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Mathematics > Quantum Algebra

arXiv:1807.09209 (math)
[Submitted on 24 Jul 2018 (v1), last revised 17 Oct 2020 (this version, v4)]

Title:Hodge Theory of the Turaev Cobracket and the Kashiwara--Vergne Problem

Authors:Richard Hain
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Abstract:In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.
Comments: Corrections to statement and proof of Prop. 9.3. To appear in JEMS: 39 pages, 1 figure
Subjects: Quantum Algebra (math.QA); Algebraic Geometry (math.AG); Geometric Topology (math.GT)
MSC classes: Primary 17B62, 58A12, Secondary 57N05, 14C30
Cite as: arXiv:1807.09209 [math.QA]
  (or arXiv:1807.09209v4 [math.QA] for this version)
  https://doi.org/10.48550/arXiv.1807.09209

Submission history

From: Richard Hain [view email]
[v1] Tue, 24 Jul 2018 16:16:15 UTC (31 KB)
[v2] Tue, 8 Jan 2019 17:22:00 UTC (35 KB)
[v3] Mon, 27 Apr 2020 15:11:20 UTC (64 KB)
[v4] Sat, 17 Oct 2020 19:38:14 UTC (64 KB)
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