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

$DG$-models of projective modules and Nakajima quiver varieties

“…×ī n ∈ Hom(C, V n ) ,j n ∈ Hom(V n , C), In the case m = 2, the varieties D n (k0,...,km−1) have been introduced recently in [18] (see loc. cit., Theorem 3) to classify the ideals of the Z 2 -invariant subring of A 1 (C).…”

Derived Categories of Noncommutative Quadrics and Hilbert Squares

“…Note that there exists a notion of Hilbert scheme of points for a general cubic Artin-Schelter regular graded algebra [6], which is a subset of all noncommutative quadrics. We do not address the comparison between these moduli spaces and the deformations constructed in this paper.…”

Artin-Schelter regular algebras of dimension five

“…Since then these algebras of dimension three and four, and their module theory, have been intensively studied, see [2], [3]. Ideal theory (see [12], [7], [8]) and deformations (see [9]) have also been studied in recent years. This paper is concerned with basic questions for AS-regular algebras of dimension five.…”