Symplectic $C_{\mathrm{\infty }}$-algebras

In this paper we show that a strongly homotopy commutative (or $C_{\mathrm{\infty }}$-) algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_{\mathrm{\infty }}$-algebra (an $\mathrm{\infty }$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). This result relies on the algebraic Hodge decomposition of the cyclic Hochschild cohomology of a $\text{ci}$-algebra and does not generalize to algebras over other operads.