𝐺-identities on associative algebras

Bahturin, Y.; Giambruno, A.; Zaicev, M.

doi:10.1090/S0002-9939-99-04530-X

$G$-identities on associative algebras
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by Y. Bahturin, A. Giambruno and M. Zaicev

Proc. Amer. Math. Soc. 127 (1999), 63-69

DOI: https://doi.org/10.1090/S0002-9939-99-04530-X

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Abstract:

Let $R$ be an algebra over a field and $G$ a finite group of automorphisms and anti-automorphisms of $R$. We prove that if $R$ satisfies an essential $G$-polynomial identity of degree $d$, then the $G$-codimensions of $R$ are exponentially bounded and $R$ satisfies a polynomial identity whose degree is bounded by an explicit function of $d$. As a consequence we show that if $R$ is an algebra with involution $*$ satisfying a $*$-polynomial identity of degree $d$, then the $*$-codimensions of $R$ are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case $R$ must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

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