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

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(1)(<rho >) of the first Weyl algebra A(1) = k < x, y >/(xy - yx - 1) under the automorphism rho(x) = -x, rho(y) = -y.