Symplectic $C_\infty$-algebras
Alastair Hamilton, Andrey Lazarev

TL;DR
This paper proves that strongly homotopy commutative algebras with invariant inner products can be uniquely extended to symplectic $C_ abla$-algebras, generalizing commutative Frobenius algebras via algebraic Hodge decomposition.
Contribution
It establishes a unique extension of $C_ abla$-algebras to symplectic $C_ abla$-algebras using cyclic Hochschild cohomology and Hodge decomposition, specific to the $C_ abla$ operad.
Findings
Extension is unique for $C_ abla$-algebras with invariant inner product.
Relies on algebraic Hodge decomposition of cyclic Hochschild cohomology.
Does not generalize to other operads.
Abstract
In this paper we show that a strongly homotopy commutative (or -) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic -algebra (an -generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a -algebra and does not generalize to algebras over other operads.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology
