Weak representability of actions of non-associative algebras
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915165055746048 |
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| author | Brox, Jose García-Martínez, Xabier Mancini, Manuel Van der Linden, Tim Vienne, Corentin |
| author_facet | Brox, Jose García-Martínez, Xabier Mancini, Manuel Van der Linden, Tim Vienne, Corentin |
| contents | We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field.
Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to study the representability of actions for them. Here we prove that the varieties of two-step nilpotent (anti-)commutative algebras and that of commutative associative algebras are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called amalgam.
Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object $E(X)$ for the actions on (or split extensions of) an object $X$. We actually obtain a partial algebra $E(X)$, which we call external weak actor of $X$, together with a monomorphism of functors ${\operatorname{SplExt}(-,X) \rightarrowtail \operatorname{Hom}(U(-),E(X))}$, which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor $\operatorname{USGA}(X)$ and that of $E(X)$ are described in detail. We end with some open questions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_02812 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Weak representability of actions of non-associative algebras Brox, Jose García-Martínez, Xabier Mancini, Manuel Van der Linden, Tim Vienne, Corentin Category Theory Rings and Algebras 08A35, 08C05, 16B50, 16W25, 17A32, 17A36, 18C05, 18E13 We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field. Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to study the representability of actions for them. Here we prove that the varieties of two-step nilpotent (anti-)commutative algebras and that of commutative associative algebras are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called amalgam. Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object $E(X)$ for the actions on (or split extensions of) an object $X$. We actually obtain a partial algebra $E(X)$, which we call external weak actor of $X$, together with a monomorphism of functors ${\operatorname{SplExt}(-,X) \rightarrowtail \operatorname{Hom}(U(-),E(X))}$, which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor $\operatorname{USGA}(X)$ and that of $E(X)$ are described in detail. We end with some open questions. |
| title | Weak representability of actions of non-associative algebras |
| topic | Category Theory Rings and Algebras 08A35, 08C05, 16B50, 16W25, 17A32, 17A36, 18C05, 18E13 |
| url | https://arxiv.org/abs/2306.02812 |