Varieties of group-graded algebras of proper central exponent greater than two
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| Format: | Preprint |
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2025
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| _version_ | 1866908360262025216 |
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| author | Benanti, F. S. Valenti, A. |
| author_facet | Benanti, F. S. Valenti, A. |
| contents | Let $F$ be a field of characteristic zero and let $ \mathcal V $ be a variety of associative $F$-algebras graded by a finite abelian group $G$. To a variety $ \mathcal V $ is associated a numerical sequence called the sequence of proper central $G$-codimensions, $c^{G,δ}_n(\mathcal V), \, n \ge 1.$ Here $c^{G,δ}_n(\mathcal V)$ is the dimension of the space of multilinear proper central $G$-polynomials in $n$ fixed variables of any algebra $A$ generating the variety $\mathcal V.$
Such sequence gives information on the growth of the proper central $G$-polynomials of $A$ and in \cite{LMR} it was proved that $exp^{G,δ}(\mathcal V)=\lim_{n\to\infty}\sqrt[n]{c_n^{G,δ}(\mathcal V)}$ exists and is an integer called the proper central $G$-exponent.
The aim of this paper is to characterize the varieties of associative
$G$-graded algebras of proper central $G$-exponent greater than two.
To this end we construct a finite list of $G$-graded algebras and we prove that $exp^{G,δ}(\mathcal V) >2$ if and only if at least one of the algebras belongs to $\mathcal V$.
Matching this result with the characterization of the varieties of almost polynomial growth given in \cite{GLP}, we obtain a characterization of the varieties of proper central $G$-exponent equal to two. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_07410 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Varieties of group-graded algebras of proper central exponent greater than two Benanti, F. S. Valenti, A. Rings and Algebras Primary 16R10, 16R50, Secondary 16P90, 16W50 Let $F$ be a field of characteristic zero and let $ \mathcal V $ be a variety of associative $F$-algebras graded by a finite abelian group $G$. To a variety $ \mathcal V $ is associated a numerical sequence called the sequence of proper central $G$-codimensions, $c^{G,δ}_n(\mathcal V), \, n \ge 1.$ Here $c^{G,δ}_n(\mathcal V)$ is the dimension of the space of multilinear proper central $G$-polynomials in $n$ fixed variables of any algebra $A$ generating the variety $\mathcal V.$ Such sequence gives information on the growth of the proper central $G$-polynomials of $A$ and in \cite{LMR} it was proved that $exp^{G,δ}(\mathcal V)=\lim_{n\to\infty}\sqrt[n]{c_n^{G,δ}(\mathcal V)}$ exists and is an integer called the proper central $G$-exponent. The aim of this paper is to characterize the varieties of associative $G$-graded algebras of proper central $G$-exponent greater than two. To this end we construct a finite list of $G$-graded algebras and we prove that $exp^{G,δ}(\mathcal V) >2$ if and only if at least one of the algebras belongs to $\mathcal V$. Matching this result with the characterization of the varieties of almost polynomial growth given in \cite{GLP}, we obtain a characterization of the varieties of proper central $G$-exponent equal to two. |
| title | Varieties of group-graded algebras of proper central exponent greater than two |
| topic | Rings and Algebras Primary 16R10, 16R50, Secondary 16P90, 16W50 |
| url | https://arxiv.org/abs/2505.07410 |