Jason Gaddis

Isomorphisms of some quantum spaces

We consider a series of questions that grew out of determining when two quantum planes are isomorphic. In particular, we consider a similar question for quantum matrix algebras and certain ambiskew polynomial rings. Additionally, we modify a result by Alev and Dumas to show that two quantum Weyl algebras are isomorphic if and only if their parameters are equal or inverses of each other.

On the discriminant of twisted tensor products

Abstract We provide formulas for computing the discriminant of noncommutative algebras over central subalgebras in the case of Ore extensions and skew group extensions. The formulas follow from a more general result regarding the discriminants of certain twisted tensor products. We employ our formulas to compute automorphism groups for examples in each case.

PBW deformations of Artin–Schelter regular algebras

We consider properties and extensions of PBW deformations of Artin–Schelter regular algebras. PBW deformations in global dimension two are classified and the geometry associated to the homogenizations of these algebras is exploited to prove that all simple modules are one-dimensional in the non-PI case. It is shown that this property is maintained under tensor products and certain skew polynomial extensions.

Two-Generated Algebras and Standard-Form Congruence

Matrix congruence can be used to mimic linear maps between homogeneous quadratic polynomials in n variables. We introduce a generalization, called standard-form congruence, which mimics affine maps between non-homogeneous quadratic polynomials. Canonical forms under standard-form congruence for three-by-three matrices are derived. This is then used to give a classification of algebras defined by two generators and one degree two relation. We also apply standard-form congruence to classify homogenizations of these algebras.

Auslander’s theorem for permutation actions on noncommutative algebras

When

Discriminants of Taft Algebra Smash Products and Applications

A = \mathbb{k}[x_1, \ldots, x_n]

The isomorphism problem for quantum affine spaces, homogenized quantized Weyl algebras, and quantum matrix algebras

and

Two-Parameter Analogs of the Heisenberg Enveloping Algebra

G

Some algebras similar to the 2×2 Jordanian matrix algebra

is a small subgroup of

PBW deformations of Artin-Schelter regular algebras and their homogenizations

\operatorname{GL}_n(\mathbb{k})