Dimension growth and Gelfand-Kirillov dimension of representations of quantum groups

TL;DR

This paper explores algebraic invariants in quantum group representations, revealing new phenomena and classifying minimal values, with implications for understanding the structure of quantum modules across different Lie types.

Contribution

It introduces new insights into the behavior of dimension growth and Gelfand-Kirillov dimensions in quantum groups, especially regarding non-integral weights and minimal invariants.

Findings

Minimal non-zero invariants determined for each Lie type

Quantum cuspidal modules only occur for types A, B, C

New phenomena in quantum case related to non-integral weights

Abstract

We consider two algebraic invariants in the representation theory of quantized enveloping algebras: the dimension growth of simple modules for the De Concini-Kac quantum group at roots of unity, and the Gelfand-Kirillov dimension of simple highest weight modules for the quantum group at generic $q$. In spite of being defined for different values of the parameter $q$, these invariants reflect closely related features in the respective contexts. We show that several new phenomena appear in the quantum case and the representations with non-integral weights contribute to both invariants in a way that cannot be ignored. Building on this, we determine the minimal non-zero value of these invariants for each Lie type. As an application we show that quantum cuspidal modules at generic $q$ can occur only when the underlying semisimple Lie algebra has simple components of type $A$, $B$, or $C$, providing a more explicit representation-theoretic distinction with the classical case.