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 Poincaré duality between Hochschild homology and cohomology of A, as for N = 2.

The Hochschild homology of cubic Artin-Schelter regular algebras of type A with generic coefficients is computed. We follow the method used by van den Bergh [K-Theory 8 (1994) 213-230] in the quadratic case and consider these algebras as deformations of a polynomial algebra with remarkable Poisson brackets. A new morphism of resolutions is introduced. The de Rham cohomology, cyclic and periodic cyclic homologies, and the Hochschild cohomology are also computed.

No abstract

We classify reflexive graded right ideals, up to isomorphism and shift, of generic cubic three-dimensional Artin-Schelter regular algebras. We also determine the possible Hilbert functions of these ideals. These results are obtained by using similar methods as for quadratic Artin-Schelter algebras [K. De Naeghel, M. Van den Bergh, Ideal classes of three-dimensional Sklyanin algebras, J. Algebra 276 (2) (2004) 515-551; K. De Naeghel, M. Van den Bergh, Ideal classes of three dimensional Artin-Schelter regular algebras, J. Algebra 283 (1) (2005) 399-429]. In particular our results apply to the enveloping algebra of the Heisenberg-Lie algebra from which we deduce a classification of right ideals of the invariant ring A ϕ 1 of the first Weyl algebra A 1 = k x, y /(xy − yx − 1) under the automorphism ϕ(x) = −x, ϕ(y) = −y.