On double brackets for marked surfaces

2024/10/08 by Michael Gekhtman, Gekhtman, Michael, Eugen Rogozinnikov +1
Computer Science · Engineering · Mathematics · #17B63 #22E40 #57K20 #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematics and Applications #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2410.06137

openalex publication_date 2024/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 GL_n(A)-, symplectic and indefinite orthogonal local systems.

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