Abstract:
this is a joint work with victor ginzburg [4] in which we study a class of associative algebras associated to finite groups acting on a vector space. these algebras are non-homogeneous n-koszul algebra generalizations of symplectic reflection algebras. we realize the extension of the n-koszul property to non-homogeneous algebras through a poincaré-birkhoff-witt property.

Abstract:
It has been shown recently, in a joint work with Michel Dubois-Violette and Marc Wambst (see math.QA/0203035), that Koszul property of $N$-homogeneous algebras (as defined in the original paper) becomes natural in a $N$-complex setting. A basic question is to define the differential of the bimodule Koszul complex of an $N$-homogeneous algebra, e.g., for computing its Hochschild homology. The differential defined here uses $N$-complexes. That puts right the wrong differential presented in the original paper in a 2-complex setting. Actually, as we shall see, it is impossible to avoid $N$-complexes in defining the differential, whereas the bimodule Koszul complex is a 2-complex.

Abstract:
The paper is devoted to graded algebras having a single homogeneous relation. Using Gerasimov's theorem, a criterion to be N-Koszul is given, providing new examples. An alternative proof of Gerasimov's theorem for N=2 is given. Some related results on Calabi-Yau algebras are proved.

Abstract:
The numerical Hilbert series combinatorics and the comodule Hilbert series combinatorics are introduced, and some applications are presented, including the MacMahon Master Theorem.

Abstract:
We define a family of 3-Calabi-Yau algebras by potentials. For some of these algebras, we explicitly compute the Hochschild homology with the help of Poisson homology. The point is that the Poisson potential has non-isolated singularities.

Abstract:
Koszul property was generalized to homogeneous algebras of degree N>2 in [5], and related to N-complexes in [7]. We show that if the N-homogeneous algebra A is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincare duality between Hochschild homology and cohomology of A, as for N=2.

Abstract:
From symplectic reflection algebras, some algebras are naturally introduced. We show that these algebras are non-homogeneous N-Koszul algebras, through a PBW theorem.

Abstract:
Recently, Bocklandt proved a conjecture by Van den Bergh in its graded version, stating that a graded quiver algebra (with relations) which is Calabi-Yau of dimension 3 is defined from a homogeneous potential W. In this paper, we prove that if we add to W any potential of smaller degree, we get a Poincare-Birkhoff-Witt deformation of A. Such PBW deformations are Calabi-Yau and are characterised among all the PBW deformations of A. Various examples are presented.

Abstract:
We give a necessary and sufficient condition for an N-Koszul algebra defined by a homogeneous potential, to be 3-Calabi-Yau. As an application, we recover two families of 3-Calabi-Yau algebras recently appeared in the literature, by studying skew polynomial algebras over non-commutative quadrics.