Let be a Koszul (or more generally, -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 , which induces a graded Lie algebra structure on the cyclic homology of ; moreover, we show that the Hochschild homology of 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 is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.
Cite this article
Xiaojun Chen, Alimjon Eshmatov, Farkhod Eshmatov, Song Yang, The derived non-commutative Poisson bracket on Koszul Calabi–Yau algebras. J. Noncommut. Geom. 11 (2017), no. 1, pp. 111–160DOI 10.4171/JNCG/11-1-4