Quantum Matrix Algebras of BMW type: Structure of the Characteristic Subalgebra

TL;DR

This paper explores the structure of characteristic subalgebras in BMW type quantum matrix algebras, generalizing classical matrix properties within a quantum algebra framework and establishing new algebraic relations.

Contribution

It introduces a detailed analysis of characteristic subalgebras in BMW type QM-algebras, including recursive relations and a generalized matrix product.

Findings

Defined generating elements of the characteristic subalgebra

Derived recursive Newton and Wronski relations

Proved $ ext{star}$-commutativity of matrix descendants

Abstract

A notion of quantum matrix (QM-) algebra generalizes and unifies two famous families of algebras from the theory of quantum groups: the RTT-algebras and the reflection equation (RE-) algebras. These algebras being generated by the components of a `quantum' matrix $M$ possess certain properties which resemble structure theorems of the ordinary matrix theory. It turns out that such structure results are naturally derived in a more general framework of the QM-algebras. In this work we consider a family of Birman-Murakami-Wenzl (BMW) type QM-algebras. These algebras are defined with the use of R-matrix representations of the BMW algebras. Particular series of such algebras include orthogonal and symplectic types RTT- and RE- algebras, as well as their super-partners. For a family of BMW type QM-algebras, we investigate the structure of their `characteristic subalgebras' --- the subalgebras where the coefficients of characteristic polynomials take values. We define three sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. We also define an associative $\star$-product for the matrix $M$ of generators of the QM-algebra which is a proper generalization of the classical matrix multiplication. We determine the set of all matrix `descendants' of the quantum matrix $M$, and prove the $\star$-commutativity of this set in the BMW type.