Symplectic $C_\infty$-algebras

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 $C_\infty$-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\infty$-algebra (an $\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a $\ci$-algebra and does not generalize to algebras over other operads.