Centers and automorphisms of PI Quantum Matrix Algebras

TL;DR

This paper investigates the structure of centers and automorphisms of PI quantum matrix algebras, revealing conditions under which automorphisms are graded and describing the automorphism groups in various parameter cases.

Contribution

It provides a detailed analysis of automorphisms and centers of PI quantum matrix algebras, including presentations of centers and classification of automorphisms in different parameter regimes.

Findings

All automorphisms are graded when the center is a polynomial ring in multi-parameter cases for n=2,3.

The center's presentation is determined in the single-parameter case.

Automorphism groups are not graded in the single-parameter case, but certain automorphisms are explicitly described.

Abstract

We study PI quantum matrix algebras and their automorphisms using the noncommutative discriminant. In the multi-parameter case at $n=2$ and $n=3$, we show that all automorphisms are graded when the center is a polynomial ring. In the single-parameter case, we determine a presentation of the center and show that the automorphism group is not graded, though are able to describe certain families of automorphisms in this case, as well as those of certain subalgebras.