Gelfand-Kirillov dimension of the algebra of regular functions on quantum groups

TL;DR

This paper proves that the Gelfand-Kirillov dimension of the algebra of regular functions on certain quantum groups equals the real dimension of the classical group, linking algebraic and geometric properties.

Contribution

It establishes a precise equality between the Gelfand-Kirillov dimension of quantum group function algebras and the classical group dimensions, for types A, C, and D.

Findings

Gelfand-Kirillov dimension equals the real dimension of G

Valid for quantum groups of types A, C, D

Connects algebraic and geometric properties of quantum groups

Abstract

Let $G_q$ be the $q$-deformation of a simply connected simple compact Lie group $G$ of type $A$, $C$ or $D$ and $\mathcal{O}_q(G)$ be the algebra of regular functions on $G_q$. In this article, we prove that the Gelfand-Kirillov dimension of $\mathcal{O}_q(G)$ is equal to the dimension of real manifold $G$.