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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2407.09666 |
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| _version_ | 1866909580333678592 |
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| author | Berele, Allan Danchev, Peter Tenner, Bridget Eileen |
| author_facet | Berele, Allan Danchev, Peter Tenner, Bridget Eileen |
| contents | We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{σ(1)} \cdots x_{σ(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $σ$ not fixing 1 or $n$ are eventually commutative in the sense that the equality $x_1\cdots x_k = x_{τ(1)} \cdots x_{τ(k)}$ holds for $k$ large enough and all permutations $τ\in S_k$. Calling the minimal such $k$ the degree of eventual commutativity, we prove that $k$ is never more than $2n-3$, and that this bound is sharp. For various natural examples, we prove that $k$ can be taken to be $n+1$ or $n+2$. In the case when $q \ne 1$, we establish that the algebra must be nilpotent.
We, moreover, demonstrate that if an algebra is eventually commutative of arbitrary characteristic, then it has a finite basis of its polynomial identities, thus confirming the Specht conjecture in this particular case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09666 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-Term Polynomial Identities Berele, Allan Danchev, Peter Tenner, Bridget Eileen Rings and Algebras Representation Theory We study algebras satisfying a two-term multilinear identity, namely one of the form $x_1 \cdots x_n= q x_{σ(1)} \cdots x_{σ(n)}$, where $q$ is a parameter from the base field. We show that such algebras with $q=1$ and $σ$ not fixing 1 or $n$ are eventually commutative in the sense that the equality $x_1\cdots x_k = x_{τ(1)} \cdots x_{τ(k)}$ holds for $k$ large enough and all permutations $τ\in S_k$. Calling the minimal such $k$ the degree of eventual commutativity, we prove that $k$ is never more than $2n-3$, and that this bound is sharp. For various natural examples, we prove that $k$ can be taken to be $n+1$ or $n+2$. In the case when $q \ne 1$, we establish that the algebra must be nilpotent. We, moreover, demonstrate that if an algebra is eventually commutative of arbitrary characteristic, then it has a finite basis of its polynomial identities, thus confirming the Specht conjecture in this particular case. |
| title | Two-Term Polynomial Identities |
| topic | Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2407.09666 |