Lie stacks and their enveloping algebras
type: Item
instance of: scholarly article
title: Lie stacks and their enveloping algebras
zbMATH Open document ID: 0917.16018
author: Lieven Le Bruyn
published in: Advances in Mathematics
publication date: 1997-11-06T00:00:00Z
review text: The purpose of this clearly written article is to introduce the notion of ``Lie stack'' and to study the first general properties of this mathematical object. A Lie stack is an algebra morphism \({\mathfrak s}\colon A\to A\otimes B\) where \(A\) and \(B\) are finite dimensional (associative) \(\mathbb{C}\)-algebras with \(B\) augmented local. Let \(\mathfrak g\) be a finite dimensional Lie algebra with basis \(g_1,\dots,g_n\) and realize it as a Lie subalgebra of \(\text{Der }A\) for some finite dimensional associative algebra \(A\), e.g. by Ado's theorem. Let \(B=\mathbb{C}[y_1,\dots,y_n]/(y_iy_j)\); then \({\mathfrak s}\colon A\to A\otimes B\), \({\mathfrak s}(a)=a\otimes 1+\sum_i g_i(a)\otimes y_i\) is a Lie stack. So the notion of ``Lie stack'' extends in a way the notion of Lie algebra. To each Lie stack \(\mathfrak s\), the author attaches an irreducible pointed Hopf algebra \(U({\mathfrak s})\); if \(\mathfrak s\) comes from a Lie algebra \(\mathfrak g\) as above then \(U({\mathfrak s})\simeq U({\mathfrak g})\). Simple conditions on \(\mathfrak s\) are given which are equivalent to the commutativity, resp. cocommutativity, of \(U({\mathfrak s})\); as a consequence new examples of non-commutative non-cocommutative Hopf algebras are presented. Further, regularity (homological) properties of \(U({\mathfrak s})\) are investigated. It is shown that \(U({\mathfrak s})\) is a domain with a PBW basis; if \(\text{GKdim }U({\mathfrak s})\) is finite then \(U({\mathfrak s})\) is an Auslander regular noetherian domain satisfying the Cohen-Macaulay property. The author also explains one of his main motivations, namely to interpret Lie stacks as deformations of Lie algebras. For more information see [\textit{L. Le Bruyn}, J. Algebra 196, No. 2, 552-577 (1997; Zbl 0892.17013)].
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 1066078
zbMATH Keywords: Lie stacks
zbMATH Keywords: Hopf algebras
zbMATH Keywords: deformations of Lie algebras
zbMATH Keywords: irreducible pointed Hopf algebras
reviewed by: Nicolás Andruskiewitsch
MaRDI profile type: Publication
full work available at URL: aima.1997.1653
OpenAlex ID: W2023465333
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DOI: 10.1006/AIMA.1997.1653
description: scientific article; zbMATH DE number 1066078