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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.10495 |
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| _version_ | 1866912707276439552 |
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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 with involution *. Associated to $ \mathcal V^* $ are three sequences: the sequence of \(*\)-codimensions \( c^{*}_n(\mathcal V^*) \), the sequence of central \(*\)-codimensions \( c^{*,z}_n(\mathcal V^*) \) and the sequence of proper central \(*\)-codimensions \( c^{*,δ}_n(\mathcal V^*) \).
These sequences provide information on the growth of, respectively, the *-polynomial identities, the central *-polynomial and the proper central *-polynomial of any generating algebra with involution $A$ of $ \mathcal V^*.$
In \cite{MR2022} it was proved that $exp^{*,δ}(\mathcal V^*)=\lim_{n\to\infty}\sqrt[n]{c_n^{*,δ}(\mathcal V^*)}$ exists and is an integer called the proper central $*$-exponent.
The aim of this paper is to study the varieties of associative
algebras with involution of proper central $*$-exponent greater than two.
To this end we construct a finite list of algebras with involution and we prove that if $exp^{*,δ}(\mathcal V^*) >2$, then at least one of these algebras belongs to $\mathcal V^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Star-Varieties of proper central exponent greater than two Benanti, F. S. Valenti, A. Rings and Algebras Let $F$ be a field of characteristic zero and let $ \mathcal V^* $ be a variety of associative $F$-algebras with involution *. Associated to $ \mathcal V^* $ are three sequences: the sequence of \(*\)-codimensions \( c^{*}_n(\mathcal V^*) \), the sequence of central \(*\)-codimensions \( c^{*,z}_n(\mathcal V^*) \) and the sequence of proper central \(*\)-codimensions \( c^{*,δ}_n(\mathcal V^*) \). These sequences provide information on the growth of, respectively, the *-polynomial identities, the central *-polynomial and the proper central *-polynomial of any generating algebra with involution $A$ of $ \mathcal V^*.$ In \cite{MR2022} it was proved that $exp^{*,δ}(\mathcal V^*)=\lim_{n\to\infty}\sqrt[n]{c_n^{*,δ}(\mathcal V^*)}$ exists and is an integer called the proper central $*$-exponent. The aim of this paper is to study the varieties of associative algebras with involution of proper central $*$-exponent greater than two. To this end we construct a finite list of algebras with involution and we prove that if $exp^{*,δ}(\mathcal V^*) >2$, then at least one of these algebras belongs to $\mathcal V^*$. |
| title | Star-Varieties of proper central exponent greater than two |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2511.10495 |