Noncommutative Poisson structures, derived representation schemes and Calabi-Yau algebras

Abstract

Recantly, William Crawley-Boevey proposed the definition of a Poisson structure on a noncommutative algebra $A$ based on the Kontsevich principle. His idea was to find the {\it weakest} possible structure on $A$ that induces standard (commutative) Poisson structures on all representation spaces $ \Rep_V(A) $. It turns out that such a weak Poisson structure on $A$ is a Lie algebra bracket on the 0-th cyclic homology $ \HC_0(A) $ satisfying some extra conditions; it was thus called in an {\it $ H_0$-Poisson structure}. This paper studies a higher homological extension of this construction. In our more general setting, we show that noncommutative Poisson structures in the above sense behave nicely with respect to homotopy (in the sense that homotopy equivalent NC Poisson structures on $A$ induce (via the derived representation functor) homotopy equivalent Poisson algebra structures on the derved representation schemes $\DRep_V(A) $). For an ordinary algebra $A$, a noncommutative Poisson structure on a semifree (more generally, cofibrant) resolution of $A$ yields a graded (super) Lie algebra structure on the full cyclic homology $ \HC_\bullet(A) $ extending Crawley-Boevey's $\H_0$-Poisson structure on $ \HC_0(A) $. We call such structures {\it derived Poisson structures} on $A$. We also show that derived Poisson structures do arise in nature: the cobar construction $\Omega(C)$ of an $(-n)$-cyclic coassociative DG coalgebra (in particular, of the linear dual of a finite dimensional $n$-cyclic DG algebra) $C$ carries a $(2-n)$-double Poisson bracket in the sense of Van den Bergh. This in turn induces a corresponding noncommutative $(2-n)$-Poisson structure on $\Omega(C)$. When (the semifree) DG algebra $\Omega(C)$ resolves an honest algebra $A$, $A$ acquires a derived $(2-n)$-Poisson structure.