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Reciprocal relations for orthogonal quantum matrices

Pavel Pyatov, Oleg Ogievetsky

TL;DR

This work develops the theory of orthogonal BMW-type quantum matrix algebras by analyzing their characteristic subalgebras and the reciprocal relations among elementary sums. It proves that in the orthogonal case the elementary sums satisfy quadratic reciprocal relations and resolves these relations for the extended algebra ${\\cal M^{\\bullet}}(R,F)$ by incorporating the inverse contraction $g^{-1}$, while deriving explicit commutation relations between the quantum determinant det$_q M$ and the generators via an operator $O$. The authors compute representations ${\\alpha}_{\\pm}$ of the QM-algebra, showing the quantum determinant and elementary sums map to scalar operators involving $O$ (e.g., ${\\alpha}_{\\pm}(\\goth{e}_k)=(-1)^{k-1}q^{k(1-k)}O^{-1}$), and establish centrality and grouping into components for even rank through central elements like $g^{-\\\ell}\\goth{e}_{2\\ell}$. Finally, they obtain a detailed resolution of reciprocal relations, including a two-component decomposition for even $k$ and a square-root construction for odd $k$, providing a complete eigenstructure-like description of orthogonal quantum matrices in this framework.

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 reciprocal relations for these generators. 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.

Reciprocal relations for orthogonal quantum matrices

TL;DR

This work develops the theory of orthogonal BMW-type quantum matrix algebras by analyzing their characteristic subalgebras and the reciprocal relations among elementary sums. It proves that in the orthogonal case the elementary sums satisfy quadratic reciprocal relations and resolves these relations for the extended algebra by incorporating the inverse contraction , while deriving explicit commutation relations between the quantum determinant det and the generators via an operator . The authors compute representations of the QM-algebra, showing the quantum determinant and elementary sums map to scalar operators involving (e.g., ), and establish centrality and grouping into components for even rank through central elements like . Finally, they obtain a detailed resolution of reciprocal relations, including a two-component decomposition for even and a square-root construction for odd , providing a complete eigenstructure-like description of orthogonal quantum matrices in this framework.

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 reciprocal relations for these generators. 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.

Paper Structure

This paper contains 17 sections, 17 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. BMW algebras
  4. Yang--Baxter representations and R-trace
  5. Compatible Yang--Baxter pairs
  6. Finite height and orthogonal Yang--Baxter matrices
  7. Finite height
  8. Orthogonal case
  9. Rank-one projectors
  10. Quantum matrix algebra in the orthogonal case
  11. Orthogonal quantum matrix algebra
  12. Characteristic subalgebra
  13. Reciprocal relations
  14. Group-like elements in the orthogonal QM-algebras
  15. Representations of quantum matrix algebras
  16. ...and 2 more sections

Key Result

Lemma 3.3

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

Theorems & Definitions (29)

  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.5
  • Definition 4.1
  • Remark 4.2
  • Remark 4.4
  • Proposition 4.5
  • Theorem 4.7
  • ...and 19 more