Cohomology and Homotopy of Lie triple systems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909145951633408 |
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| author | Xia, Haobo Sheng, Yunhe Tang, Rong |
| author_facet | Xia, Haobo Sheng, Yunhe Tang, Rong |
| contents | In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that $2$-term homotopy Lie triple systems is equivalent to Lie triple $2$-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple $2$-systems. We show that skeletal Lie triple $2$-systems can be classified the third cohomology group, and strict Lie triple $2$-systems are equivalent to crossed modules of Lie triple systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10862 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cohomology and Homotopy of Lie triple systems Xia, Haobo Sheng, Yunhe Tang, Rong Rings and Algebras In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that $2$-term homotopy Lie triple systems is equivalent to Lie triple $2$-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple $2$-systems. We show that skeletal Lie triple $2$-systems can be classified the third cohomology group, and strict Lie triple $2$-systems are equivalent to crossed modules of Lie triple systems. |
| title | Cohomology and Homotopy of Lie triple systems |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2312.10862 |