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arXiv · 2011.11888

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

Natalia Iyudu·Maxim Kontsevich
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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.

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BibTeXRIS

Natalia Iyudu, Maxim Kontsevich. 2020-11-24. Pre-Calabi-Yau algebras and noncommutative calculus on higher cyclic Hochschild cohomology. https://arxiv.org/abs/2011.11888

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