Symmetries of algebras captured by actions of weak Hopf algebras
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866909380940660736 |
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| author | Calderón, Fabio Huang, Hongdi Wicks, Elizabeth Won, Robert |
| author_facet | Calderón, Fabio Huang, Hongdi Wicks, Elizabeth Won, Robert |
| contents | In this paper, we present a generalization of well-established results regarding symmetries of $\Bbbk$-algebras, where $\Bbbk$ is a field. Traditionally, for a $\Bbbk$-algebra $A$, the group $\Bbbk$-algebra automorphisms of $A$ captures the symmetries of $A$ via group actions. Similarly, the Lie algebra of derivations of $A$ captures the symmetries of $A$ via Lie algebra actions. In this paper, given a category $\mathcal{C}$ whose objects possess $\Bbbk$-linear monoidal categories of modules, we introduce an object $\operatorname{Sym}_{\mathcal{C}}(A)$ that captures the symmetries of $A$ via actions of objects in $\mathcal{C}$. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected $\Bbbk$-algebra $A$, some of its symmetries are naturally captured within the weak Hopf framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_11903 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Symmetries of algebras captured by actions of weak Hopf algebras Calderón, Fabio Huang, Hongdi Wicks, Elizabeth Won, Robert Quantum Algebra Category Theory Rings and Algebras 18B40, 16T05, 18M05 In this paper, we present a generalization of well-established results regarding symmetries of $\Bbbk$-algebras, where $\Bbbk$ is a field. Traditionally, for a $\Bbbk$-algebra $A$, the group $\Bbbk$-algebra automorphisms of $A$ captures the symmetries of $A$ via group actions. Similarly, the Lie algebra of derivations of $A$ captures the symmetries of $A$ via Lie algebra actions. In this paper, given a category $\mathcal{C}$ whose objects possess $\Bbbk$-linear monoidal categories of modules, we introduce an object $\operatorname{Sym}_{\mathcal{C}}(A)$ that captures the symmetries of $A$ via actions of objects in $\mathcal{C}$. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected $\Bbbk$-algebra $A$, some of its symmetries are naturally captured within the weak Hopf framework. |
| title | Symmetries of algebras captured by actions of weak Hopf algebras |
| topic | Quantum Algebra Category Theory Rings and Algebras 18B40, 16T05, 18M05 |
| url | https://arxiv.org/abs/2209.11903 |