Quantum exceptional group $G_2$ and its conjugacy classes

TL;DR

This paper constructs quantum analogs of semisimple conjugacy classes in the exceptional group G_2, using highest weight modules of the quantum group U_q(g) to realize their quantized polynomial algebras.

Contribution

It introduces a method to quantize conjugacy classes of G_2 via representations in highest weight modules, establishing isomorphisms for classes in the same Weyl orbit.

Findings

Quantization of semisimple conjugacy classes of G_2 achieved.

Realization of quantized polynomial algebras via linear operators.

Isomorphism of quantizations for points in the same Weyl orbit.

Abstract

We construct quantization of semisimple conjugacy classes of the exceptional group $G=G_2$ along with and by means of their exact representations in highest weight modules of the quantum group $U_q(\mathfrak{g})$. With every point $t$ of a fixed maximal torus we associate a highest weight module $M_t$ over $U_q(\mathfrak{g})$ and realize the quantized polynomial algebra of the class of $t$ by linear operators on $M_t$. Quantizations corresponding to points of the same orbit of the Weyl group are isomorphic.