Ideals of cubic algebras and an invariant ring of the Weyl algebra

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 of the first Weyl algebra A1=k x,y /(xy−yx−1) under the automorphism φ(x)=−x, φ(y)=−y.