Lie identities for skew and symmetric elements of semiprime superalgebras with superinvolution.
type: Item
instance of: scholarly article
title: Lie identities for skew and symmetric elements of semiprime superalgebras with superinvolution.
zbMATH Open document ID: 1269.16037
author: F. Catino
author: Roberto Rizzo
author: Ernesto Spinelli
published in: Journal of Algebra
publication date: 2013-04-17T00:00:00Z
review text: For a semiprime associative superalgebra \(A\) the intersection of the prime ideals of \(A\) is zero and so \(A\) is a subdirect product of its prime images. When each prime image of \(A\) is a central order in a simple superalgebra at most \(n^2\)-dimensional over its center, we say that \(A\) is \(S(n)\). For an algebra \(A\) with involution \(*\) denote by \(H\) (respectively \(K\)) the set of all elements \(x\in A\) such that \(x^*=x\) (respectively \(x^*=-x\)). Now for an associative superalgebra \(A\) over a (commutative associative and unital) ring \(R\), denote by \(A^-\) the Lie superalgebra whose underlying \(R\)-module agrees with that of \(A\) and whose Lie superbracket satisfies \([a,b]=ab-(-1)^{\overline a\overline b}ba\) for any homogeneous elements \(a,b\) of degrees \(\overline a\) and \(\overline b\), respectively. For any \(m\geq 3\) and elements \(x_1,\dots,x_m\in A\) we can inductively define \([x_1,\dots,x_m]:=[[x_1,\dots,x_{m-1}],x_m]\) and so a subspace \(V\) of \(A\) is said to be Lie nilpotent if there exists an \(m\) such that \([x_1,\dots,x_m]=0\) for any \(x_1,\dots,x_m\in V\). In a similar way we can define Lie solvability of the subspace \(V\). The first main result of this paper is that if \(A\) is a nontrivial semiprime associative superalgebra over a commutative ring of scalars \(R\) with \({1\over 2}\in R\) and \(A\) has a superinvolution, then if \(H\) is Lie solvable \(A\) is \(S(2)\). The second main result is that if \(A\) is a nontrivial semiprime associative superalgebra over a commutative ring of scalars \(R\) with \({1\over 2}\in R\) and \(A\) has a superinvolution then: (a) \(H\) is Lie nilpotent if and only if the elements of \(H\) commute. (b) \(K\) is Lie nilpotent if and only if the elements of \(K\) commute. Some other interesting results can be found in section 3 of the paper. So for instance the authors characterize nontrivial prime associative superalgebras with superinvolution whose skew elements or symmetric elem…
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 6154886
zbMATH Keywords: semiprime superalgebras
zbMATH Keywords: superinvolutions
zbMATH Keywords: skew elements
zbMATH Keywords: symmetric elements
zbMATH Keywords: Lie superalgebras
zbMATH Keywords: Lie superbrackets
zbMATH Keywords: Lie nilpotence
zbMATH Keywords: Lie solvability
reviewed by: Candido Martín González
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.07.007
OpenAlex ID: W2050414951
Wikidata QID: Q115351082
cites work: Lie nilpotence of group rings
cites work: On the jordan H-submodules of a simple superalgebra with superinvolution∗
cites work: On the lie structure of the skew elements of a prime superalgebra with superinvolution
cites work: On herstein's constructions relating jordan and associative superalgebras
cites work: On the Lie structure of the skew elements of a simple superalgebra with superinvolution
cites work: Q4112071
cites work: Lie superalgebras
cites work: Classification of simple z-graded lie superalgebras and simple jordan superalgebras
cites work: Q4063291
cites work: Lie structure in semiprime superalgebras with superinvolution.
cites work: On Certain Semiprime Associative Superalgebras
cites work: Regularity conditions on skew and symmetric elements in superalgebras with superinvolution
cites work: Group algebras whose symmetric and skew elements are Lie solvable
cites work: On the lie structure of associative superalgebras
cites work: Constructing simple Lie superalgebras from associative graded algebras
DOI: 10.1016/J.JALGEBRA.2012.07.007
Recommended article: On the lie structure of the skew elements of a prime superalgebra with superinvolution
Recommended article: On the Lie structure of the skew elements of a simple superalgebra with superinvolution
Recommended article: Lie structure in semiprime superalgebras with superinvolution.
Recommended article: The derived superalgebra of skew elements of a semiprime superalgebra with superinvolution
Recommended article: Lie superalgebras with semisimple even part
Recommended article: On the Lie structure of a prime associative superalgebra
Recommended article: Lie superhomomorphisms in superalgebras with superinvolution
Recommended article: Regularity conditions on skew and symmetric elements in superalgebras with superinvolution
Recommended article: IDENTITIES IN THE UNIVERSAL ENVELOPING ALGEBRA FOR A LIE SUPERALGEBRA
Recommended article: Skew-symmetric identities in special Lie algebras
description: scientific article; zbMATH DE number 6154886