Research

My interest is in noncommutative algebra, especially noncommutative invariant theory, noncommutative algebraic geometry, and certain classification problems in the field.

Some items of (potential) interest.

Below are some of my completed and ongoing projects. Also consider checking out my list of mathematicians I follow. Please see my page on student research for information on undergraduate projects I have mentored.

Preprints

  • (w/ Ellen Kirkman, W. Frank Moore, and Robert Won) Auslander's Theorem for permutation actions on noncommutative algebras.

    When A = k[x_1,..., x_n] and G is a small subgroup of GL_n(k), Auslander's Theorem says that the skew group algebra A#G is isomorphic to End_{A^G}(A) as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on (-1)-skew polynomial rings, (-1)-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain homogeneous down-up algebra. We also show that certain fixed rings A^G are graded isolated singularities in the sense of Ueyama.

  • (w/ Robert Won and Daniel Yee) Discriminants of Taft algebra smash products and applications.

    A general criterion is given for when the center of a Taft algebra smash product is the fixed ring. This is applied to the study of the noncommutative discriminant. Our method relies on the Poisson methods of Nguyen, Trampel, and Yakimov, but also makes use of Poisson Ore extensions. Specifically, we fully determine the inner faithful actions of Taft algebras on quantum planes and quantum Weyl algebras. We compute the discriminant of the corresponding smash product and apply it to compute the Azumaya locus and restricted automorphism group.

  • Isomorphisms of graded path algebras.

    We prove that if two path algebras with homogeneous relations are isomorphic as algebras, then they are isomorphic as graded path algebras. This extends a result by Bell and Zhang in the connected case.

Publications

  • The isomorphism problem for quantum affine spaces, homogenized quantized Weyl algebras, and quantum matrix algebras. Journal of Pure and Applied Algebra, 221 (2017), no. 10, 2511-2524.

    Bell and Zhang have shown that if A and B are two connected graded algebras finitely generated in degree one that are isomorphic as ungraded algebras, then they are isomorphic as graded algebras. We exploit this result to solve the isomorphism problem in the cases of quantum affine spaces, quantum matrix algebras, and homogenized multiparameter quantized Weyl algebras. Our result involves determining the degree one normal elements, factoring out, and then repeating. This creates an iterative process that allows one to determine relationships between relative parameters.

  • (w/ Ellen Kirkman and W. Frank Moore) On the discriminant of twisted tensor products. Journal of Algebra 477 (2017), 29-55.

    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.

  • (w/ Ken Price) Some algebras similar to the 2x2 Jordanian matrix algebra. Communications in Algebra, 45(5), 2017, 2091-2104.

    The impetus for this study is the work of Dumas and Rigal on the Jordanian deformation of the ring of coordinate functions on 2x2 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.

  • Two-parameter analogs of the Heisenberg enveloping algebra. Communications in Algebra, 44(11), 2016, 4637-4653.

    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.

  • PBW deformations of Artin-Schelter regular algebras. Journal of Algebra and Its Applications, 15(4), 2016.

    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.

  • Two-generated algebras and standard-form congruence. Communications in Algebra, 43(4), 2015, 1668-1686.

    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.

  • Isomorphisms of some quantum spaces. Ring Theory and Its Applications, Contemp. Math., 609, Amer. Math. Soc., 2014, 107-116.

    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.

  • Dissertation: PBW deformations of Artin-Schelter regular algebras and their homogenizations (Slides)

Current Projects

  • (w/ Dan Rogalski) Quivers supporting graded global dimension 3 Calabi-Yau algebras of finite GK dimension (in preparation)
  • (w/ Dan Rogalski) Blowups of the four-dimensional Sklyanin algebras (in preparation)
  • (w/ S.P. Smith) A birational equivalence between non-commutative analogues of ℙ2 and ℙ1 x ℙ1 (in preparation)