On double brackets for marked surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909342772494336 |
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| author | Gekhtman, Michael Rogozinnikov, Eugen |
| author_facet | Gekhtman, Michael Rogozinnikov, Eugen |
| contents | We propose a construction of a double quasi-Poisson bracket on the group algebra associated to the twisted fundamental group of a marked oriented surface $(S,P)$ with boundary, where $P$ is a finite set of marked points on the boundary of the surface $S$ such that on every boundary component there is at least one point of $P$. We show that this double bracket is a noncommutative generalization of the well-known Goldman bracket, defined on the space of free homotopy classes of loops on $S$. For an algebra $A$ without polynomial identities, we construct a double bracket on the space of decorated twisted $\mathrm{GL}_n(A)$-, symplectic and indefinite orthogonal local systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On double brackets for marked surfaces Gekhtman, Michael Rogozinnikov, Eugen Differential Geometry Geometric Topology Rings and Algebras Representation Theory 57K20, 17B63, 22E40 We propose a construction of a double quasi-Poisson bracket on the group algebra associated to the twisted fundamental group of a marked oriented surface $(S,P)$ with boundary, where $P$ is a finite set of marked points on the boundary of the surface $S$ such that on every boundary component there is at least one point of $P$. We show that this double bracket is a noncommutative generalization of the well-known Goldman bracket, defined on the space of free homotopy classes of loops on $S$. For an algebra $A$ without polynomial identities, we construct a double bracket on the space of decorated twisted $\mathrm{GL}_n(A)$-, symplectic and indefinite orthogonal local systems. |
| title | On double brackets for marked surfaces |
| topic | Differential Geometry Geometric Topology Rings and Algebras Representation Theory 57K20, 17B63, 22E40 |
| url | https://arxiv.org/abs/2410.06137 |