The derived non-commutative Poisson bracket on Koszul Calabi–Yau algebras

Abstract

Let A be a Koszul (or more generally, N -Koszul) Calabi–Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on A , which induces a graded Lie algebra structure on the cyclic homology of A ; moreover, we show that the Hochschild homology of A is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz–Loday bracket associated to the derived non-commutative Poisson structure on A is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of A itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.