Identities of graded algebras
type: Item
instance of: scholarly article
title: Identities of graded algebras
zbMATH Open document ID: 0920.16011
author: Yuri Bahturin
author: Mikhail Zaicev
published in: Journal of Algebra
publication date: 1999-01-22T00:00:00Z
review text: Let \(G\) be a semigroup with unity and cancellation, let \(H\) be a finite subset of \(G\) and let \(A=\sum_{g\in H}A_g\) be a nonassociative algebra over a field \(F\) with a (finite) \(G\)-grading. The algebra \(A\) is of Lie type if for any \(g,h,k\in G\) there exist constants \(\alpha,\beta\in F\), \(\alpha\not=0\), such that \(a(bc)=\alpha(ab)c+\beta(ac)b\) for all \(a\in A_g\), \(b\in A_h\), \(c\in A_k\). The class of algebras of Lie type includes all associative and Lie algebras, quantum Lie algebras, (colour) Lie superalgebras, etc. The paper under review is inspired by a result of \textit{J. Bergen} and \textit{M. Cohen} [Bull. Lond. Math. Soc. 18, 159-164 (1986; Zbl 0563.16003)] that if the unity component \(A_1\) of an associative algebra graded by a finite group \(G\) satisfies a polynomial identity, then the algebra \(A\) also satisfies a polynomial identity. Recently, a new proof of this result has been given [in \textit{Yu. Bahturin, A. Giambruno} and \textit{D. Riley}, Isr. J. Math. 104, 145-155 (1998; see the preceding review Zbl 0920.16010)] using only combinatorial arguments and showing that the degree of the polynomial identity of \(A\) can be expressed in terms of the degree of the identity of \(A_1\) and the order of \(G\). The main result of the paper under review gives that an algebra \(A\) of Lie type and with finite \(G\)-grading over any field, where \(G\) is a semigroup with unity and cancellation satisfies a polynomial identity of a special kind provided that the component \(A_1\) satisfies a polynomial identity of the same kind. The main applications of this result are for Lie algebras. In particular, if \(L\) is a Lie algebra graded by a finite group and the unity component of \(L\) satisfies a Lie polynomial identity, then so does \(L\). Another important consequence is that the Lie algebra \(L\) satisfies a polynomial identity if the algebra of fixed points \(L^G\) of a finite solvable automorphism group \(G\) of order invertible in…
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1218703
zbMATH Keywords: algebras with polynomial identities
zbMATH Keywords: graded Lie algebras
zbMATH Keywords: Lie superalgebras
zbMATH Keywords: algebras of Lie type
zbMATH Keywords: graded algebras
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
full work available at URL: jabr.1997.7017
OpenAlex ID: W2091388493
cites work: Q3996716
cites work: Group-graded algebras with polynomial identity
cites work: Infinite dimensional Lie superalgebras
cites work: Actions of Commutative Hopf Algebras
cites work: The fractional-graded extension of the Virasoro algebra
cites work: Q4063291
cites work: Q4009416
cites work: Fixed rings of finite automorphism groups of associative rings
cites work: Contractions of Lie algebras: Generalized Inönü–Wigner contractions versus graded contractions
cites work: Q4314482
DOI: 10.1006/JABR.1997.7017
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description: scientific article; zbMATH DE number 1218703