Cayley--Hamilton Theorem for Orthogonal Quantum Matrix Algebras

TL;DR

This paper establishes a Cayley-Hamilton theorem analogue for orthogonal quantum matrix algebras, with distinct forms for odd and even cases, and introduces spectral parameterization of coefficients by quantum eigenvalues.

Contribution

It provides the first formulation of Cayley-Hamilton identities tailored for orthogonal quantum matrix algebras, differentiating between cases based on matrix height and component.

Findings

Different Cayley-Hamilton forms for odd and even heights.

Two versions of the theorem for positive and negative components.

Spectral parameterization of coefficients by quantum eigenvalues.

Abstract

For a family of the orthogonal $O(k)$ type Quantum Matrix algebras we establish an analogue of the Cayley--Hamilton theorem. The form of the Cayley-Hamilton identity is different in three cases. First, the cases of odd ($k=2\ell -1$) and even ($k=2\ell$) heights are different. Second, for even height orthogonal Quantum Matrix algebra we derive two versions of the Cayley--Hamilton theorem, one for its positive component $O^+(2\ell)$ and another one for the negative component $O^-(2\ell)$. In each case we introduce the spectral parameterization of the coefficients of the Cayley--Hamilton identity by the `eigenvalues' of the quantum matrices.