The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring
type: Item
instance of: scholarly article
title: The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring
zbMATH Open document ID: 0793.16016
DOI: 10.1016/0022-4049(93)90106-4
author: Christopher R. Stover
published in: Journal of Pure and Applied Algebra
publication date: 1994-03-09T00:00:00Z
review text: For \(R\) a unital commutative ring, a twisted Lie algebra over \(R\) (or \(R\Sigma_ x\)-Lie algebra) is a graded \(R\)-module \(L = (L_ 0,L_ 1,L_ 2,\dots)\) with a bracket \(L_ p \otimes L_ q \to L_{p+q}\), and an action of the symmetric group \(\Sigma_ n\) on \(L_ n\) such that certain ``Lie identities'' hold. \(L\) is connected if \(L_ 0 = 0\). Similarly, one defines twisted algebras, (here the 0-th component is \(R\)), and twisted Hopf algebras. One also has twisted modules for these various twisted types of algebras. The author generalizes the Poincaré- Birkhoff-Witt theory of Lie algebras to twisted Lie algebras. Every twisted Lie algebra has a universal enveloping twisted Hopf algebra, and the category of twisted connected Lie algebras is shown to be equivalent to a category of connected twisted Hopf algebras with cocommutative comultiplication. A PBW-like decomposition is obtained for the enveloping algebra of a connected twisted Lie algebra, and free twisted Lie algebras are shown to admit a basis of simple (left-normed) brackets of generators.
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 179080
zbMATH Keywords: Lie identities
zbMATH Keywords: twisted Lie algebra
zbMATH Keywords: twisted algebras
zbMATH Keywords: twisted Hopf algebras
zbMATH Keywords: twisted modules
zbMATH Keywords: Poincaré-Birkhoff-Witt theory
zbMATH Keywords: universal enveloping twisted Hopf algebra
zbMATH Keywords: twisted connected Lie algebras
zbMATH Keywords: cocommutative comultiplication
zbMATH Keywords: free twisted Lie algebras
zbMATH Keywords: generators
Wikidata QID: Q115364420
reviewed by: Earl J. Taft
MaRDI profile type: Publication
full work available at URL: 0022-4049(93)90106-4
OpenAlex ID: W2087078124
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description: scientific article; zbMATH DE number 179080