arXiv subjects

Jason Gaddis

Publications and source records attributed to Jason Gaddis.

41 records · Page 3Linked to original sources

arXiv2016

Some algebras similar to the 2x2 Jordanian matrix algebras

The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on $2\times 2$ matrices. We are also motivated by current interest in birational equivalence of noncommutative rings. Recognizing the construction of the Jordanian matrix algebra as a skew polynomial ring, we construct a family of algebras relative to differential operator rings over a polynomial ring in one variable which are birationally equivalent to the Weyl algebra over a polynomial ring in two variables.

math.RA↗

arXiv2015

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.

math.RA↗

arXiv2014

Two-parameter analogs of the Heisenberg enveloping algebra

One-parameter analogs of the Heisenberg enveloping algebra were studied previously by Kirkman and Small. In particular, they demonstrated how one may obtain Hayashi's analog of the Weyl algebra as a primitive factor of this algebra. We consider various two-parameter versions of this problem. Of particular interest is the case when the parameters are dependent. Our study allows us to consider the representation theory of a two-parameter version of the Virasoro enveloping algebra.

math.RA↗

arXiv2013

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.

math.QA↗

arXiv2013

PBW deformations of Artin-Schelter regular algebras

We consider algebras that can be realized as PBW deformations of (Artin-Schelter) regular algebras. This is equivalent to the homogenization of the algebra being regular. It is shown that the homogenization, when it is a geometric algebra, contains a component whose points are in 1-1 correspondence with the simple modules of the deformation. We classify all PBW deformations of 2-dimensional regular algebras and give examples of 3-dimensional deformations. Other properties, such as the skew Calabi-Yau property and closure under tensor products, are considered.

math.RA↗