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\(G\)-identities of non-associative algebras. (English. Russian original) Zbl 0939.17002

Sb. Math. 190, No. 11, 1559-1570 (1999); translation from Mat. Sb. 190, No. 11, 3-14 (1999).
Let \(Q\) be an abelian group and \(A=\sum_{q\in Q}A_q\) be an algebra over a field \(F\) graded by \(Q\). A \(Q\)-grading is called finite if there is a finite subset \(P\) of \(Q\) with \(A_q=\{0\}\) for any \(q\notin P\). An algebra \(A\) is called an algebra of Lie type if for any \(g,h,k\in Q\) there are scalars \(\alpha,\beta\in F\) such that \(\alpha\neq 0\) and \(a(bc)=\alpha(ab)c+ \beta(ac)b\) for all \(a\in A_g\), \(b\in A_h\), \(c\in A_k\). The class of algebras of Lie type includes, in particular, associative algebras, Lie algebras and superalgebras, Leibniz algebras, quantum Lie algebras, and many others. The authors study algebras of Lie type with a finite \(Q\)-grading on which a finite group \(G\) acts by automorphisms and anti-automorphisms.
The main result of the article (Theorem 2) states that if such an algebra \(A\) satisfies an essential \((G,Q)\)-identity of degree \(d\), then \(A\) satisfies an ordinary polynomial identity of degree bounded by a function of \(d\), \(|G|\) and \(|P|\), where \(|P|\) is the number of non-zero summands in the \(Q\)-grading of \(A\). Let \(A^G\) (respectively \({}^GA\)) denote the subspace of (skew) invariants of the action of \(G\) on \(A\). The second result of the article (Theorem 3) states that if \(A\) is a Lie algebra over a field \(F\), \(\operatorname{char} F\neq 2\), \((\operatorname{char} F,|G|)= 1\) and \({}^GA\) satisfies a non-trivial identity of degree \(d\) then the whole \(A\) satisfies a non-trivial identity of degree depending only on \(d\) and \(|G|\).
The authors note that an analogue of Theorem 3 with replacing of \({}^GA\) by \(A^G\) is not true. But if \(A\) is an associative algebra then an analogue of Theorem 3 holds for \({}^GA\) and for \(A^G\) as well (Corollary 1).
Theorem 4 states that if \(A\) is an alternative algebra with involution over a field \(F\), \(\operatorname{char} F\neq 2\) and the subspace of symmetric (or skew-symmetric) elements of \(A\) satisfies a non-trivial identity then \(A\) is an alternative PI-algebra. This fact generalizes Amitsur’s theorem on polynomial identities of associative algebras with involution.
The last result of the article (Corollary 2) is an analogue of Corollary 1 by replacing \({}^GA\) by \(A^G\) in the case when \(A\) is an alternative algebra and the group \(G\) is solvable.

MSC:

17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
17B01 Identities, free Lie (super)algebras
17D05 Alternative rings
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