The Bogomolov-Tian-Todorov Theorem Of Cyclic $A_{\infty }$-Algebras

Abstract

Let $A$ be a finite-dimensional smooth unital cyclic $A_{\infty }$-algebra. Assume furthermore that $A$ satisfies the Hodge-to-de-Rham degeneration property. In this short note, we prove the non-commutative analogue of the Bogomolov-Tian-Todorov theorem: the deformation functor associated with the differential graded Lie algebra of Hochschild cochains of $A$ is smooth. Furthermore, the deformation functor associated with the DGLA of cyclic Hochschild cochains of $A$ is also smooth.

7 pages. Reference added. Title changed. Proof of Corollary 2.2 improved. part (A) of Theorem 1.1 was proved by Isamu Iwanari in arXiv:1604.08283