Multiparameter quantum Cauchy-Binet formulas

TL;DR

This paper develops multiparameter quantum Cauchy-Binet formulas for coefficients related to the reflection equation algebra, providing explicit and computationally accessible formulas for quantum matrix invariants.

Contribution

It introduces Cauchy-Binet formulas for coefficients of the reflection equation algebra within the multiparameter quantized function algebra, including formulas for matrix inverses.

Findings

Derived explicit Cauchy-Binet formulas for quantum coefficients

Provided formulas for the inverse of quantum matrices involving these generators

Extended classical matrix identities to the multiparameter quantum setting

Abstract

The quantum Cayley-Hamilton theorem for the generator of the reflection equation algebra has been proven by Pyatov and Saponov, with explicit formulas for the coefficients in the Cayley-Hamilton formula. However, these formulas do not give an \emph{easy} way to compute these coefficients. Jordan and White provided an elegant formula for the coefficients given with respect to the generators of the reflection equation algebra. In this paper, we provide Cauchy-Binet formulas for these coefficients with respect to generators of $\mathcal{O}_{q,Q}^{\mathbb{R}}(M_{N}(\mathbb{C}))$, the multiparameter quantized $^{*}$-algebra of functions on $M_{N}(\mathbb{C})$ as a real variety, which contains the reflection equation algebra as a subalgebra. We also prove a Cauchy-Binet formula for the inverse of a matrix involving these generators.