2020/11/23 by Natalia Iyudu, Maxim Kontsevich, Iyudu, Natalia +1 · 1 citation
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2011.11888
We prove L∞-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 ξδ-monomials. The Lie structure on this subcomplex is combinatorially described in terms of ξδ-monomials. This subcomplex and a basis of ξδ-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of ξδ-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.