1202.2717 (Yuri Berest et al.)

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

Yuri Berest, Xiaojun Chen, Farkhod Eshmatov, Ajay Ramadoss

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.

View original: http://arxiv.org/abs/1202.2717