On the representability of actions of unital algebras
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909034363224064 |
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| author | Mancini, Manuel Piazza, Federica Vienne, Corentin |
| author_facet | Mancini, Manuel Piazza, Federica Vienne, Corentin |
| contents | Working in the setting of ideally exact categories, we investigate the representability of actions of unital non-associative algebras over a field. We show that, in general, such categories fail to be action representable: for instance, the category of all unital algebras is not even action accessible.
We then consider this problem in the context of operadic, action accessible, unit-closed varieties. Using the construction of the external weak actor, we prove that for any algebra $X$ in such a variety $\mathsf{V}$, the canonical map into its external weak actor is an isomorphism if and only if $X$ is unital. Consequently, the ideally exact category $\mathsf{V}_1$ of unital algebras in $\mathsf{V}$ is action representable, and the actor of $X$ is $X$ itself.
Finally, we prove action representability for unital Poisson algebras via an explicit construction of the universal strict general actor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04488 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the representability of actions of unital algebras Mancini, Manuel Piazza, Federica Vienne, Corentin Category Theory Rings and Algebras 08A35, 08C05, 16B50, 16W25, 17A36, 17B63, 18E13 Working in the setting of ideally exact categories, we investigate the representability of actions of unital non-associative algebras over a field. We show that, in general, such categories fail to be action representable: for instance, the category of all unital algebras is not even action accessible. We then consider this problem in the context of operadic, action accessible, unit-closed varieties. Using the construction of the external weak actor, we prove that for any algebra $X$ in such a variety $\mathsf{V}$, the canonical map into its external weak actor is an isomorphism if and only if $X$ is unital. Consequently, the ideally exact category $\mathsf{V}_1$ of unital algebras in $\mathsf{V}$ is action representable, and the actor of $X$ is $X$ itself. Finally, we prove action representability for unital Poisson algebras via an explicit construction of the universal strict general actor. |
| title | On the representability of actions of unital algebras |
| topic | Category Theory Rings and Algebras 08A35, 08C05, 16B50, 16W25, 17A36, 17B63, 18E13 |
| url | https://arxiv.org/abs/2503.04488 |