Let $G$ be a finite abelian group and $A$ a $G$-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by $A$. We study the asymptotic behavior of $c_n^G(A), \ n=1,2, \ldots,$ the sequence of graded codimensions of $A$ and we prove that if $A$ satisfies an ordinary polynomial identity, $\lim_{n\to \infty}\sqrt[n]{c_n^G(A)}$ exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple $G\times \mathbb{Z}_2$-graded algebra related to $A$.

Giambruno, A., La Mattina, D. (2010). Graded polynomial identities and codimensions: computing the exponential growth. ADVANCES IN MATHEMATICS, 225, 859-881 [10.1016/j.aim.2010.03.013].

Graded polynomial identities and codimensions: computing the exponential growth

GIAMBRUNO, Antonino;LA MATTINA, Daniela
2010-01-01

Abstract

Let $G$ be a finite abelian group and $A$ a $G$-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by $A$. We study the asymptotic behavior of $c_n^G(A), \ n=1,2, \ldots,$ the sequence of graded codimensions of $A$ and we prove that if $A$ satisfies an ordinary polynomial identity, $\lim_{n\to \infty}\sqrt[n]{c_n^G(A)}$ exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple $G\times \mathbb{Z}_2$-graded algebra related to $A$.
2010
Giambruno, A., La Mattina, D. (2010). Graded polynomial identities and codimensions: computing the exponential growth. ADVANCES IN MATHEMATICS, 225, 859-881 [10.1016/j.aim.2010.03.013].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/54659
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