We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a c
onjecture of Hesselholt and Rains, producing new $p$-torsion classes in degrees 2p^l, l >= 1, We relate these classes by p-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. An important tool is a generalization of the Diamond Lemma to modules over commutative rings, which we give in the appendix. In the previous version, additional results are included, such as: the Poisson center of $text{Sym } HH_0(Pi)$ for all quivers, the BV algebra structure on Hochschild cohomology, including how the Lie algebra structure $HH_0(Pi_Q)$ naturally arises from it, and the cyclic homology groups of $Pi_Q$.
The Darboux-Weinstein decomposition is a central result in the theory of Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the point and a fo
rmal slice. Recently, conical symplectic resolutions, and more generally, Poisson cones, have been very actively studied in representation theory and algebraic geometry. This motivates asking for a C*-equivariant version of the Darboux-Weinstein decomposition. In this paper, we develop such a theory, prove basic results on their existence and uniqueness, study examples (quotient singularities and hypertoric varieties), and applications to noncommutative algebra (their quantization). We also pose some natural questions on existence and quantization of C*-actions on slices to conical symplectic leaves.
We prove a conjecture of Etingof and the second author for hypertoric varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that thi
s conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson-de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson-de Rham-Poincare polynomial, and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham. We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.
We consider the Hamiltonian flow on complex complete intersection surfaces with isolated singularities, equipped with the Jacobian Poisson structure. More generally we consider complete intersections of arbitrary dimension equipped with Hamiltonian f
low with respect to the natural top polyvector field, which one should view as a degenerate Calabi-Yau structure. Our main result computes the coinvariants of functions under the Hamiltonian flow. In the surface case this is the zeroth Poisson homology, and our result generalizes those of Greuel, Alev and Lambre, and the authors in the quasihomogeneous and formal cases. Its dimension is the sum of the dimension of the top cohomology and the sum of the Milnor numbers of the singularities. In other words, this equals the dimension of the top cohomology of a smoothing of the variety. More generally, we compute the derived coinvariants, which replaces the top cohomology by all of the cohomology. Still more generally we compute the D-module which represents all invariants under Hamiltonian flow, which is a nontrivial extension (on both sides) of the intersection cohomology D-module, which is maximal on the bottom but not on the top. For cones over smooth curves of genus g, the extension on the top is the holomorphic half of the maximal extension.
We study the existence of symplectic resolutions of quotient singularities V/G where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form $K r
times S_2$ where $K < SL_2(C)$, for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for $dim V eq 4$, we classify all quotient singularities $V/G$ admitting a projective symplectic resolution which do not decompose as a product of smaller-dimensional quotient singularities, except for at most four explicit singularities, that occur in dimensions at most 10, for whom the question of existence remains open.
These are significantly expanded lecture notes for the authors minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of the deformati
on theory of associative algebras in terms of the Hochschild cochain complex as well as quantization of Poisson structures, and Kontsevichs formality theorem in the smooth setting. We then discuss quantization and deformation via Calabi-Yau algebras and potentials. Examples discussed include Weyl algebras, enveloping algebras of Lie algebras, symplectic reflection algebras, quasihomogeneous isolated hypersurface singularities (including du Val singularities), and Calabi-Yau algebras.
We prove that the space of coinvariants of functions on an affine variety by a Lie algebra of vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theorem include Poisson (or more generally Jacobi) varieties wit
h finitely many symplectic leaves under Hamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities under the flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preserving groups under the incompressible vector fields, and arbitrary varieties with isolated singularities under the flow of all vector fields. We compute this quotient explicitly in many of these cases. The proofs involve constructing a natural D-module representing the invariants under the flow of the vector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicity we study in more detail). We give many counterexamples to naive generalizations of our results. These examples have been a source of motivation for us.
We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham diffe
rential, which replaces the Connes differential, and another operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative cyclic homology of a flat family of associative algebras over a central base ring. We introduce and study `free-product deformations of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups.
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendiecks notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by D
Der(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain `Fock space associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in some `twisted sense. The resulting double Poisson structure on gr D(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that gr D(F(A))=F(T_A(DDer(A)), provided A is smooth. It is crucial for our construction that the Fock space F(A) carries an extra-structure of a wheelgebra, a new notion closely related to the notion of a wheeled PROP. There are also notions of Lie wheelgebras, and so on. In that language, D(F(A)) becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on DDer(A) gives rise to a second order (wheeled) differential operator, a noncommutative analogue of the BV-operator.
