Higher structures for Lie $H$-pseudoalgebras
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929279298699264 |
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| author | Das, Apurba |
| author_facet | Das, Apurba |
| contents | Let $H$ be a cocommutative Hopf algebra. The notion of Lie $H$-pseudoalgebra is a multivariable generalization of Lie conformal algebras. In this paper, we study some higher structures related to Lie $H$-pseudoalgebras where we increase the flexibility of the Jacobi identity. Namely, we first introduce $L_\infty$ $H$-pseudoalgebras (also called strongly homotopy Lie $H$-pseudoalgebras) as the homotopy analogue of Lie $H$-pseudoalgebras. We give several equivalent descriptions of such homotopy algebras and show that some particular classes of these homotopy algebras are closely related to the cohomology of Lie $H$-pseudoalgebras and crossed modules of Lie $H$-pseudoalgebras. Next, we introduce another higher structure, called Lie-$2$ $H$-pseudoalgebras which are the categorification of Lie $H$-pseudoalgebras. Finally, we show that the category of Lie-$2$ $H$-pseudoalgebras is equivalent to the category of certain $L_\infty$ $H$-pseudoalgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher structures for Lie $H$-pseudoalgebras Das, Apurba Representation Theory Category Theory 17B56, 17B69, 18N40, 18N25 Let $H$ be a cocommutative Hopf algebra. The notion of Lie $H$-pseudoalgebra is a multivariable generalization of Lie conformal algebras. In this paper, we study some higher structures related to Lie $H$-pseudoalgebras where we increase the flexibility of the Jacobi identity. Namely, we first introduce $L_\infty$ $H$-pseudoalgebras (also called strongly homotopy Lie $H$-pseudoalgebras) as the homotopy analogue of Lie $H$-pseudoalgebras. We give several equivalent descriptions of such homotopy algebras and show that some particular classes of these homotopy algebras are closely related to the cohomology of Lie $H$-pseudoalgebras and crossed modules of Lie $H$-pseudoalgebras. Next, we introduce another higher structure, called Lie-$2$ $H$-pseudoalgebras which are the categorification of Lie $H$-pseudoalgebras. Finally, we show that the category of Lie-$2$ $H$-pseudoalgebras is equivalent to the category of certain $L_\infty$ $H$-pseudoalgebras. |
| title | Higher structures for Lie $H$-pseudoalgebras |
| topic | Representation Theory Category Theory 17B56, 17B69, 18N40, 18N25 |
| url | https://arxiv.org/abs/2403.10777 |