A combinatorial approach to quantification of Lie algebras

TL;DR

This paper introduces a quantum universal enveloping algebra for any Lie algebra, utilizing a quantum Lie operation, and explores its basis structures and primitive elements, especially for classical nilpotent algebras.

Contribution

It defines a new quantum universal enveloping algebra framework with a PBW basis and analyzes its primitive elements for classical nilpotent Lie algebras.

Findings

The enveloping algebra admits a PBW basis with Kashiwara crystallization.

All skew primitive elements are described for classical nilpotent algebras.

PBW-generators coincide with Lalonde-Ram basis in these cases.

Abstract

We propose a notion of a quantum universal enveloping algebra for an arbitrary Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits the Kashiwara crystalization. We describe all skew primitive elements of the quantum universal enveloping algebra for the classical nilpotent algebras of the infinite series defined by the Serre relations and prove that the set of PBW-generators for each of these enveloping algebras coincides with the Lalonde-Ram basis of the ground Lie algebra with a skew commutator in place of the Lie operation. The similar statement is valid for Hall-Shirshov basis of any Lie algebra defined by one relation, but it is not so in general case.