F$ is strict skew invertible as well (for proofs, see [OP1], proposition 3.6). For a compatible pair {$R, F$} of the skew invertible matrices the relation

$$F_1 D_{R1} D_{R2} = D_{R1} D_{R2} F_1, \qquad (2.31)$$

is satisfied (see [OP1], corollary 3.4).

# 3 Finite height and orthogonal Yang–Baxter matrices

## 3.1 Finite height

In [OP2] we have introduced a notion of the *finite height* BMW type Yang–Baxter matrix. Namely, assuming that the parameters $q$ and $\mu$ of the BMW type Yang–Baxter matrix satisfy conditions (the union of conditions (2.5), (2.10))

$$q \neq 0, \quad q^{2i} \neq 1; \quad \mu \notin \{0, q\}, \quad \mu \neq -q^{-2i+1} \quad \forall i = 1, \dots, k, \qquad (3.1)$$

we consider Yang–Baxter matrix realizations $\rho_R(a^{(i)})$ of the $q$-antisymmetrizers (2.8). The skew invertible BMW type Yang–Baxter matrix R is called the *Yang–Baxter matrix of height k* if its $q$-antisymmetrizers satisfy conditions

$$\mathrm{rk} \rho_R(a^{(i)}) \neq 0 \quad \forall i = 2, 3, \dots, k, \quad \text{and} \quad \rho_R(a^{(k)\uparrow 1} \sigma_1^- (q^{-2k}) a^{(k)\uparrow 1}) \equiv 0 \qquad (3.2)$$

for some $k \ge 2$.

As discussed in [OP2], section 2.4, it is impossible to construct the height k BMW type Yang–Baxter matrix unless $\mu \in \{-q^{-1-2k}, \pm q^{1-k}\}$. Taking $\mu = -q^{-1-2k}$ one obtains series of the so called
symplectic Sp(2k) type Yang-Baxter matrices. The corresponding quantum matrix algebras were investigated in [OP2].

The choice $\mu = -q^{1-k}$ for even values of $k$ contradicts conditions (3.1). Besides, for odd values of $k$ it can be transformed to the case $\mu = q^{1-k}$ by the map: $R \mapsto -R$. Therefore, a consideration of the case $\mu = q^{1-k}$ completes the investigation of the finite height BMW type Yang-Baxter matrices and their corresponding QM-algebras. In the present work we are dealing with a family of the Yang-Baxter matrices belonging to the case $\mu = q^{1-k}$.

## 3.2 Orthogonal case

**Definition 3.1** Let $R$ be the skew invertible BMW type Yang-Baxter matrix of the height $k \ge 2$ whose two spectral values $q$ and $\mu$ are related by equality $\mu = q^{1-k}$. Assume additionally that

$$rk \rho_R(a^{(k)}) = 1. \qquad (3.3)$$

Then $R$ is called an orthogonal Yang-Baxter matrix of the type $O(k)$.

**Remark 3.2** We remind that for the $GL(k)$ type Yang-Baxter matrices, that is, the Hecke type Yang-Baxter matrices of the height $k$, the strict skew invertibility implies the relation (3.3) [G]. However, for the BMW type finite height Yang-Baxter matrices this is not the case. E.g., for the family of symplectic $Sp(2k)$ type Yang-Baxter matrices, where the parameters of the BMW algebra are related by $\mu = -q^{1-2k}$ (see [OP2], section 2.4), the rank one condition is not valid for $k > 1$. In the case $\mu = q^{1-k}$ a widely known family of Yang-Baxter matrices satisfying condition (3.3) is related to the orthogonal quantum groups [RTF]. We believe that in this case the finiteness of the height together with the skew invertibility implies the rank one property. This is an interesting open question.

▶

## 3.3 Rank-one projectors

In this subsection we investigate specific properties of certain rank-one projectors which can be constructed from the orthogonal Yang-Baxter matrices. Later on, these projectors will play a central role in the proof of the reciprocal relations (theorem 4.7) and in the investigation of the quantum determinant (proposition 4.12).

First, we observe that any skew invertible BMW type Yang-Baxter matrix gives rise to a series of rank-one projectors.

**Lemma 3.3** Assume that $R$ is a skew invertible Yang-Baxter matrix of the BMW type. Then $\rho_R(c^{(2i)}) \in \text{End}(V^{\otimes 2i})$ (for the definition of the idempotents $c^{(2i)}$, see eqs. (2.11))$ are rank-one projectors fulfilling relations

$$\mathrm{Tr}_{R(2i)} \rho_R(c^{(2i)}) = \eta^{-1} \mu \rho_R(c^{(2i-2)\uparrow 1}) \quad \forall i \ge 1, \qquad (3.4)$$

$$\mathrm{Tr}_{R(i+1, i+2, \dots, 2i)} \rho_R(c^{(2i)}) = (\eta^{-1} \mu)^i I^{\otimes i}. \qquad (3.5)$$

**Proof.** For $i = 1$, the relation (3.4) is just the equality (2.26). In case $i > 1$, we check the equation (3.4) by the direct calculation

$$
\begin{align*}
\mathrm{Tr}_{R(2i)} \rho_R(c^{(2i)}) &= \eta^{-1} \rho_R(c^{(2i-2)\uparrow 1}) (\mathrm{Tr}_{R(2i)} K_{2i-1}) \rho_R((\kappa_{2i-2}\kappa_{2i-3} \dots \kappa_{i+1})(\kappa_1\kappa_2 \dots \kappa_i)) \\
&= \eta^{-1} \mu \rho_R(c^{(2i-2)\uparrow 1} \kappa_{2i-2}\kappa_{2i-3} \dots \kappa_{i+1}(\kappa_1\kappa_2 \dots \kappa_i)) \\
&= \eta^{-1} \mu \rho_R(c^{(2i-2)\uparrow 1} \kappa_{i-1} \dots \kappa_3\kappa_2(\kappa_1\kappa_2 \dots \kappa_i)) \\
&= \eta^{-1} \mu \rho_R(c^{(2i-2)\uparrow 1} \kappa_{i-1}\kappa_i) = \eta^{-1} \mu \rho_R(c^{(2i-2)\uparrow 1}).
\end{align*}
$$
Here in the first line we substituted the expression (2.12) for 2i-th order contractors. In the second line we used the relation (2.26). In the third line we applied repeatedly the relations (2.14) for the elements $\kappa_{2i-2}, \dots, \kappa_{i+1}$ (they are quadratic polynomials in $\sigma$'s); after each application of the relation (2.14) we move the resulting $\kappa$ to the right. In the last line we used the relation (2.4) ($i-2$) times. Then, noticing that the element $\kappa_i$ divides $c^{(2i-2)\uparrow 1}$ from the right (see (2.12)) and applying once again eq. (2.4) we complete the transformation.

The relation (3.5) follows by a repeated application of the relation (3.4).

As we mentioned before, the rank one property of the 2-contractor $\rho_R(c^{(2)}) = \eta^{-1}K$ associated to a skew invertible BMW type Yang-Baxter matrix was proved in [IOP3]. To calculate the rank of the operators $\rho_R(c^{(2i)}) \in \text{End}(V^{\otimes 2i})$, $i > 1$, we notice that using iteratively the relations (2.13) one can rewrite the formula (2.11) for the contractors in the form

$$c^{(2i)} = \eta^{-\varepsilon(i)} c^{(2i-2)\uparrow 1} \left( \prod_{j=1}^{i} \kappa_{2j-1} \right) c^{(2i-2)\uparrow 1}, \qquad (3.6)$$

where $\varepsilon(i)$ is the parity function: $\varepsilon(i) = 0/1$ if $i$ is even/odd. The appearance of the factor $\eta^{-1}$ for odd values of $i$ is due to the fact that for odd $i$ the iterative procedure terminates with $c^{(2)\uparrow(i-1)} = \eta^{-1}K_i$. We leave further details to the reader.

By the rank-one property of the operator $K \in \text{End}(V^{\otimes 2})$, a composite operator $\rho_R(\prod_{j=1}^{i} \kappa_{2j-1}) = \prod_{j=1}^{i} K_{2j-1} \in \text{End}(V^{\otimes 2i})$ also has rank one. Hence, the operator $\rho_R(c^{(2i)})$ is either of rank-one or identically vanishes. The latter possibility contradicts the relation (3.5). ■

We next describe specific properties of the rank-one projectors.

**Lemma 3.4** 1) Let R be a skew invertible Yang-Baxter matrix of the BMW type.<fnref n="5" /> Then

$$(\prod_{j=1}^{2i} D_{R_j}) \rho_R(c^{(2i)}) = \mu^{2i} \rho_R(c^{(2i)}) \quad \forall i \ge 1. \qquad (3.7)$$

Let R be a Yang-Baxter matrix of the type O(k). Then

$$(\prod_{j=1}^{k} D_{R_j}) \rho_R(a^{(k)}) = q^{k(1-k)} \rho_R(a^{(k)}). \qquad (3.8)$$

2) Let additionally F be a strict skew invertible Yang-Baxter matrix and assume that $\{R, F\}$ is a compatible pair. Then one has, respectively,

$$(\prod_{j=1}^{2i} D_{(R_F)_j}) \rho_R(c^{(2i)}) = \mu^{2i} \rho_R(c^{(2i)}) \quad \forall i \ge 1 \qquad (3.9)$$

if the Yang-Baxter matrix R is strict skew invertible of the BMW type, and

$$(\prod_{j=1}^{k} D_{(R_F)_j}) \rho_R(a^{(k)}) = q^{k(1-k)} \rho_R(a^{(k)}) \qquad (3.10)$$

if R is of the type O(k).

**Proof.** We first prove the equality (3.7). Substitute the expression (3.6) into the left hand side of (3.7). By the left equality in (2.25), we can change the order of operators $\rho_R(c^{(2i-2)\uparrow 1})$ and $\prod_{j=1}^{2i} D_{R_j}$. Then, by the right equality in (2.25), the operator $\prod_{j=1}^{2i} D_{R_j}$ in the product $\prod_{j=1}^{2i} D_{R_j} \cdot \rho_R(\prod_{j=1}^{i} \kappa_{2j-1})$ can be replaced by the factor $\mu^{2i}$.

The equality (3.8) is confirmed by the calculation

$$\begin{aligned} (\prod_{j=1}^{k} D_{R_j}) \rho_R(a^{(k)}) &= \rho_R(a^{(k)}) (\prod_{j=1}^{k} D_{R_j}) \rho_R(a^{(k)}) = (\text{Tr}_{R(1, 2, \dots, k)} \rho_R(a^{(k)})) \rho_R(a^{(k)}) \\ &= q^{k(1-k)} \rho_R(a^{(k)}). \end{aligned}$$

<footnote n="5">⁵Recall that for the BMW type the skew invertibility is equivalent to the strict skew invertibility.</footnote>
Here the idempotency of the operator $\rho_R(a^{(k)})$ and its commutativity with the product $\prod_{j=1}^k D_{R_j}$ (by the relation (2.24)) were taken into account in the first equality; the second equality follows by the rank-one property of the operator $\rho_R(a^{(k)})$. Eqs.(2.28) were used for the evaluation of the R-traces in the second line. Namely,

$$ \mathrm{Tr}_{R(1, 2, \dots, i)} \rho_R(a^{(i)}) = \prod_{j=1}^{i} \delta_j(q, \mu)|_{\mu=q^{1-i}} = q^{-ki} \frac{q^{2i} + q^k}{1 + q^k} \frac{k_q!}{i_q!(k-i)_q!}, \quad (3.11) $$

where $j_q! := 1_q 2_q 3_q \dots j_q$.

The conditions of the second part of the proposition guarantee that the twisted Yang-Baxter matrix $R_F = F^{-1}RF$ is skew invertible of the BMW type. Hence, the relations (3.7) and (3.8) are satisfied for $R_F$ as well. Now, to prove the equalities (3.9) and (3.10), it is enough to demonstrate that the two operators $\rho_{R_F}(c^{(2i)})$ and $\rho_R(c^{(2i)})$ (respectively, $\rho_{R_F}(a^{(k)})$ and $\rho_R(a^{(k)})$) are related by a similarity transformation, which commutes with $\prod_{j=1}^{2i} D_{(R_F)_j}$ (respectively, $\prod_{j=1}^k D_{(R_F)_j}$).

Define a set of operators $Z^{(i)} \in \mathrm{Aut}(V^{\otimes i})$, $i = 1, 2, \dots$, by the iteration<fnref n="6" />

$$ Z^{(1)} := I, \quad Z^{(2)} := F_1, \quad Z^{(i+1)} := (F_1 F_2 \dots F_i) Z^{(i)} = Z^{(i)} (F_i \dots F_2 F_1). \quad (3.12) $$

By induction, one can readily prove the relations

$$ R_i Z^{(k)} = Z^{(k)} (R_F)_{(k-i)} \quad \forall i = 1, 2, \dots, k-1. \quad (3.13) $$

Namely, for $i = 2, \dots, k-1$, one has, by induction on $k$,

$$ \begin{aligned} R_i Z^{(k)} &= R_i F_1 F_2 \dots F_{k-1} Z^{(k-1)} = F_1 F_2 \dots F_{k-1} R_{i-1} Z^{(k-1)} \\ &= F_1 F_2 \dots F_{k-1} Z^{(k-1)} (R_F)_{(k-i)} = Z^{(k)} (R_F)_{(k-i)}. \end{aligned} $$

Here we used the twist relation for the compatible pair $\{R, F\}$ and the first iterative definition of the element $Z^{(k)}$ in eq.(3.12).

For $i = 1$, one has, again by induction on $k$,

$$ \begin{aligned} R_1 Z^{(k)} &= R_1 Z^{(k-1)} F_{k-1} \dots F_2 F_1 = Z^{(k-1)} (R_F)_{(k-2)} F_{k-1} \dots F_2 F_1 \\ &= Z^{(k-1)} F_{k-1} \dots F_2 F_1 (R_F)_{(k-1)} = Z^{(k)} (R_F)_{(k-1)}. \end{aligned} $$

Here we used the twist relation for the pair $\{R_F, F\}$ and the second iterative definition of the element $Z^{(k)}$ in eq.(3.12).

The relations (3.13) imply following equalities

$$ \rho_R(a^{(k)}) Z^{(k)} = Z^{(k)} \rho_{R_F}(a^{(k)}), \quad (3.14) $$

$$ \rho_R(c^{(2i)}) Z^{(2i)} = Z^{(2i)} \rho_{R_F}(c^{(2i)}) \quad \forall i = 1, 2, \dots \quad (3.15) $$

To verify eq.(3.14), substitute different expressions from eq.(2.8) for the $q$-antisymmetrizers, one expression for $\rho_R(a^{(k)})$ and another one for $\rho_{R_F}(a^{(k)})$; then use eq.(3.13). As for eq.(3.15), note that the expression (2.11) for the contractor $c^{(2i)}$ is invariant with respect to the reflection $\sigma_j \leftrightarrow \sigma_{2i-j}$. Thus it can be used in both sides of eq.(3.15).

Due to the relations (2.31) for the compatible pair $\{R_F, F\}$, the operators $Z^{(l)}$ and $\prod_{j=1}^l D_{(R_F)_j}$ commute for any $l$ and, hence, the operator $Z^{(2i)}$ (respectively, the operator $Z^{(k)}$) realizes the similarity transformation we are looking for. ■

<footnote n="6">⁶Two expressions for the element $Z^{(i+1)}$ in eq.(3.12) correspond to two different decompositions of the lift, to the braid group $B_{i+1}$, of the longest element of the symmetric group $S_{i+1}$.</footnote>
**Remark 3.5** For a strict skew invertible Yang-Baxter matrix of the Hecke type of height $k$ the identity (3.8) holds as well. If the Yang-Baxter matrix $R$ is strict skew invertible of the Hecke type, $F$ is strict skew invertible Yang-Baxter matrix and $\{R, F\}$ is a compatible pair then the identity (3.10) also holds. The above proof holds in the Hecke case. ►

# 4 Quantum matrix algebra in the orthogonal case

In this section we introduce the main objects of our study, the quantum matrix (shortly, QM-) algebras of the orthogonal type.

In subsections 4.1-4.2 we collect some known results about the BMW type QM-algebras, in particular, about the structure of their distinguished commutative subalgebra, called the characteristic subalgebra. Additional reciprocal relations, specific to the orthogonal type QM-algebras, satisfied by generators of the characteristic subalgebra are derived in subsection 4.3 and resolved in subsection 4.6. Subsections 4.4 and 4.5 present results, respectively, on the properties of the quantum determinant (analogue of the determinant for the quantum matrix) and on two specific representations of the orthogonal QM-algebra. These results are used in the solution of the reciprocal relations.

## 4.1 Orthogonal quantum matrix algebra

Here we remind necessary definitions and few basic results about quantum matrix algebras of BMW type. For a detailed exposition the reader is referred to [OP1]. We remind that $V$ is a finite dimensional $\mathbb{C}$-linear space, $\dim V = n$; the operators $R$ and $F$ belong to $\mathrm{Aut}(V^{\otimes 2})$.

**Definition 4.1** Assume that $\{R, F\}$ is a compatible pair of the strict skew invertible Yang-Baxter matrices. The QM-algebra $\mathcal{M}(R, F)$ is a quotient algebra of the free associative unital algebra $W = \mathbb{C}\langle M_b^a \rangle_{a,b=1}^N$ by a two-sided ideal generated by entries of the matrix relation

$$R_1 M_{\bar{1}} M_{\bar{2}} = M_{\bar{1}} M_{\bar{2}} R_1. \qquad (4.1)$$

Here $M = \|M_b^a\|_{a,b=1}^N$ is the matrix of generators; the matrix copies $M_{\bar{i}}$, $i \ge 1$, are constructed by the following iteration

$$M_{\bar{1}} := M_1, \quad M_{\bar{i}} := F_{i-1} M_{\bar{i}-1} F_{i-1}^{-1}. \qquad (4.2)$$

The QM-algebra $\mathcal{M}(R, F)$ is called BMW/O($k$) type if, respectively, the Yang-Baxter matrix $R$ is of BMW/O($k$) type.

We call the matrix $M$ whose entries are the generators of the QM-algebra $\mathcal{M}(R, F)$ the quantum matrix.

For any given value of the index $i \ge 1$ the set of relations

$$R_i M_{\bar{i}} M_{\bar{i}+1} = M_{\bar{i}} M_{\bar{i}+1} R_i \qquad (4.3)$$

is equivalent to (4.1) (see [OP1], lemma 4.3).

For the BMW type QM-algebra the relations

$$K_i M_{\bar{i}} M_{\bar{i}+1} = M_{\bar{i}} M_{\bar{i}+1} K_i = \mu^{-2} g K_i \quad \forall i \ge 1, \qquad (4.4)$$

where

$$g := \mu^2 \mathrm{Tr}_{(1,2)} (M_{\bar{1}} M_{\bar{2}} \rho_R (c^{(2)})) = \eta^{-1} \mathrm{Tr}_{R(1,2)} (M_{\bar{1}} M_{\bar{2}} K_1), \qquad (4.5)$$

are satisfied as a consequence of eqs. (4.3), (2.25) and the rank-one property of the matrix $K$ (see [OP1], lemma 4.6). The element $g$ is called the contraction of $M$.
**Remark 4.2** (i) For orthogonal group $O(n)$, $n \ge 2$, the tensor square of the vector representation $V$ decomposes into the direct sum of three irreducible representations for each $n$ except $n = 4$. For $n = 4$ the tensor square of the vector representation is the direct sum of four irreducible representations: the wedge square of $V$ is a direct sum of self-dual and anti-self-dual tensors. This geometry quantizes and there are four $q$-deformed projectors (see [OSWZ1, OSWZ2]). This is also explained by the $q$-deformation of accidental isomorphisms of semi-simple Lie groups [JO]. In such situation the QM-algebra must be defined by the union of equations

$$ \Pi_{\alpha} M_{\bar{1}} M_{\bar{2}} = M_{\bar{1}} M_{\bar{2}} \Pi_{\alpha} \qquad (4.6) $$

for all projectors $\Pi_{\alpha}$ $q$-deforming the decomposition of the tensor square of $V$ into the direct sum of irreducibles.

(ii) We believe that for any $O(4)$-type Yang-Baxter matrix, admitting the classical limit, there exists a non-trivial decomposition of the second $q$-antisymmetrizer (acting in the tensor square of the vector representation) into a sum of two projectors, generalizing the situation with the standard $q$-deformation of $O_4$.

(iii) The fact that in general the QM-algebra is a quotient of the algebra defined by the eq.(4.1) does not affect the main objective of our study – the characteristic equation for the quantum matrices. This should be compared with the perfectly classical situation – the characteristic polynomial $\chi_A(t)$ of a matrix $A$ is always just $\chi_A(t) := \det(tI - A)$, regardless of the type of the matrix $A$ (general linear, symplectic, orthogonal, ...) Note however that the type of the matrix affects the properties of its eigenvalues.

(iv) Let $g$ be a semi-simple Lie algebra, $Ug$ its universal enveloping algebra and $U_{qg}$ its standard $q$-deformation. Let $\rho$ be an irreducible representation of $g$ in a vector space $V$, and $\rho_q$ its $q$-deformation. The image of the universal $\mathcal{R}$-matrix may not distinguish all the projectors in the tensor square of $V$, as it is the case for $O(4)$. In such a situation the quantum matrix algebra should be defined by the union of equations (4.6) as above. However, in the $O(4)$ case the situation is quite beautiful: one can construct, out of projectors $\Pi_{\alpha}$ two Yang-Baxter matrices [OSWZ1] (which, of course commute, since they are built of the projectors entering the decomposition of unity). These two matrices fully serve the geometry of the $q$-deformed 4-dimensional orthogonal space. We believe that this will be the case for all $g$ and $\rho$ for which the image of the universal $\mathcal{R}$-matrix does not distinguish all the projectors in the tensor square of $V$ (maybe there will be more than two Yang-Baxter matrices) but this is an open question.

▶

## 4.2 Characteristic subalgebra

**Definition-Proposition 4.3** [IOP1] Let $\mathcal{C}(R, F)$ be a subspace in $\mathcal{M}(R, F)$ which is spanned linearly by the unity and elements

$$ \mathrm{ch}(\alpha^{(n)}) := \mathrm{Tr}_{R(1, \dots, n)} (M_{\bar{1}} \dots M_{\bar{n}} \rho_R(\alpha^{(n)})), \quad n = 1, 2, \dots, \qquad (4.7) $$

where $\alpha^{(n)}$ is an arbitrary element of the braid group $B_n$. The space $\mathcal{C}(R, F)$ is a commutative subalgebra in $\mathcal{M}(R, F)$. It is called the characteristic subalgebra of $\mathcal{M}(R, F)$.

**Remark 4.4** For the family $\mathcal{M}(R, R)$ of the so-called reflection equation algebras their characteristic subalgebras lie in their centers: $\mathcal{C}(R, R) \subset Z[\mathcal{M}(R, R)]$ (see [1], proposition 5). ▶

**Proposition 4.5** ([IOP1], propositions 4.7, 4.8) The characteristic subalgebra of the BMW type QM-algebra is generated by either of the sets:

1. $\{g, \mathfrak{p}_i\}_{i \ge 0}$, where elements

$$ \mathfrak{p}_0 := \mathrm{Tr}_R I = \mu\eta, \quad \mathfrak{p}_1 := \mathrm{Tr}_R M, \quad \mathfrak{p}_i := \mathrm{ch}(\sigma_{i-1} \dots \sigma_2 \sigma_1), \quad \forall i \ge 2, \qquad (4.8) $$

are called power sums;
2. $\{g, \mathfrak{h}_i\}_{i \ge 0}$, where elements
$$ \mathfrak{h}_0 := 1, \quad \mathfrak{h}_i := \mathrm{ch}(s^{(i)}), \forall i \ge 1, \qquad (4.9) $$
are called complete sums;

3. $\{g, \epsilon_i\}_{i \ge 0}$, where elements
$$ \epsilon_0 := 1, \quad \epsilon_i := \mathrm{ch}(a^{(i)}), \forall i \ge 1, \qquad (4.10) $$
are called elementary sums.

Here in the definitions of the last two sets we assume additionally that conditions (2.10) on the parameters $q$ and $\mu$ are fulfilled.

Our definition of the QM-algebra does not assume invertibility of the quantum matrix $M$ with respect to the usual matrix product. Now we introduce a natural extension of the BMW type QM-algebra containing the inverse of the matrix $M$.

**Definition-Proposition 4.6** ([OP1], lemma 4.13 and proposition 4.14) Let $\mathcal{M}^*(R, F)$ and $\mathcal{C}^*(R, F)$ denote localizations of the BMW type QM-algebra $\mathcal{M}(R, F)$ and of its characteristic subalgebra $\mathcal{C}(R, F)$ by adding the inverse $g^{-1}$ of the contraction

$$ g^{-1}g = gg^{-1} = 1. \qquad (4.11) $$

The contraction and its inverse satisfy the following permutation relations with the QM-algebra generators

$$ Mg = g(G^{-1}MG), \quad \text{and so} \quad g^{-1}M = (G^{-1}MG)g^{-1}, \qquad (4.12) $$

where the numeric mutually inverse matrices $G^{\pm 1} \in \mathrm{Aut}(V)$ are given by the formulas

$$ G_1 := \mathrm{Tr}_{(23)} K_2 F_1^{-1} F_2^{-1}, \quad G_1^{-1} = \mathrm{Tr}_{(23)} F_2 F_1 K_2. \qquad (4.13) $$

The algebra $\mathcal{M}^*(R, F)$ contains the inverse to the quantum matrix $M$,

$$ (M^{-1})_1 = \mu \mathrm{Tr}_{R(2)} (M_2 K_1) g^{-1}, \qquad (4.14) $$

so that

$$ MM^{-1} = M^{-1}M = I. $$

## 4.3 Reciprocal relations

For the QM-algebra $\mathcal{M}(R, F)$ where $R$ is of finite height, the set of the elementary sums becomes finite. Moreover, for the orthogonal $O(k)$ type QM-algebra, the rank-one condition (3.3) for the operator $\rho_R(a^{(k)})$ gives rise to algebraic equalities among the elementary sums $\{\epsilon_i\}_{0 \le i \le k}$ and the contraction $g$. These equalities — we call them *reciprocal relations* — are described in the next theorem.

**Theorem 4.7** Let the QM-algebra $\mathcal{M}(R, F)$ be of the orthogonal $O(k)$ type. Then the following reciprocal relations for the generators $\{g, \epsilon_i\}_{i=0}^k$ of its characteristic subalgebra

$$ g^{k-i} \epsilon_i = \epsilon_k \epsilon_{k-i} \quad \forall i = 0, 1, \dots, k-1, \qquad (4.15) $$

are satisfied.

**Proof.** For the proof, we use three auxiliary statements.
**Lemma 4.8** Let the QM-algebra $\mathcal{M}(R, F)$ be of the BMW type, respectively, of the $O(k)$ type. Then we have, respectively,

$$ M_1^\top M_2 \dots M_{2i} \rho_R(c^{(2i)}) = (\mu^{-2}g)^i \rho_R(c^{(2i)}) \quad (4.16) $$

in the BMW case and

$$ M_1^\top M_2 \dots M_k \rho_R(a^{(k)}) = q^{k(k-1)} \mathfrak{e}_k \rho_R(a^{(k)}) \quad (4.17) $$

in the $O(k)$ case.

**Proof.** The relations (4.16) and (4.17) are proved in exactly the same way as the relations (3.7) and (3.8) of the lemma 3.4. One has to use additionally the relation (4.4) for the proof of the relation (4.16). For the relation (4.17), the equality

$$ \mathrm{Tr}_{(1,2,\dots,k)}(M_1^\top M_2 \dots M_k \rho_R(a^{(k)})) = q^{k(k-1)} \mathrm{Tr}_{R(1,2,\dots,k)}(M_1^\top M_2 \dots M_k \rho_R(a^{(k)})) = q^{k(k-1)} \mathfrak{e}_k, \quad (4.18) $$

following from the relation (3.8), also should be taken into account. ►

**Remark 4.9** For a strict skew invertible Yang-Baxter matrix of the Hecke type of height $k$ the identity (4.18) holds as well, with the same proof. ►

**Lemma 4.10** Let $Z^{(i)} \in \mathrm{Aut}(V^{\otimes i})$ be the operators defined in the formulas (3.12). For a given matrix $M \in \mathrm{Mat}_N(W)$, consider yet another set of its "copies" defined as (c.f. with eq. (4.2))

$$ M_1 := M_1, \quad M_{i+1} := F_i^{-1} M_i F_i, \quad i = 2, 3, \dots \quad (4.19) $$

Then relations

$$ M_j^- Z^{(i)} = Z^{(i)} M_{i-j+1} \quad (4.20) $$

hold for any $i = 1, 2, \dots$ and for any $j = 1, 2, \dots, i$.

**Proof.** The equalities (4.20) can be proved by induction. First, one considers the case $j = i$ and carries out induction on $i$. Then, performing induction on $(i-j)$, one proves the equalities (4.20) in the full generality. ►

**Lemma 4.11** The following relation

$$ a^{(j)\uparrow j} c^{(2j)} = a^{(j)} c^{(2j)} \quad (4.21) $$

holds.

**Proof.** The proof is by induction on $j$ (the base of induction, $j = 1$, is obvious):

$$ \begin{align} a^{(j+1)\uparrow j+1} c^{(2j+2)} &= \frac{q^j}{(j+1)_q} a^{(j)\uparrow j+1} \sigma_{2j+1}^{-} (q^{-2j}) a^{(j)\uparrow j+1} c^{(2j+2)} \nonumber \\ &= \frac{q^j}{(j+1)_q} a^{(j)\uparrow j+1} \sigma_{2j+1}^{-} (q^{-2j}) a^{(j)\uparrow 1} c^{(2j+2)} = \frac{q^j}{(j+1)_q} a^{(j)\uparrow j+1} a^{(j)\uparrow 1} \sigma_1^{-} (q^{-2j}) c^{(2j+2)} \tag{4.22} \\ &= \frac{q^j}{(j+1)_q} a^{(j)\uparrow 1} \sigma_1^{-} (q^{-2j}) a^{(j)\uparrow 1} c^{(2j+2)} = a^{(j+1)} c^{(2j+2)}. \nonumber \end{align} $$

In the first equality we used the left definition of the $q$-antisymmetrizers in eq.(2.8); in the second equality we used the induction assumption (the contractor $c^{(2j)\uparrow 1}$ divides the contractor $c^{(2j+2)}$, see eq.(2.13)); in the third equality we moved the element $\sigma_{2j+1}^{-}(q^{-2j})$ rightwards to the contractor and used the relation (2.14); in the fourth equality we moved the $q$-antisymmetrizer $a^{(j)\uparrow j+1}$ rightwards
to the contractor and again used the induction assumption; in the fifth equality we used the right definition of the q-antisymmetrizers in eq.(2.8). ▶

Now, we will prove the relations (4.15) by expressing elements $ch(c^{(2k-2i)}a^{(k)\uparrow k-i}) \in C(R, F)$ in terms of elementary generators in two different ways.

The first way is

$$
\begin{align*}
& \mathrm{ch}(c^{(2k-2i)} a^{(k)\uparrow k-i}) \\
&= (\mu^{-2} g)^{k-i} \mathrm{Tr}_{R(1,2,\dots,2k-i)} (M_{\overline{2k-2i+1}} \dots M_{\overline{2k-i-1}} M_{\overline{2k-i}} \rho_R(c^{(2k-2i)} a^{(k)\uparrow k-i})) \\
&= (\mu^{-2} g)^{k-i} \mathrm{Tr}_{R(1,2,\dots,2k-i)} (Z^{(2k-i)} M_{\underline{i}} \dots M_{\underline{2}} M_{\underline{1}} \rho_{R_F} (c^{(2k-2i)\uparrow i} a^{(k)})(Z^{(2k-i)})^{-1}) \\
&= (\mu^{-2} g)^{k-i} \mathrm{Tr}_{R(1,2,\dots,i)} M_{\underline{i}} \dots M_{\underline{2}} M_{\underline{1}} \left( \mathrm{Tr}_{R_F^{(i+1,\dots,2k-i)}} \rho_{R_F} (c^{(2k-2i)\uparrow i} a^{(k)}) \right) \\
&= (\eta^{-1} \mu^{-1} g)^{k-i} \mathrm{Tr}_{R(1,2,\dots,i)} M_{\underline{i}} \dots M_{\underline{2}} M_{\underline{1}} \left( \mathrm{Tr}_{R_F^{(i+1,\dots,k)}} \rho_{R_F} (a^{(k)}) \right) \\
&= \left( \prod_{j=1}^{k-i} \eta^{-1} \mu^{-1} \delta_{k-j+1} g \right) \mathrm{Tr}_{R(1,2,\dots,i)} \left( (Z^{(i)})^{-1} M_{\overline{1}} M_{\overline{2}} \dots M_{\overline{i}} \rho_R(a^{(i)}) Z^{(i)} \right) \\
&= \left( \prod_{j=1}^{k-i} \eta^{-1} \mu^{-1} \delta_{k-j+1} \right) g^{k-i} \mathfrak{e}_i. \tag{4.23}
\end{align*}
$$

Here the relation (4.16) was applied in the first equality; the relations (4.20) as well as direct generalizations

$$
\begin{align*}
\rho_R(c^{(2i)}) Z^{(j)} &= Z^{(j)} \rho_{R_F} (c^{(2i)\uparrow j-2i}) && \quad \forall j \ge 2i \ge 0, \\
\rho_R(a^{(i)\uparrow j-i}) Z^{(j)} &= Z^{(j)} \rho_{R_F} (a^{(i)}) && \quad \forall j \ge i \ge 0
\end{align*}
$$

of the relations (3.14-3.15) were used in the second equality; the commutativity of $\prod_{j=1}^{2k-i} D_{Rj}$ with $Z^{(2k-i)}$ (which follows from the relations (2.31)) allows to cancel the latter when passing to third equality; the relations (3.7) and (3.9) were used to substitute the R-trace $\mathrm{Tr}_R$ by the R-trace $\mathrm{Tr}_{R_F}$ in the spaces with labels $i+1, i+2, \dots, 2k-i$ in the third equality; the relations (3.5) and (2.28) were applied for the evaluation of the R-traces in the fourth and fifth equalities; a similarity transformation by $Z^{(i)}$ was performed in the argument of the R-trace and, then, the relations (4.20) and (3.14-3.15) were used to complete the transformation.

The second way to express the element $ch(c^{(2k-2i)}a^{(k)\uparrow k-i})$ in terms of elementary generators is

$$
\begin{align}
\mathrm{ch}(c^{(2k-2i)} a^{(k)\uparrow k-i}) &= q^{k(k-1)} \mathrm{Tr}_{R(1,2,\dots,2k-i)} (M_{\overline{1}} M_{\overline{2}} \dots M_{\overline{k-i}} \rho_R(a^{(k)\uparrow k-i} c^{(2k-2i)})) \mathfrak{e}_k \nonumber \\
&= q^{k(k-1)} \left( \prod_{j=k-i+1}^{k} \delta_j \right) \mathrm{Tr}_{R(1,2,\dots,2k-2i)} (M_{\overline{1}} M_{\overline{2}} \dots M_{\overline{k-i}} \rho_R(a^{(k-i)} c^{(2k-2i)})) \mathfrak{e}_k \nonumber \\
&= q^{k(k-1)} \left( \prod_{j=k-i+1}^{k} \delta_j \right) (\eta^{-1} \mu)^{k-i} \mathrm{Tr}_{R(1,2,\dots,k-i)} (M_{\overline{1}} M_{\overline{2}} \dots M_{\overline{k-i}} \rho_R(a^{(k-i)})) \mathfrak{e}_k \tag{4.24} \\
&= \left( \prod_{j=1}^{k-i} \eta^{-1} \mu \delta_j^{-1} \right) \mathfrak{e}_{k-i} \mathfrak{e}_k. \nonumber
\end{align}
$$

Here (a version of) the relation (4.17) was used in the first line; eq. (2.28) was applied for the evaluation of the R-traces in the second line. There also, we substituted, due to the lemma 4.11, the element $a^{(k-i)\uparrow k-i} c^{(2k-2i)}$ by the element $a^{(k-i)} c^{(2k-2i)}$ in the argument of $\rho_R$; in the third line, some R-traces were evaluated with the help of the relation (3.5); the relation (3.11) for $i=k$ were used to transform the expression to its final form in the fourth line.
Comparing the results of the calculations (4.23) and (4.24) and substituting the values, which $\mu$ and $\delta_j$ take in the $O(k)$ case (see the definition 3.1 and eq.(2.28)), we finish the verification of the relations (4.15).

## 4.4 Group-like elements in the orthogonal QM-algebras

A particular subfamily of the QM-algebras — $M(R, P)$ — is often called FRT-algebras as it was first considered in details in [RTF]. These algebras carry a natural bialgebra structure with the coproduct given on generators by$^7$

$$ \Delta(M_b^a) = \sum_c M_c^a \otimes M_b^c. \qquad (4.25) $$

In this subsection we investigate algebraic properties of the group like elements (that is, elements $b$ such that $\Delta b = b \otimes b$) in the FRT-algebras. A generalization of the coproduct (4.25) for a general QM-algebra $M(R, F)$ is discussed in the appendix A.

Let $b^{(i)} \in B_i$ be such that $\rho_R(b^{(i)}) \in \text{End}(V^{\otimes i})$ is a rank-one projector. Then the element $b \in M(R, P)$, defined by

$$ b = \text{Tr}_{(1,2,...,i)}(M_{\bar{1}} M_{\bar{2}} \dots M_{\bar{i}} \rho_R(b^{(i)})) \qquad (4.26) $$

is group-like.

Thus, in any BMW type FRT-algebra $M(R, P)$ the rescaled contraction $\mu^{-2}g$ associated with the rank-one projector $\rho_R(c^{(2)})$ is group-like.$^8$ Algebraic properties of this element, i.e., its permutation relations with the algebra generators are described in the proposition 4.6.

With the orthogonal $O(k)$ type Yang-Baxter matrix $R$ one associates yet another rank-one projector $\rho_R(a^{(k)})$. The corresponding group-like element in the FRT-algebra $M(R, P)$ is the $k$-th order polynomial in the algebra generators called the *quantum determinant of M*

$$ \det_M M := \text{Tr}_{(1,2,...,k)}(M_{\bar{1}} M_{\bar{2}} \dots M_{\bar{k}} \rho_R(a^{(k)})) = q^{k(k-1)} \epsilon_k \qquad (4.27) $$

(for the latter equality see (4.18)). By a slight abuse of notation we will call element $\epsilon_k$, in the $O(k)$ type QM-algebra $M(R, F)$, the quantum determinant of $M$.

In the rest of this subsection we derive the permutation relations between the quantum determinant $\epsilon_k$ and the components of the quantum matrix $M$. Our method of derivation does not depend on the choice of the Yang-Baxter matrix $F$, so we carry out the consideration for a general orthogonal QM-algebra.

**Proposition 4.12** Let $M(R, F)$ be the QM-algebra of the orthogonal $O(k)$ type. Define two operators $O, O^{-1} \in \text{End}(V)$:

$$ O_1 := \frac{k_q(1+q^{2-k})}{(q+q^{1-k})} \text{Tr}_{(2,...,k+1)} \left( \rho_R(a^{(k)\uparrow 1}) F_1^{-1} F_2^{-1} \dots F_k^{-1} \right) \qquad (4.28) $$

and

$$ O_1^{-1} = \frac{k_q(1+q^{2-k})}{(q+q^{1-k})} \text{Tr}_{(2,...,k+1)} \left( F_k \dots F_2 F_1 \rho_R(a^{(k)\uparrow 1}) \right). \qquad (4.29) $$

Then

(a) the commutation relations of the quantum determinant with the quantum matrix elements are given by

$$ M \epsilon_k = \epsilon_k (O^{-1} M O); \qquad (4.30) $$

(b) the operators $O$ and $O^{-1}$ are inverse to one another.

<footnote>$^7$In this article summation is always assumed to be performed from lower to upper indices; that is, for example, if $A$ and $B$ are two endomorphisms of a vector space $V$, then $(AB)^{\dagger}_j := \sum_l A_l^\dagger B_j^l$.</footnote>

<footnote>$^8$ The series of the higher order rank-one projectors $\rho_R(c^{(2i)})$, $i > 1$, do not give rise to the new group-like elements. The projector $\rho_R(c^{(2i)})$ is associated with the $i$-th power of $\mu^{-2}g$.</footnote>
**Proof.** (a) A check of the commutation relation (4.30) is similar to the proof of formula (4.12) (see lemma 4.13 in [OP1]), although a bit more sophisticated. For the calculation we use a version of the “vanishing” condition in (3.2), involving the operator $\rho_R(a^{(k)\uparrow 1}a^{(k)}a^{(k)\uparrow 1})$. To derive it we use the next Lemma.

**Lemma 4.13** Let R be the Yang-Baxter matrix of the BMW type. Assume that the conditions (3.1) for the parameters $q$ and $\mu$ are satisfied. Then

$$ j_q^2 \rho_R(a^{(j)\uparrow 1} a^{(j)} a^{(j)\uparrow 1}) = \rho_R(a^{(j)\uparrow 1}) + q^j (j-1)_q \rho_R(a^{(j)\uparrow 1} \sigma_1^- (q^{-2j}) a^{(j)\uparrow 1}) \\ + \frac{\mu^{-1} q^{-2j+2} (q-q^{-1})^2 (j-1)_q j_q}{(\mu^{-1} q^{1-2j} + 1)(\mu^{-1} q^{3-2j} + 1)} \rho_R(a^{(j)\uparrow 1} \kappa_1 a^{(j)\uparrow 1}) \quad (4.31) $$

for all $j = 1, \dots, k$.

**Proof.** Using the second expression for $a^{(j)}$ in eq.(2.8) and the relations (see [OP1], proposition 2.2)

$$ a^{(j)} a^{(j-1)} = a^{(j-1)} a^{(j)} = a^{(j)}, \quad (4.32) $$

we find

$$ \rho_R(a^{(j)\uparrow 1} a^{(j)} a^{(j)\uparrow 1}) = \frac{q^{j-1}}{j_q} \rho_R(a^{(j)\uparrow 1} (1 + \frac{q^{2-2j}-1}{q-q^{-1}} \sigma_1 + \frac{q^{2-2j}-1}{\mu^{-1}q^{3-2j}+1} \kappa_1) a^{(j)\uparrow 1}) . \quad (4.33) $$

On the other hand, by eq.(2.6) and the idempotency of $a^{(j)}$,

$$ \begin{aligned} \rho_R(a^{(j)\uparrow 1} \sigma_1 a^{(j)\uparrow 1}) = & \frac{q^j}{j_q} \rho_R(a^{(j)\uparrow 1}) - \frac{q^j}{j_q} \rho_R(a^{(j)\uparrow 1} \sigma_1^- (q^{-2j}) a^{(j)\uparrow 1}) \\ & - \frac{q-q^{-1}}{\mu^{-1}q^{1-2j}+1} \rho_R(a^{(j)\uparrow 1} \kappa_1 a^{(j)\uparrow 1}) . \end{aligned} \quad (4.34) $$

Substituting the expression for the operator $\rho_R(a^{(j)\uparrow 1}\sigma_1 a^{(j)\uparrow 1})$ from eq.(4.34) into eq.(4.33), we arrive at eq. (4.31). ▶

Assume that the Yang-Baxter matrix R is of the type $O(k)$. Then, taking into account the conditions $\mu = q^{1-k}$ and (3.2), the identity (4.31) for $j=k$ specifies to

$$ \rho_R(a^{(k)\uparrow 1}) = k_q^2 \rho_R(a^{(k)\uparrow 1} a^{(k)} a^{(k)\uparrow 1}) - \frac{q^{1-k} (q-q^{-1})^2 k_q (k-1)_q}{(1+q^{-k})(1+q^{2-k})} \rho_R(a^{(k)\uparrow 1} \kappa_1 a^{(k)\uparrow 1}) . \quad (4.35) $$

This is the needed version of the vanishing condition.

Now we calculate

$$ \begin{aligned} M_1 (\varepsilon_k \rho_R(a^{(k)\uparrow 1})) &= M_1(F_1 \dots F_k) (\varepsilon_k \rho_R(a^{(k)})) (F_1 \dots F_k)^{-1} = q^{k(1-k)} M_{\overline{1}} M_{\overline{2}} \dots M_{\overline{k+1}} \rho_R(a^{(k)\uparrow 1}) \\ &= k_q^2 \varepsilon_k \rho_R(a^{(k)\uparrow 1} a^{(k)}) M_{\overline{k+1}} \rho_R(a^{(k)\uparrow 1}) - \frac{q^{1-k} (q-q^{-1})^2 k_q (k-1)_q}{(1+q^{-k})(1+q^{2-k})} \rho_R(a^{(k)\uparrow 1} \kappa_1) M_1 \rho_R(a^{(k)\uparrow 1}) \varepsilon_k. \end{aligned} \quad (4.36) $$

Here in the second equality we used the relation (4.17) and the “shifting” relation (see, e.g., [OP1], lemma 4.3)

$$ F_1 F_2 \dots F_k \cdot M_{\overline{1}} M_{\overline{2}} \dots M_{\overline{k}} = M_{\overline{2}} M_{\overline{3}} \dots M_{\overline{k+1}} \cdot F_1 F_2 \dots F_k. $$

In the third equality in the computation (4.36) we used the condition (4.35), then applied permutation relations (4.3) and $R_i M_{\overline{j}} = M_{\overline{j}} R_i \quad \forall j \neq i, i+1$, and then used again the relation (4.17).
We continue the calculation by taking the traces in the spaces from 2 to $k+1$ in the equality (4.36). Due to the rank 1 property of the operator $\rho_R(a^{(k)})$, $\text{Tr}_{(2,...,k+1)}(\rho_R(a^{(k)\uparrow 1})) = I_1$, and we find

$$ M_1 \epsilon_k = k_q^2 \epsilon_k \text{Tr}_{(2,...,k+1)} (\rho_R(a^{(k)\uparrow 1} a^{(k)}) M_{\overline{k+1}}) \\ - \frac{q^{1-k}(q-q^{-1})^2 k_q (k-1)_q}{(1+q^{-k})(1+q^{2-k})} \text{Tr}_{(2,...,k+1)} (K_1 \rho_R(a^{(k)\uparrow 1})) M_1 \epsilon_k. \qquad (4.37) $$

The traces in the last term in the left hand side of eq.(4.37) can be calculated with the help of the identity

$$ \text{Tr}_{(2,...,k)} (\rho_R(a^{(k)})) = q^{k(k-1)} D_{R1} \text{Tr}_{R(2,...,k)} (\rho_R(a^{(k)})) = \frac{q^k(1+q^{-k})}{(1+q^{2-k})k_q} D_{R1}. \qquad (4.38) $$

Here in the first equality we used the relation (3.8); in the second equality we used eq.(2.28) $k-1$ times and substituted $\mu = q^{1-k}$.

Therefore,

$$ \text{Tr}_{(2,...,k+1)} (K_1 \rho_R(a^{(k)\uparrow 1})) = \frac{q^k(1+q^{-k})}{(1+q^{2-k})k_q} \text{Tr}_{R(2)} K_1 = \frac{q(1+q^{-k})}{(1+q^{2-k})k_q} I_1 \qquad (4.39) $$

due to the relation (2.26) with $\mu = q^{1-k}$.

After substituting the expression (4.39) and rearranging similar terms, eq.(4.37) simplifies to

$$ \frac{(q+q^{1-k})^2}{(1+q^{2-k})^2} M_1 \epsilon_k = k_q^2 \text{Tr}_{(2,...,k+1)} (\rho_R(a^{(k)\uparrow 1} a^{(k)}) M_{\overline{k+1}}). \qquad (4.40) $$

It is left to evaluate the traces in the right hand side of eq.(4.40).

Due to the rank 1 property of the operator $\rho_R(a^{(k)})$, we have

$$ \rho_R(a^{(k)\uparrow 1}) X_{1,...,k+1} \rho_R(a^{(k)\uparrow 1}) = \text{Tr}_{(2,...,k+1)} (\rho_R(a^{(k)\uparrow 1}) X_{1,...,k+1}) \rho_R(a^{(k)\uparrow 1}) \qquad (4.41) $$

for any operator $X \in \text{End}(V^{\otimes(k+1)})$.

Let $Y$ be an operator acting in the space $V$. Then

$$ \begin{aligned} \text{Tr}_{(2,...,k+1)} (\rho_R(a^{(k)\uparrow 1} a^{(k)}) Y_{\overline{k+1}}) &= \text{Tr}_{(2,...,k+1)} (\rho_R(a^{(k)}) Y_{\overline{k+1}} \rho_R(a^{(k)\uparrow 1})) \\ &= \text{Tr}_{(2,...,k+1)} (\rho_R(a^{(k)}) \cdot F_k \dots F_2 F_1 \cdot Y_1 \cdot F_1^{-1} F_2^{-1} \dots F_k^{-1} \cdot \rho_R(a^{(k)\uparrow 1})) \\ &= \text{Tr}_{(2,...,k+1)} (F_k \dots F_2 F_1 \cdot Y_1 \rho_R(a^{(k)\uparrow 1}) \cdot F_1^{-1} F_2^{-1} \dots F_k^{-1} \cdot \rho_R(a^{(k)\uparrow 1})) \\ &= \frac{q+q^{1-k}}{k_q(1+q^{2-k})} \text{Tr}_{(2,...,k+1)} (F_k \dots F_2 F_1 \cdot Y_1 O_1 \rho_R(a^{(k)\uparrow 1})) \\ &= \frac{q+q^{1-k}}{k_q(1+q^{2-k})} \text{Tr}_{(2,...,k+1)} (\rho_R(a^{(k)\uparrow 1}) \cdot F_k \dots F_2 F_1 \cdot \rho_R(a^{(k)\uparrow 1})) Y_1 O_1 \\ &= \frac{(q+q^{1-k})^2}{k_q^2(1+q^{2-k})^2} \text{Tr}_{(2,...,k+1)} (\rho_R(a^{(k)\uparrow 1})) O_1^{-1} Y_1 O_1 = \frac{(q+q^{1-k})^2}{k_q^2(1+q^{2-k})^2} O_1^{-1} Y_1 O_1. \end{aligned} \qquad (4.42) $$

Here in the first equality we moved the operator $\rho_R(a^{(k)\uparrow 1})$ cyclically to the right under the trace; in the second equality we substituted the definition of $Y_{\overline{k+1}}$; in the third equality we used the twist relations (2.29) to move the operator $\rho_R(a^{(k)})$ rightwards; in the fourth equality we used eq.(4.41) with $X_{1,...,k+1} = F_1^{-1} F_2^{-1} \dots F_k^{-1}$ and the definition (4.28) of the operator $O$; in the fifth equality we replaced $\rho_R(a^{(k)\uparrow 1})$ by $\rho_R(a^{(k)\uparrow 1})\rho_R(a^{(k)\uparrow 1})$ using the idempotency of the operator $\rho_R(a^{(k)\uparrow 1})$, moved
one of the operators $\rho_R(a^{(k)\uparrow 1})$ cyclically to the left under the trace and moved $Y_1O_1$ out of the trace; in the sixth equality we used eq.(4.41) with $X_{1,...,k+1} = F_k...F_2F_1$ and the definition (4.29) of the operator $O^{-1}$; in the seventh equality the rank 1 property of the operator $\rho_R(a^{(k)})$ was used again.

Substituting the result of the calculation (4.42) with $Y = M$ into eq.(4.40), we finally establish the relation (4.30).

(b) To check the relation $O^{-1}O = I$ we use the same calculation (4.42), but now with $Y = I$: in this case

$$ \begin{aligned} \mathrm{Tr}_{(2,...,k+1)}(\rho_R(a^{(k)\uparrow 1} a^{(k)}) Y_{\overline{k+1}})\Big|_{Y=I} &= \mathrm{Tr}_{(2,...,k+1)}(\rho_R(a^{(k)\uparrow 1} a^{(k)})) \\ &= \frac{q^{k-1}}{k_q} \mathrm{Tr}_{(2,...,k+1)}(\rho_R(\sigma_1(q^{2-2k}) a^{(k)\uparrow 1})) = \frac{q^{-1}(1+q^{-k})}{k_q^2(1+q^{2-k})} \mathrm{Tr}_{R(2)}(\rho_R(\sigma_1(q^{2-2k}))) \quad (4.43) \\ &= \frac{(q+q^{1-k})^2}{k_q^2(1+q^{2-k})^2} I_1. \end{aligned} $$

Here in the second equality we used the second expression for $a^{(k)}$ in the definition (2.8), the cyclic property of the trace and the identities (4.32) for the $q$-antisymmetrizers; in the third equality we used the calculation (4.38); in the fourth equality we evaluated the R-traces with the help of the relations (2.20), (2.26) and (2.27).

We are done by comparing the results of calculations (4.43) and (4.42) for $Y = I$.

**Remark 4.14** Using a similar technique, one can verify independently relation $OO^{-1} = I$ which is of course equivalent to $O^{-1}O = I$ for a finite-dimensional $V$. This is done by a calculation, parallel to (4.42), where one uses the copies $Y_j$, defined in eq.(4.19), instead of the copies $Y_j^-$.

**Remark 4.15** For the compatible pair $\{R, R\}$ of Yang-Baxter matrices, we can evaluate all traces in the expression (4.29) moving cyclically the operator $\rho_R(a^{(k)\uparrow 1})$ under the trace and using characteristic properties of the $q$-antisymmetrizers $a^{(n)}\sigma_i = \sigma_i a^{(n)} = -q^{-1}a^{(n)} \forall 1 \le i < n$ (see [OP1], proposition 2.2) and (4.38). Thus we obtain $O^{-1} = O = (-1)^{k-1} I$. Likewise, the operator $G$ (4.13) trivializes for the compatible pair $\{R, R\}$: we have $G = I$.

Scalar forms of the operators $O$ and $G$ in the reflection equation algebra $M(R, R)$ was expected since the elements $\epsilon_k$ and $g$ both belong to the center of the algebra (see remark 4.4).

## 4.5 Representations of quantum matrix algebras

In the next subsection we will use two representations of the QM-algebra $M(R, F)$ on the same underlying vector space $V$. In the present subsection these representations – we denote them by $\alpha_+$ and $\alpha_-$ – are introduced and investigated. We use the label 0 for the copy of the vector space $V$, on which $\alpha_\pm$ are defined. Here the symbol 0 carries an auxiliary role; 0 does not mean a number preceding 1; there is no matrix $M_0$. The space $V_0$ is just another space in which our operators act.

**Proposition 4.16** Let $M(R, F)$ be the QM-algebra of the general type (no additional conditions on the compatible pair $\{R, F\}$ of Yang-Baxter matrices).

The following formulas:

$$ \alpha_{\epsilon}(M_1)_0 = R_{01}^{\epsilon} F_{01}, \text{ or, shortly, } R_0^{\epsilon} F_0 \quad (4.44) $$

where $\epsilon$ is the sign, $\epsilon = \pm 1$, define two representations $\alpha_\epsilon$ of the QM-algebra $M(R, F)$ on the vector space $V$.

**Proof.** A straightforward calculation, with the help of the Yang-Baxter equations (2.17) and the twist relations (2.29) for the compatible pair $\{R, F\}$ of Yang-Baxter matrices.<fnref n="9" />

<footnote n="9">⁹The solution $\alpha_+$ was noted in [IOP2]. The solution $\alpha_-$ becomes trivial in the case of the reflection equation algebra, when $F = R$.</footnote>
**Remark 4.17** There are many ways to construct representations of the QM-algebra $\mathcal{M}(R, F)$ using the compatible Yang-Baxter matrix pair $\{R, F\}$. In particular, for $\mathcal{M}(R, F)$ of the general type, there is yet another representation $\beta$ on the vector space $V$. It is given by

$$ \beta(M_1)_0 = F_{01}^2, \text{ or } F_0^2 \qquad \text{▶} $$

The quantum matrix string $M_1^\top M_2 \dots M_j^\top$, which we always use to construct elements of the characteristic subalgebra $\mathcal{C}(R, F)$ acquires a quite simple expression in the representations $\alpha_\pm$.

**Lemma 4.18** We have

$$ \alpha_{\epsilon}(M_{1}^{\top} M_{2} \dots M_{j}^{\top})_{0} = R_{0}^{\epsilon} R_{1}^{\epsilon} \dots R_{j-1}^{\epsilon} F_{j-1} \dots F_{1} F_{0} \quad (4.45) $$

for any $j > 0$.

**Proof.** Induction on $j$:

$$ \begin{aligned} \alpha_{\epsilon}(M_{1}^{\top} M_{2} \dots M_{j+1}^{\top})_{0} &= (R_{0}^{\epsilon} R_{1}^{\epsilon} \dots R_{j-1}^{\epsilon} F_{j-1} \dots F_{1} F_{0}) F_{j} \dots F_{2} F_{1} R_{0}^{\epsilon} F_{0} (F_{j} \dots F_{2} F_{1})^{-1} \\ &= R_{0}^{\epsilon} R_{1}^{\epsilon} \dots R_{j-1}^{\epsilon} (F_{j-1} F_{j}) (F_{j-2} F_{j-1}) \dots (F_{0} F_{1}) R_{0}^{\epsilon} F_{0} (F_{j} \dots F_{2} F_{1})^{-1} \\ &= R_{0}^{\epsilon} R_{1}^{\epsilon} \dots R_{j-1}^{\epsilon} R_{j}^{\epsilon} F_{j} (F_{j-1} F_{j}) (F_{j-2} F_{j-1}) \dots (F_{0} F_{1}) (F_{j} \dots F_{2} F_{1})^{-1} \\ &= R_{0}^{\epsilon} R_{1}^{\epsilon} \dots R_{j}^{\epsilon} F_{j} \dots F_{1} F_{0}. \end{aligned} $$

In the first equality we used the induction assumption and the definition of the matrix $M_{j+1}^\top$. In the second equality we shifted the elements of the string $F_{j-1} \dots F_{1} F_{0}$ as far as possible to the right into the string $F_{j} \dots F_{2} F_{1}$; as a result they formed pairs. In the third equality we used the compatibility relations for the $\{R, F\}$ pair to move the product $R_0^\epsilon F_0$ leftwards. In the fourth equality we reassembled the product of pairs of the operators $F$ back into the product of two strings $F_{j-1} \dots F_1 F_0 \cdot F_j \dots F_2 F_1$. The proof is finished. ■

We shall now calculate the image of the contraction $g$ in the representations $\alpha_\pm$.

**Proposition 4.19** For QM-algebras of the BMW type the images of the contraction $g$ in the representations $\alpha_\pm$ are proportional to the operator $G^{-1}$,

$$ \alpha_{\pm}(g) = \mu^2 G^{-1}. \qquad (4.46) $$

**Proof.** The proof is a direct calculation:

$$ \begin{aligned} \alpha_{\pm}(g)_0 &= \eta^{-1} \mathrm{Tr}_{R(1,2)} (K_1 R_0^\epsilon R_1^\epsilon F_1 F_0) = \eta^{-1} \mu^2 \mathrm{Tr}_{(1,2)} (K_1 K_0 F_1 F_0) \\ &= \eta^{-1} \mu^2 \mathrm{Tr}_{(1,2)} (K_0 F_1 F_0 K_1) = \mu^2 \mathrm{Tr}_{(1,2)} (K_0 F_1 F_0) = \mu^2 G_0^{-1}. \end{aligned} $$

Here in the first equality we used the definition (4.5) of the contraction $g$ and expressed the string $M_1^\top M_2$ as in the lemma 4.18; in the second equality we used relations (2.3) and (2.25); in the third equality we moved $K_1$ cyclically to the right under the trace; in the fourth equality we used the twist relation and the identity $K^2 = \eta K$; the fifth equality reproduces the formula (4.13). ■

**Proposition 4.20** For orthogonal QM-algebras of the type $O(k)$ the images of the quantum determinant $\epsilon_k \in \mathcal{C}(R, F)$ in the representations $\alpha_+$ and $\alpha_-$ are given by

$$ \alpha_{\pm}(\epsilon_k) = (-1)^{k-1} q^{k(1-k)} O^{-1}. \qquad (4.47) $$
**Proof.** Evaluating formula (4.17) in the representation $\alpha_+$ we find, with the help of the relation (4.45), that

$$ \rho_R(a^{(k)}) R_0 R_1 \dots R_{k-1} F_{k-1} \dots F_1 F_0 = q^{k(k-1)} \rho_R(a^{(k)}) (\alpha_+(\mathfrak{e}_k))_0 . $$

Multiplying both sides of this equality by the operator $F_0^{-1}$ from the right, by the operator $C_{R0}$ (2.19) from the left, taking the trace in the space $V_0$ and using eq.(2.20), we find

$$ \rho_R(a^{(k)}) F_{k-1} \dots F_2 F_1 = (-1)^{k-1} q^{(k+1)(k-1)} \rho_R(a^{(k)}) \mathrm{Tr}_{(0)}(C_{R0} (\alpha_+(\mathfrak{e}_k))_0 F_0^{-1}), $$

because $\rho_R(a^{(k)}) R_j = (-q)^{-1} \rho_R(a^{(k)})$ for all $j = 1, \dots, k-1$.

Taking the traces in the spaces from 2 to $k$ and using eq.(4.38), we obtain, upon the replacement $i \mapsto i+1$ of the labels of all the spaces, except $V_0$:

$$ \begin{aligned} & \mathrm{Tr}_{(3, \dots, k+1)} (\rho_R(a^{(k)\uparrow 1}) F_k \dots F_3 F_2) \\ & = (-1)^{k-1} \frac{q^{k^2+k-1}(1+q^{-k})}{k_q(1+q^{2-k})} D_{R2} \mathrm{Tr}_{(0)}(C_{R0} (\alpha_+(\mathfrak{e}_k))_0 F_{02}^{-1}). \end{aligned} \quad (4.48) $$

By the lemma 3.3, [OP1] $(\Psi_{F^{-1}})_{21} = (D_R^{-1})_1 F_{12} D_{R2}$. Multiplying the relation (4.48) from the right by a combination $(D_R^{-1})_1 F_{12}$, we take the trace in the space 2, moving the operator $D_{R2}$ cyclically to the right under the trace in the right hand side and applying eq.(2.18) for the Yang-Baxter matrix $F^{-1}$. All traces in the right hand side can now be evaluated and we find

$$ (D_R^{-1})_1 \mathrm{Tr}_{(2, \dots, k+1)} (\rho_R(a^{(k)\uparrow 1}) F_k \dots F_2 F_1) = (-1)^{k-1} \frac{q^{k^2+k-1}(1+q^{-k})}{k_q(1+q^{2-k})} C_{R1} (\alpha_+(\mathfrak{e}_k))_1 . \quad (4.49) $$

Using the definition (4.29) of the matrix $O^{-1}$ and the relation (2.22), we establish the relation (4.47) for the representation $\alpha_+$.

The calculations for the representation $\alpha_-$ are similar, we do not repeat them.

In the end of the subsection we apply the representation $\alpha_+$ to clarify properties of the operator $O$.

**Proposition 4.21** Let {$R, F$} be a compatible Yang-Baxter matrix pair, where $R$ is $O(k)$ type. The corresponding matrices $O, G, D_R$ satisfy following relations

$$ O_1 F_{12}^{\epsilon} = F_{12}^{\epsilon} O_2, \quad \text{where } \epsilon = \pm 1, \quad (4.50) $$

$$ R_{12} O_1 O_2 = O_1 O_2 R_{12}, \quad (4.51) $$

$$ [D_R, O] = 0 = [G, O], \quad (4.52) $$

$$ O^2 = G^k. \quad (4.53) $$

**Proof.** For an arbitrary integer $j > 0$ and an operator $X = X_{1, \dots, j} \in \mathrm{End}(V^{\otimes j})$, let $\tilde{X} \in \mathrm{End}(V)$ be an operator defined by $\tilde{X}_1 = \mathrm{Tr}_R(2, \dots, j)(X_{1, \dots, j})$. It follows then from lemma 3.2, [OP1] that

$$ \mathrm{Tr}_R(3, \dots, j+1) (F_2^{-\epsilon} F_3^{-\epsilon} \dots F_j^{-\epsilon} X_{1, \dots, j} F_j^{\epsilon} \dots F_3^{\epsilon} F_2^{\epsilon}) = \tilde{X}_1 I_2 $$

for $\epsilon = \pm 1$.

Therefore, for the operator $X_{1, \dots, k+1} = \rho_R(a^{(k)\uparrow 1}) \cdot F_k \dots F_2 F_1$, we have

$$ \begin{aligned} \tilde{X}_1 F_1^{\epsilon} &= \mathrm{Tr}_R(3, \dots, k+2) (F_2^{-\epsilon} F_3^{-\epsilon} \dots F_{k+1}^{-\epsilon} \rho_R(a^{(k)\uparrow 1}) F_k \dots F_2 F_1 F_{k+1}^{-\epsilon} \dots F_2 F_1^{\epsilon}) \\ &= \mathrm{Tr}_R(3, \dots, k+2) (F_2^{-\epsilon} F_3^{-\epsilon} \dots F_{k+1}^{-\epsilon} \rho_R(a^{(k)\uparrow 1}) F_{k+1}^{-\epsilon} \dots F_2^{\epsilon} F_1^{\epsilon} F_{k+1} \dots F_3 F_2) \\ &= F_1^{\epsilon} \mathrm{Tr}_R(3, \dots, k+2) (\rho_R(a^{(k)\uparrow 2}) F_{k+1} \dots F_3 F_2) = F_1^{\epsilon} \tilde{X}_2. \end{aligned} \quad (4.54) $$
Here we used the compatibility relations for the {$R, F$} pair.

Noticing that for the chosen $X$ the operator $\tilde{X}$ is proportional to the operator $O^{-1}$, (see the definition (4.29) and the relation (3.8)) we see that the calculation (4.54) proves the relation (4.50).

The commutation relation (4.30) between the quantum matrix and the quantum determinant, in the representation $\alpha_+$, reads, by the proposition 4.20,

$$R_0 F_0 O_0^{-1} = O_0^{-1} O_1^{-1} R_0 F_0 O_1. \qquad (4.55)$$

Applying the relation (4.50) to eq.(4.55), we establish the relation (4.51).

By eq.(2.31) the following identity

$$\begin{aligned} F_j \dots F_2 F_1 D_{R1} &= F_j \dots F_2 F_1 (D_{R1} D_{R2} \dots D_{Rj+1}) (D_{R_2}^{-1} D_{R_3}^{-1} \dots D_{R_{j+1}}^{-1}) \\ &= (D_{R1} D_{R2} \dots D_{Rj+1}) F_j \dots F_2 F_1 (D_{R_2}^{-1} D_{R_3}^{-1} \dots D_{R_{j+1}}^{-1}) \end{aligned} \qquad (4.56)$$

holds for any $j > 0$.

The cyclic properties of the trace, together with eq.(4.56), shows that the expression (4.29) commutes with $D_{R1}$, which proves the left equality in (4.52).

The right equality in (4.52) is a consequence of the commutativity of the characteristic subalgebra $C(R, F)$ because the images of the elements $g, \mathfrak{e}_k \in C(R, F)$ in the representations $\alpha_\pm$ are given by the formulas (4.46) and (4.47).

The relation (4.53) is the matrix image of the reciprocal relation (4.15) with $i = 0$, $g^k = \mathfrak{e}_k^2$, in the representation $\alpha_+$.

The proof is finished.

## 4.6 Resolution of reciprocal relations

In this subsection we consider the extended QM-algebra $M^*(R, F)$ of the orthogonal type $O(k)$. We shall use shorthand notation $O_k(R, F)$ for these algebras from now on.

We employ the invertibility of the contraction $g$ to resolve the reciprocal relations with respect to the first $\lfloor k/2 \rfloor$ elementary sums.

The case $k = 2\ell$ requires a more detailed investigation. For $k = 2\ell$ we can resolve the reciprocal relations under the assumption that the element $g^{-\ell}\mathfrak{e}_{2\ell}$ is central. To motivate this assumption we first notice that $(g^{-\ell}\mathfrak{e}_{2\ell})^2 = 1$ by eq.(4.15) in which $k = 2\ell$ and $i = 0$. Next, the following proposition holds.

**Proposition 4.22** Assume, for the (extended) QM-algebra $O_{2\ell}(R, F)$, that at least one of the matrices $\alpha_\pm(\mathfrak{e}_\ell)$ is non-degenerate. Then the element $g^{-\ell}\mathfrak{e}_{2\ell}$ is central in $O_{2\ell}(R, F)$.

**Remark 4.23** In the classical case ($R = F = P$), both matrices $\alpha_\pm(\mathfrak{e}_\ell)$ are proportional to the identity matrix, the coefficient of proportionality equals $(2\ell)!/(\ell!)^2$ (the rank of the $\ell$-th q-antisymmetrizer), so the assumption of Proposition 4.22 holds on the perturbation level. ►

**Proof.** By the propositions 4.6 and 4.12 the element $g^{-\ell}\mathfrak{e}_{2\ell}$ commutes with the components of the quantum matrix $M$ if and only if the matrix $G^\ell O^{-1}$ is scalar.

Taking the images of the reciprocal relations (4.15) in the representations $\alpha_\epsilon, \epsilon = \pm 1$ we find, in particular, that $\alpha_\epsilon(g)^\ell \alpha_\epsilon(\mathfrak{e}_\ell) = \alpha_\epsilon(\mathfrak{e}_{2\ell})\alpha_\epsilon(\mathfrak{e}_\ell)$ which, by the propositions 4.19, 4.20 and under the assumption of the present proposition, implies that $G^{-\ell} = -O^{-1}$. It follows then that the element $g^{-\ell}\mathfrak{e}_{2\ell}$ is central. ■

**Definition 4.24** Assume that the element $g^{-\ell}\mathfrak{e}_{2\ell}$ belongs to the center of the QM-algebra $O_{2\ell}(R, F)$. Then, its quotient algebra by the relation

$$g^{-\ell}\mathfrak{e}_{2\ell} = 1, \quad \text{respectively} \quad g^{-\ell}\mathfrak{e}_{2\ell} = -1, \qquad (4.57)$$

is called the component $O_{2\ell}^+(R, F)$, respectively $O_{2\ell}^-(R, F)$.
In fact we defined the decomposition of the algebra $O_{2\ell}(R, F)$ into the direct sum of its two components $O_{2\ell}^{\pm}(R, F)$. Note that classically, that is, for $R = F = P$ and under the restriction $g = 1$, $O_{2\ell}^{\pm}$ become the algebras of functions on two connected components of the orthogonal group $O_{2\ell}$.

Now we are ready to resolve the reciprocal relations.

**Theorem 4.25** In the assumption of the definition 4.24, the reciprocal relations (4.15) in the component algebras $O_{2\ell}^{\pm}(R, F)$ can be resolved with respect to $\mathfrak{e}_j$, $j \ge \ell$:

$$ \mathfrak{e}_{\ell+i} = \pm g^i \mathfrak{e}_{\ell-i} \quad \forall i = 0, 1, \dots, \ell, \qquad (4.58) $$

respectively. Note that in the $O_{2\ell}^{-}$ case, this implies that $\mathfrak{e}_{\ell} = 0$.<fnref n="10" />

For the $O_{2\ell-1}(R, F)$ the reciprocal relations (4.15) admit the resolution

$$ \mathfrak{e}_{\ell+i} = g^{i+1/2} \mathfrak{e}_{\ell-1-i} \quad \forall i = 0, 1, \dots, \ell-1, \qquad (4.59) $$

where we introduce

$$ g^{1/2} := g^{1-\ell} \mathfrak{e}_{2\ell-1}, \qquad (4.60) $$

so that $g = (g^{1/2})^2$ and $\mathfrak{e}_{2\ell-1} = (g^{1/2})^{2\ell-1}$.

**Proof.** Reciprocal relations (4.15) are not independent. Since the element $g$ is invertible, the relations are invariant under the involution $i \leftrightarrow k - i$. The only unpaired relation corresponds to $i = 0$. Resolution of this relation depends on the parity of $k$.

If $k = 2\ell$, eq.(4.15) for $i = 0$ splits in two factors

$$ g^{2\ell} - \mathfrak{e}_{2\ell}^2 = (g^{\ell} + \mathfrak{e}_{2\ell})(g^{\ell} - \mathfrak{e}_{2\ell}) = 0, $$

which implies the formulas (4.58) for the component algebras $O_{2\ell}^{\pm}(R, F)$.

If $k = 2\ell - 1$, eq.(4.15) for $i = 0$ assumes a form

$$ g = (g^{1-\ell} \mathfrak{e}_{2\ell-1})^2 $$

which allows one to define square root of $g$ by the formula (4.60).

The check of the rest of the proposition is straightforward.

<figure><img src="image_2.png" /></figure>

**Corollary 4.26** Characteristic subalgebras of the extended orthogonal QM-algebras $O_{2\ell-1}(R, F)$ and (in the assumption of definition 4.24) of the component algebras $O_{2\ell}^{\pm}(R, F)$ and $O_{2\ell}^{-}(R, F)$ are generated, respectively, by the sets of elements $g^{1/2} \cup \{\mathfrak{e}_i\}_{i=0}^{\ell-1}$, $g \cup \{\mathfrak{e}_i\}_{i=0}^{\ell}$ and $g \cup \{\mathfrak{e}_i\}_{i=0}^{\ell-1}$.

**Proof.** Finite height condition (3.2) implies that for the orthogonal $O(k)$ type QM-algebras $\mathfrak{e}_{k+i} = 0$ $\forall i \ge 1$. The corollary then follows directly from proposition 4.5 and theorem 4.25.

<figure><img src="image_3.png" /></figure>

## Acknowledgments

The authors thank Pavel Saponov for fruitful discussions and valuable remarks. The work of the second author (P. P.) was supported by the Basic Research Program of HSE University.

<footnote n="10">¹⁰A classical counterpart is: if $X$ is an orthogonal operator acting in a space $V$ of dimension $2\ell$ and $\det X = -1$ then $\mathrm{tr}_{\wedge^{\ell}V}(X) = 0$, where $\wedge^j V$, $j = 2, 3, \dots$, are the wedge tensor powers of the space $V$.</footnote>
# A Braided coproduct on quantum matrix algebras

The $M(R, P)$ FRT-algebras possess the coproduct structure. In this appendix we discuss its generalization, usually called the *braided coproduct*, to QM-algebras $M(R, F)$. The braided coproduct structure is known, mainly in the case of reflection equation algebras $M(R, R)$, in the quantum group theory community. We describe it here for general QM-algebras.

Before we begin, we note that the construction of the group-like elements described in subsection 4.4 for orthogonal/BMW type FRT-algebras carries over without any changes to general orthogonal/BMW type QM-algebras.

(i) To define the braided coproduct on $M(R, F)$ let us interpret the coproduct on the algebra $M(R, P)$ as follows. The matrices $M \otimes I$ and $I \otimes M$ element-wise commute in the vector space $M(R, P) \otimes M(R, P)$ endowed with the canonical algebra structure; the canonical algebra structure on the tensor product of the vector spaces underlying two associative algebras $A$ and $B$ means exactly that any elements $x \otimes 1$ and $1 \otimes y$, $x \in A$ and $y \in B$, commute (while, as algebras, $A \otimes 1$ is isomorphic to $A$ and $1 \otimes B$ is isomorphic to $B$). So the algebra $M(R, P) \otimes M(R, P)$ is the same as the unital algebra generated by two commuting quantum matrices that is by the elements $\|\dot{M}_a^b\|_{a,b=1}^N$ and $\|\ddot{M}_a^b\|_{a,b=1}^N$ with the defining relations

$$R_1 \dot{M}_1 \dot{M}_2 = \dot{M}_1 \dot{M}_2 R_1, \quad R_1 \ddot{M}_1 \ddot{M}_2 = \ddot{M}_1 \ddot{M}_2 R_1, \qquad (A.1)$$

and

$$\ddot{M}_1 \dot{M}_2 = \dot{M}_2 \ddot{M}_1. \qquad (A.2)$$

The coproduct is the homomorphism $\Delta: M(R, P) \to M(R, P)^{\otimes 2}$ defined on generators by

$$\Delta(M_b^a) = \sum_c \dot{M}_c^a \ddot{M}_b^c. \qquad (A.3)$$

We shall henceforth refer to the algebra $M(R, P)^{\otimes 2}$ as $M^{(2;P)}(R)$, to avoid the tensor sign in the name of the algebra with a view to endow the space $M(R, F)^{\otimes 2}$ with a non-canonical algebra structure. The upper '2' in the symbol $M^{(2;P)}(R)$ means that the algebra is generated by two copies of the quantum matrix, while the upper 'P' means that we use the flip operator $P$ to create the second copy $\dot{M}_2 = P \dot{M}_1 P$, of the quantum matrix $\dot{M}_1$, which, according to (A.2), commutes with the quantum matrix $\ddot{M}_1$.

(ii) Now for a general compatible pair $\{R, F\}$ define the unital algebra generated by the two quantum matrices — by the elements $\|\dot{M}_a^b\|_{a,b=1}^N$ and $\|\ddot{M}_a^b\|_{a,b=1}^N$ which satisfy the relations

$$R_1 \dot{M}_1 \dot{M}_2 = \dot{M}_1 \dot{M}_2 R_1, \quad R_1 \ddot{M}_1 \ddot{M}_2 = \ddot{M}_1 \ddot{M}_2 R_1, \qquad (A.4)$$

but the copies $\dot{M}$ and $\ddot{M}$ no longer commute; instead, they have the following commutation relations

$$\ddot{M}_1 \dot{M}_2 = \dot{M}_2 \ddot{M}_1. \qquad (A.5)$$

For a strict skew-invertible $F$ these commutation relations allow to express any product $\dot{M}_b^a \ddot{M}_d^c$ as a linear combination of products $\ddot{M}_d^{c'} \dot{M}_b^{a'}$ and, vice versa any product $\ddot{M}_b^a \dot{M}_d^c$ as a linear combination of products $\dot{M}_d^{c'} \ddot{M}_b^{a'}$.

Let us use the symbol $M^{(2;F)}(R)$ for the algebra with the $\|\dot{M}_a^b\|_{a,b=1}^N$ and $\|\ddot{M}_a^b\|_{a,b=1}^N$ and the defining relations (A.4)-(A.5).

Let $\Phi_F$ be the algebra generated by the elements $\|\dot{M}_a^b\|_{a,b=1}^N$ and $\|\ddot{M}_a^b\|_{a,b=1}^N$ which satisfy just the commutation relations relations (A.5); the subalgebras of $\Phi_F$ generated by any of two sets of elements, either $\|\dot{M}_a^b\|_{a,b=1}^N$ or $\|\ddot{M}_a^b\|_{a,b=1}^N$, is a free algebra. Thus, the algebra $M^{(2;F)}(R)$ is the quotient of the algebra $\Phi_F$.

It is not difficult to check that

$$\begin{gathered} \textit{all matrix components of the copy } \dot{M} \qquad (A.6) \\ \textit{commute with all the components of the tensor } R_1 \ddot{M}_1 \ddot{M}_2 - \dot{M}_1 \ddot{M}_2 R_1 \end{gathered}$$
in the algebra $\Phi_F$. The convenient way of checking it is to consider the expression $(R_1 \ddot{M}_1 \ddot{M}_2 - \dot{M}_1 \dot{M}_2 R_1) \dot{M}_3$ and to move $\ddot{M}_3$ to the left. Again, by the strict skew invertibility of $F$ this is sufficient to check the statement (A.6).

Similarly, it is not difficult to check that all matrix components of the copy $\ddot{M}$ commute with all the components of the tensor $R_1 \dot{M}_1 \dot{M}_2 - \dot{M}_1 \ddot{M}_2 R_1$; here it is convenient to consider the combination $\ddot{M}_1(R_2 \dot{M}_2 \dot{M}_3 - \dot{M}_2 \ddot{M}_3 R_2)$ and to move $\ddot{M}_1$ to the right (this is again sufficient).

This argument shows that the underlying vector space of the algebra $\mathcal{M}^{(2;F)}(R)$ is the tensor square $\mathcal{M}(R, F) \otimes \mathcal{M}(R, F)$ of the vector space underlying the algebra $\mathcal{M}(R, F)$.

(iii) To construct the braided coproduct $\Delta$ on the algebra $\mathcal{M}(R, F)$ we first define the map from the generators of the algebra $\mathcal{M}(R, F)$ to the algebra $\mathcal{M}^{(2;F)}(R)$ by the same formula (A.3)

$$M_b^a \mapsto \sum_c \dot{M}_c^a \ddot{M}_b^c.$$

It is now straightforward to check that this map extends to the homomorphism $\Delta$ from the algebra $\mathcal{M}(R, F)$ to the algebra $\mathcal{M}^{(2;F)}(R)$.

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