Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0002149

A combinatorial approach to quantification of Lie algebras

V. K. Kharchenko
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

V. K. Kharchenko. 2000-02-17. A combinatorial approach to quantification of Lie algebras. https://arxiv.org/abs/math/0002149

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On singular vectors and associated Weyl elements for simply-laced universal affine vertex operator algebras

Given a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ and a complex number $κ$, let $V^κ(\mathfrak{g})$ be the associated universal affine vertex algebra. Gorelik and Kac [GK07] gave a sufficient and necessary condition for $V^κ(\mathfrak{g})$ to be simple. In this paper, for simply-laced $\mathfrak{g}$ and non-critical $κ$, we determine the weights of singular vectors of $V^κ(\mathfrak{g})$ with minimal conformal weights, when $V^κ(\mathfrak{g})$ is not simple. We further determine all the longest Weyl elements in the Kashiwara-Tanisaki character theorem [KT00] which correspond to the weights of the singular vectors.

math.QA↗
arXiv2026

Modular invariants and NIM-reps

Given a pivotal module category over a spherical fusion category, we introduce the encircling module, a module over the fusion algebra defined using the pivotal structure, and prove that it is isomorphic to the NIM-rep as a fusion algebra module. When applied to the $\mathcal{TM}$ realisation of the modular invariant partition function (arXiv:1911.09024), this identifies the NIM-rep multiplicities with the entries $Z_{I,I^\vee}$ of the modular invariant (its diagonal relative to the charge-conjugation invariant, which is the one given by the regular module category), providing a categorical generalisation of Böckenhauer, Evans and Kawahigashi's results (arXiv:math/9907149). We also show that for indecomposable module categories the dimension condition on $\mathcal{TM}$ required for modular invariance is automatically satisfied, and that $\mathcal{TM}$ is, as an algebra, the full centre of the module category in the sense of Davydov, Kong and Runkel (arXiv:1307.5956), which for module categories of algebras is the full centre of Fjelstad, Fuchs, Runkel and Schweigert (arXiv:hep-th/0612306). Consequently $\mathcal{TM}$ gives rise to Cardy algebras in the sense of Kong and Runkel (arXiv:0807.3356).

math.QA↗
arXiv2026

A diagrammatic presentation for every pivotal pointed fusion category

We provide a generators and relations presentation of pivotal pointed fusion categories, $Vec(G,ω,π)$. Unlike the well-known skeletal model, our presentation is strict and allows multiple isomorphic objects. Our main tool is skein theory, which allows us to apply topological tools to understand the relations of morphisms in the category. This expository paper provides details of skein-theoretic work in the setup that allows multiple isomorphic labels in the generators, which didn't previously exist in the literature.

math.QA↗