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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2203.13627 |
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| _version_ | 1866910285391986688 |
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| author | Militaru, G. |
| author_facet | Militaru, G. |
| contents | We prove that any Bernstein algebra $(A, ω)$ is isomorphic to a semidirect product $V \ltimes_{(\cdot, \, Ω)} \, k$ associated to a commutative algebra $(V, \cdot)$ such that $(x^2)^2 = 0$, for all $x\in A$ and an idempotent endomorphism $Ω= Ω^2 \in {\rm End}_k (V)$ of $V$ satisfying two compatibility conditions. The set of types of $(1 + |I|)$-dimensional Bernstein algebras is parametrized by an explicitely constructed (using linear algebra tools) classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups $(V, +) \ltimes {\rm GL}_k (V)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_13627 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the structure and classification of Bernstein algebras Militaru, G. Rings and Algebras 17A60, 17A30, 17D92 We prove that any Bernstein algebra $(A, ω)$ is isomorphic to a semidirect product $V \ltimes_{(\cdot, \, Ω)} \, k$ associated to a commutative algebra $(V, \cdot)$ such that $(x^2)^2 = 0$, for all $x\in A$ and an idempotent endomorphism $Ω= Ω^2 \in {\rm End}_k (V)$ of $V$ satisfying two compatibility conditions. The set of types of $(1 + |I|)$-dimensional Bernstein algebras is parametrized by an explicitely constructed (using linear algebra tools) classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups $(V, +) \ltimes {\rm GL}_k (V)$. |
| title | On the structure and classification of Bernstein algebras |
| topic | Rings and Algebras 17A60, 17A30, 17D92 |
| url | https://arxiv.org/abs/2203.13627 |