Identities of graded algebras
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 \(F\) satisfies an identity. Applications for (colour) Lie superalgebras are given as well. (Also submitted to MR).
scientific article; zbMATH DE number 1218703
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