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Modular vector fields in non-commutative geometry

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arxiv 2410.24064 v3 pith:NJ3QDMKN submitted 2024-10-31 math.QA math.ATmath.GT

classification math.QAmath.ATmath.GT
keywords connectionmodularnon-commutativevectoralekseev-kawazumi-kuno-naefalgebraicanalogueapplication
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abstract

We construct a non-commutative analogue of the modular vector field on a Poisson manifold for a given pair of a double bracket and a connection on a space of 1-forms. The key ingredient, the triple divergence map, is directly constructed from a connection on a linear category to deal with multiple base points. As an application, we give an algebraic description of the framed, groupoid version of Turaev's loop operation $\mu$ similar to the one obtained by Alekseev-Kawazumi-Kuno-Naef and the author.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The formality of the Goldman-Turaev Lie bialgebra on a closed surface

    math.QA 2025-02 accept novelty 8.0 of 10

    The pro-unipotent automorphism group of the associated graded Goldman-Turaev Lie bialgebra on a closed surface is explicitly described in terms of a divergence map and the kernel of a reduced coproduct.

  2. A family of algebraic operations extending the Turaev cobracket

    math.QA 2025-02 accept novelty 7.0 of 10

    A new family of algebraic operations, the k-divergences, generalizes the Turaev cobracket and, for free associative algebras, coincides with ribbon graph operations and the standard cohomology generators of gl_n.

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