Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology

TL;DR

This paper proves $L_{ abla}$-formality for the higher cyclic Hochschild complex over free algebras, introduces a noncommutative calculus with $\xi ext{delta}$-monomials, and interprets pre-Calabi-Yau structures as noncommutative Poisson structures.

Contribution

It establishes $L_{ abla}$-formality for the higher cyclic Hochschild complex and develops a noncommutative calculus framework using $\xi ext{delta}$-monomials.

Findings

Cohomologies of the complex are concentrated in degree zero for free algebras.

A combinatorial description of the Lie structure on the subcomplex is provided.

Pre-Calabi-Yau structures are interpreted as noncommutative Poisson structures.

Abstract

We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show that cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $\xi\delta$-monomials. The Lie structure on this subcomplex is combinatorially described in terms of $\xi\delta$-monomials. This subcomplex and a basis of $\xi\delta$-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of $\xi\delta$-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.