Cocommutative elements form a maximal commutative subalgebra in quantum matrices

TL;DR

This paper proves that cocommutative elements form a maximal commutative subalgebra in the quantized coordinate rings of matrices and related groups when the deformation parameter is not a root of unity.

Contribution

It establishes that the subalgebras of cocommutative elements are the centralizers of the trace and are maximal commutative subalgebras in these quantum algebras.

Findings

Cocommutative subalgebras are centralizers of the trace.

These subalgebras are maximal commutative.

Results hold for non-root of unity q.

Abstract

In this paper we prove that the subalgebras of cocommutative elements in the quantized coordinate rings of $M_{n}$, $GL_{n}$ and $SL_{n}$ are the centralizers of the trace $x_{1,1}+\dots+x_{n,n}$ in each algebra, for $q\in\mathbb{C}^{\times}$ being not a root of unity. In particular, it is not only a commutative subalgebra as it was known before, but it is a maximal one.