Noncommutative Algebraic Geometry Research Articles

Down–Up Algebras and Their Representations

The down–up algebras were introduced in [G. Benkart and T. Roby, 1998, J. Algebra 209, 305–344, and G. Benkart, 1998, Contemp. Math. 224, 29–45]. Their representations were studied via category O and Verma modules. Here we prove that when β ≠ 0 they are hyperbolic rings, and we study their representations via their left spectrum as defined in [A. L. Rosenberg, 1995, “Noncommutative Algebraic Geometry and Representations of Quantized Algebras,” Kluwer Academic, Dordrecht]. The center of these algebras is computed using results from hyperbolic rings which are also proved here. The case when the down–up algebra is a finite module over its center is considered in detail and its relation to the Clifford algebra of a binary form of arbitrary degree is established.

Presentation of critical modules of GK-dimension 2 over elliptic algebras

We show that critical modules of Gelfand-Kirillov dimension 2 and multiplicity d d over an elliptic algebra have (up to modules of lower GK-dimension and shifting) a presentation by d × d d\times d -matrices of linear forms. In the language of non-commutative algebraic geometry this amounts to a generic description of “curves” of degree d d in a projective quantum plane.

Orderings of higher level on noetherian rings

The noncommutative algebraic geometry has found fruitful applications in quantum geometry. Similar applications are expected to be found for its younger sister the noncommutative real algebraic geometry One of the basic results in real algebraic geometry is the Positivestellensatz. The original results of Dubois and Risler (see section 3.3 of [13]) have been extended in many directions. We refer to [14], [1], [2], [3] for commutative rings and [9], [4] for associative rings. The aim of this paper is to prove the higher level Posit ivstellensatz for noncommutative Noetherian rings. Our proof depends on the intersection theorem for orderings of higher level on skew fields ([11], Theorem 3.13). The general case of orderings of higher level on associative rings remains open.

Noncommutative geometry based on commutator expansions

We develop an approach to noncommutative algebraic geometry ``in the perturbative regime around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal neighborhood of M=Spec(A) in the (nonexistent) space Spec(R). This is a ringed space (M,O) where O is a certain sheaf of noncommutative rings on M. Such ringed spaces can be glued together to form more global objects called NC-schemes. We are especially interested in NC-manifolds, NC-schemes for which the completion of O at every point of M is isomorphic to the algebra of noncommutative power series (completion of the free associative algebra). An explicit description of the simplest NC-manifold, the affine space, is given by using the Feynman-Maslov calculus of ordered operators. We show that many familiar algebraic varieties can be naturally enlarged to NC-manifolds. Among these are all the classical flag varieties and all the smooth moduli spaces of vector bundles.

Noncommutative geometry based on commutator expansions

We develop an approach to noncommutative algebraic geometry ``in the perturbative regime around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal neighborhood of M=Spec(A) in the (nonexistent) space Spec(R). This is a ringed space (M,O) where O is a certain sheaf of noncommutative rings on M. Such ringed spaces can be glued together to form more global objects called NC-schemes. We are especially interested in NC-manifolds, NC-schemes for which the completion of O at every point of M is isomorphic to the algebra of noncommutative power series (completion of the free associative algebra). An explicit description of the simplest NC-manifold, the affine space, is given by using the Feynman-Maslov calculus of ordered operators. We show that many familiar algebraic varieties can be naturally enlarged to NC-manifolds. Among these are all the classical flag varieties and all the smooth moduli spaces of vector bundles.

Some quantum P3s with infinitely many points

We consider graded Clifford algebras on n generators in the spirit of Artin, Tate and Van den Bergh’s non-commutative algebraic geometry. We give an algorithm for counting the point modules over such an algebra, and prove that a generic graded Clifford algebra on four generators has defining relations which are determined by their zero locus in P3 × P3 Furthermore, we find a 1-parameter family of iterated Ore extensions on four generators which are deformations of a graded Clifford algebra such that the generic member has precisely one point module and a 1-dimensional family of line modules.

Noncommutative algebraic geometry: from pi-algebras to quantum groups

The main purpose of this paper is to provide a survey of dierent notions of algebraic geometry, which one may associate to an arbitrary noncommutative ring R. In the rst part, we will mainly deal with the prime spectrum of R, endowed both with the Zariski topology and the stable topology. In the second part we focus on quantum groups and, in particular, on schematic algebras and show how a noncommutative site may be associated to the latter. In the last part, we concentrate on regular algebras, and present a rather complete up to date overview of their main properties.

Algebraic deformations of toric varieties II: noncommutative instantons

We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parameterizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build instanton gauge bundles with canonical connections on the noncommutative four-sphere that satisfy appropriate anti-selfduality equations. We construct projective moduli spaces for the torsion free sheaves and demonstrate that they are smooth. We define equivariant partition functions of these moduli spaces, finding that they coincide with the usual instanton partition functions for supersymmetric gauge theories on $\mathbb{C}2$.