Dimension and enumeration of primitive ideals in quantum algebras

TL;DR

This paper investigates the structure of primitive ideals in quantum algebras with torus actions, proving a dimension theorem, providing a combinatorial criterion for primitivity, and enumerating primitive ideals in quantum matrices.

Contribution

It introduces a quantum analogue of Dixmier's theorem, a combinatorial criterion for primitive ideals, and an explicit enumeration formula for primitive ideals in quantum matrices.

Findings

Gelfand-Kirillov dimension of primitive factors is always even.

A combinatorial criterion for torus-invariant prime ideals to be primitive.

An explicit formula for counting primitive ideals in 2×n quantum matrices.

Abstract

In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of $2\times n$ quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the ``variety of $2\times n$ quantum matrices''.