Orthogonal and Symplectic Quantum Matrix Algebras and Cayley-Hamilton Theorem for them

TL;DR

This paper extends the Cayley-Hamilton theorem to orthogonal and symplectic quantum matrix algebras, explores characteristic subalgebras in Birman-Murakami-Wenzl type QM-algebras, and establishes recursive relations among generators.

Contribution

It introduces quantum analogues of the Cayley-Hamilton theorem for specific algebra families and analyzes the structure of characteristic subalgebras with new recursive relations.

Findings

Derived Cayley-Hamilton theorems for orthogonal and symplectic QM-algebras

Identified generators and recursive relations in characteristic subalgebras

Established reciprocal relations for orthogonal QM-algebra generators

Abstract

For families of orthogonal and symplectic types quantum matrix (QM-) algebras, we derive corresponding versions of the Cayley-Hamilton theorem. For a wider family of Birman-Murakami-Wenzl type QM-algebras, we investigate a structure of its characteristic subalgebra (the subalgebra in which the coefficients of characteristic polynomials take values). We define 3 sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. For the family of the orthogonal type QM-algebras, additional reciprocal relations for the generators of the characteristic subalgebra are obtained.