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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 $xidelta$-monomials. The Lie structure on this subcomplex is combinatorially described in terms of $xidelta$-monomials. This subcomplex and a basis of $xidelta$-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of $xidelta$-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.
We give an explicit formula showing how the double Poisson algebra introduced in cite{VdB} appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan eq
The Calabi-Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal en
Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hoc
We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a cycle, we show that the center is trivial, and henc
Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper we study the theory of (formal) deformation of coisotropic algebra