Infinitesimal 2-braidings from 2-shifted Poisson structures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915203264806912 |
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| author | Kemp, Cameron Laugwitz, Robert Schenkel, Alexander |
| author_facet | Kemp, Cameron Laugwitz, Robert Schenkel, Alexander |
| contents | It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal dg-category of finitely generated semi-free $A$-dg-modules. This provides a concrete realization, to first order in the deformation parameter $\hbar$, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, Toën, Vaquié and Vezzosi. Of particular interest is the case when $A$ is the Chevalley-Eilenberg algebra of a Lie $N$-algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_00391 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinitesimal 2-braidings from 2-shifted Poisson structures Kemp, Cameron Laugwitz, Robert Schenkel, Alexander Quantum Algebra Mathematical Physics Algebraic Geometry 14A30, 17B37, 18N10, 53D55 It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal dg-category of finitely generated semi-free $A$-dg-modules. This provides a concrete realization, to first order in the deformation parameter $\hbar$, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, Toën, Vaquié and Vezzosi. Of particular interest is the case when $A$ is the Chevalley-Eilenberg algebra of a Lie $N$-algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'. |
| title | Infinitesimal 2-braidings from 2-shifted Poisson structures |
| topic | Quantum Algebra Mathematical Physics Algebraic Geometry 14A30, 17B37, 18N10, 53D55 |
| url | https://arxiv.org/abs/2408.00391 |