Shifted Poisson structures on higher Chevalley-Eilenberg algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917279899320320 |
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| author | Kemp, Cameron Laugwitz, Robert Schenkel, Alexander |
| author_facet | Kemp, Cameron Laugwitz, Robert Schenkel, Alexander |
| contents | This paper develops a graphical calculus to determine the $n$-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley-Eilenberg algebra of an ordinary Lie algebra, we recover Safronov's result that the $(n=1)$- and $(n=2)$-shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley-Eilenberg algebra of a Lie $2$-algebra and obtain $n\in\{1,2,3,4\}$ shifted Poisson structures in this case, which we interpret as semi-classical data of `higher quantum groups'. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_12804 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shifted Poisson structures on higher Chevalley-Eilenberg algebras Kemp, Cameron Laugwitz, Robert Schenkel, Alexander Quantum Algebra Mathematical Physics Algebraic Geometry 14A30, 17B55, 17B62 This paper develops a graphical calculus to determine the $n$-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley-Eilenberg algebra of an ordinary Lie algebra, we recover Safronov's result that the $(n=1)$- and $(n=2)$-shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley-Eilenberg algebra of a Lie $2$-algebra and obtain $n\in\{1,2,3,4\}$ shifted Poisson structures in this case, which we interpret as semi-classical data of `higher quantum groups'. |
| title | Shifted Poisson structures on higher Chevalley-Eilenberg algebras |
| topic | Quantum Algebra Mathematical Physics Algebraic Geometry 14A30, 17B55, 17B62 |
| url | https://arxiv.org/abs/2412.12804 |