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

Iyudu, Natalia and Kontsevich, Maxim (2020) Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology. arXiv.org.

Text (2011.11888v1)
2011.11888v1.pdf - Published Version

## 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.

Item Type:
Journal Article
Journal or Publication Title:
arXiv.org
33 pages
Subjects:
Departments:
ID Code:
155833
Deposited By:
Deposited On:
08 Jun 2021 11:10
Refereed?:
No
Published?:
Published