Cumulants, Koszul brackets and homological perturbation theory for commutative BVoo and IBLoo algebras

Abstract: n the first part of this talk we shall review the classical homotopy transfer theorem in the context of Aoo and Loo algebras. We shall explain how the usual proof of this result for Aoo algebras, based on the tensor trick and the homological perturbation lemma, can be adapted to Loo algebras using a symmetrized version of the tensor trick. In the course of the discussion we shall review the construction of cumulants and Koszul brackets (as well as their coalgebraic analogs): these are graded symmetric multilinear maps associated respectively to a morphism of graded commutative algebras $f\colon A \to B$ or to an endomorphism $d\colon A \to A$, measuring the deviation of f from being an algebra morphism in the first case, and the deviation of d from being an algebra derivation in the second case. A key technical lemma will be that under certain assumptions on the involved contraction, these are compatible with homotopy transfer in an appropriate sense. In the second part of the talk we shall review commutative BVoo algebra in the sense of Kravchenko: as an application of our previous discussion we shall introduce a new definition of morphisms between these objects in terms of cumulants. Moreover, we shall explain how to use homological perturbation theory to get a homotopy transfer theorem for commutative BVoo algebras, under certain assumptions on the involved contraction. Finally, IBLoo algebras, that is, commutative BVoo algebras whose underlying algebra is free, are known to be a model for involutive Lie bialgebras up to coherent homotopies, and have recently found several applications in string topology and symplectic field theory. As an application of our results, we shall explain how to obtain a homotopy transfer theorem for IBLoo algebras via the symmetrized tensor trick and the homological perturbation lemma.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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