Lowen, Wendy Van den Bergh, Michel
Published in
Advances in Mathematics

This paper continues the development of the deformation theory of abelian categories introduced in a previous paper by the authors. We show first that the deformation theory of abelian categories is controlled by an obstruction theory in terms of a suitable notion of Hochschild cohomology for abelian categories. We then show that this Hochschild co...

Reichstein, Z. Rogalski, D. Zhang, J.J.
Published in
Advances in Mathematics

An infinite-dimensional N -graded k-algebra A is called projectively simple if dim k A / I

Backelin, Erik Kremnizer, Kobi
Published in
Advances in Mathematics

Let O q ( G ) be the algebra of quantized functions on an algebraic group G and O q ( B ) its quotient algebra corresponding to a Borel subgroup B of G. We define the category of sheaves on the “quantum flag variety of G” to be the O q ( B ) -equivariant O q ( G ) -modules and prove that this is a proj-category. We construct a category of equivaria...

Nevins, T.A. Stafford, J.T.
Published in
Advances in Mathematics

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P 2 . The generic noncommutative plane corre...

Rogalski, D.
Published in
Advances in Mathematics

We describe some interesting graded rings which are generated by degree-3 elements inside the Sklyanin algebra S, and prove that they have many good properties. Geometrically, these rings R correspond to blowups of the Sklyanin P 2 at 7 or fewer points. We show that the rings R are exactly those degree-3-generated subrings of S which are maximal or...

Craw, Alastair Velez, Alexander Quintero

This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer-Sturmfels in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of toric algebras in dimension three by associating a superpotential and a notion of consistency...

Burban, Igor Kalck, Martin
Published in
Advances in Mathematics

In this article, we study a triangulated category associated with a non-commutative resolution of singularities. In particular, we give a complete description of this category in the case of a curve with nodal singularities, classifying its indecomposable objects and computing its Auslander–Reiten quiver and K-group.

Gavarini, Fabio
Published in
Forum Mathematicum

I present a construction of connected affine algebraic supergroups 𝐆V associated with simple Lie superalgebras 𝔤 of Cartan type and with 𝔤-modules V. Conversely, I prove that every connected affine algebraic supergroup whose tangent Lie superalgebra is of Cartan type is necessarily isomorphic to one of the supergroups 𝐆V that I introduced. In parti...

Backelin, Erik Kremnizer, Kobi
Published in
Advances in Mathematics

We quantize parabolic flag manifolds and describe categories of quantum D-modules on them at a singular central character. We compute global sections of generators for these categories for any q∈C⁎. For generic q we prove a singular version of Beilinson–Bernstein localization for a quantized enveloping algebra.

Holdaway, Cody Sisodia, Gautam
Published in
Journal of Algebra

Let k be a field, Q a finite directed graph, and kQ its path algebra. Make kQ an N-graded algebra by assigning each arrow a positive degree. Let I be an ideal in kQ generated by a finite number of paths and write A=kQ/I. Let QGrA denote the quotient of the category of graded right A-modules modulo the Serre subcategory consisting of those graded mo...