Steinberg-Leibniz algebras and superalgebras
type: Item
instance of: scholarly article
title: Steinberg-Leibniz algebras and superalgebras
zbMATH Open document ID: 1071.17001
author: Dong Liu
published in: Journal of Algebra
publication date: 2005-04-07T00:00:00Z
review text: The Steinberg Lie algebra \({\mathfrak s}{\mathfrak t}(n,A)\), \(n\geq 3\), over a unital associative algebra \(A\) is the universal central extension of the matrix Lie algebra \({\mathfrak s}{\mathfrak l}(n,A)\), and the Leibniz algebra \({\mathfrak s}{\mathfrak t}{\mathfrak l}(n,A)\) has a similar property in the category of Leibniz algebras. These facts motivate the author to study the Leibniz algebra \({\mathfrak s}{\mathfrak l}(n,D)\) over a unital associative dialgebra \(D\). He constructs the Steinberg Leibniz algebra \({\mathfrak s}{\mathfrak t}{\mathfrak l}(n,D)\) and establishes its basic properties. He proves that it is the universal central extension of \({\mathfrak s}{\mathfrak l}(n,D)\) and gives a description of the kernel which is a quotient of the first Hochschild homology group of the dialgebra \(D\). The results are connected with Leibniz algebras graded by finite root systems and with Leibniz \(K\)-theory. Finally, the author extends his results to the Steinberg Leibniz superalgebras \({\mathfrak s}{\mathfrak t}{\mathfrak l}(m,n,D)\).
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17A32
Mathematics Subject Classification ID: 17B68
Mathematics Subject Classification ID: 17D99
zbMATH DE Number: 2153360
zbMATH Keywords: Steinberg-Leibniz algebras
zbMATH Keywords: dialgebras
zbMATH Keywords: central extensions
Wikidata QID: Q123126390
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.08.005
OpenAlex ID: W1964183521
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DOI: 10.1016/J.JALGEBRA.2004.08.005
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description: scientific article; zbMATH DE number 2153360