Some quantum analogues of solvable Lie groups

TL;DR

This paper explores quantum analogues of solvable Lie groups by analyzing subalgebras of quantized enveloping algebras, revealing their structure and representation theory, and confirming links to Poisson geometry.

Contribution

It introduces a detailed analysis of subalgebras related to unipotent and solvable subgroups, highlighting their non-commutative algebraic structures and representation dimensions.

Findings

Maximal dimensions of irreducible representations identified

Non-commutative structures resemble iterated twisted polynomial algebras

Representation theory linked to Poisson geometry

Abstract

In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras of twisted polynomials with a derivation, an object which has often appeared in the general theory of non-commutative rings. In particular, we find maximal dimensions of their irreducible representations. Our results confirm the validity of the general philosophy that the representation theory is intimately connected to the Poisson geometry.