My mathematical research is in non-commutative ring theory, and in
general I am interested in all things non-commutative. An operation like
addition is called commutative because a+b=b+a. In contrast, subtraction
is non-commutative because (in most cases) a-b is not b-a. Other familiar
non-commutative operations are the composition of functions and matrix
multiplication.
In the world of algebraic structures like groups or rings, commutativity
is a rarity. However, many properties known to hold for commutative rings
can be extended to non-commutative rings if one employs the proper definitions.
It is sometimes possible that the assumption of commutativity hides deeper
facts that can be understood in the more general non-commutative setting.
For a more technical account of my research, take a look at the publications
listed below.
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Publications
1. Global Dimension four Extensions of Artin-Schelter
Regular Algebras, in the Journal of Algebra, 220 (1999),
225-254.
Artin-Schelter (AS) regular algebras are of particular interest to those
working in non-commutative algebraic geometry. AS Regular algebras of global
dimension three were classified in 1987 and 1990 into a quadratic and a cubic
family. The AS regular algebras of global dimension four have been less tractable.
In 1996, Le
Bruyn, Smith and Van den Bergh classified the global dimension four AS
regular algebras which are extensions of the quadratic family by a central
element of degree one. Their method, however, is not applicable to the cubic
family, since it relies on the fact that the quadratic AS regular algebras
are Koszul algebras. I classify the global dimension four AS algebras which
are extensions of the cubic family by a central element of arbitrary degree.
Moreover, I extend my result and the result of Le Bruyn, Smith and Van den
Bergh to include extensions by normal elements of arbitrary degree.
2. Central Extensions
of Stephenson's Algebras, in Communications in Algebra, 31
(2003), 1615-1632.
In 1996 Stephenson completed
the classification of AS regular algebras of dimension three by including
algebras generated in degrees other than one. In Central Extensions of
Stephenson's Algebras (Communications in Algebra 2003) I classify the
global dimension four AS regular which are central extensions of the AS regular
algebras studied by Stephenson.
3. Homogenized Down-Up Algebras,
in Communications in Algebra, 31 (2003), 1765-1775.
Down-Up algebras where introduced
by Benkart and Roby in
1998, and have subsequently been studied by Carvalho, Jordan, Kirkman, Kuzmanovich,
Musson, Passman, Zhao and others. This paper answers questions about homogenizations
of down-up algebras which were posed by Benkart and Roby in their 1998 paper.
4. Basic Properties of
Generalized Down-Up Algebras, (with B. Shelton), in the Journal of
Algebra, 279 (2004), 402-421.
This paper generalizes the definition
of down-up algebras to include algebras studied by Smith, Rueda, Le Bruyn
and others. We prove that this larger family of algebras shares many properties
of down-up algebras. We determine the global (homological) dimension for all
the algebras in this larger family, and we relate the representation theory
of the Noetherian algebras to dynamical systems.
5. Generalized Laurent Polynomial Rings as Quantum Projective 3-Spaces, (with P. Goetz and B. Shelton), in the Journal of
Algebra, 303 (2006), 358-372.
We introduce the notion of a generalized Laurent polynomial ring over a
ground ring R. This class includes the generalized Weyl algebras.
We show that these rings inherit many properties from the R. This
construction is then used to create two new families of quadratic global dimension
four Artin-Schelter regular algebras. We show that in most cases the second
family has a finite point scheme and a defining automorphism of finite order.
Nonetheless, a generic algebra in this family is not finite over its center.
6. PBW-deformation theory and regular central extensions, (with B. Shelton), in Crelle's Journal, 610 (2007), 1-12.
A deformation U, of a graded K-algebra A is said to be of PBW type if grU is A. It has been shown for Koszul and N-Koszul algebras that the deformation is PBW if and only if the relations of U satisfy a Jacobi type condition. In particular, for these algebras the determination of the PBW property is a finite and explicitly determined linear algebra problem. We extend these results to an arbitrary graded K-algebra, using the notion of central extensions of algebras and a homological constant attached to A which we call the complexity of A.
7. Generalizing the notion of Koszul Algebra, (with B. Shelton), in Mathematische Zeitschrift, 260 (2008), no. 1, 93-114.
We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.
8. Generalizations of graded Clifford algebras and of complete intersections, (with M. Vancliff), to appear in the Journal of the London Mathmatical Society.
We introduce graded skew Clifford algebras and associate to such an algebra the notion of a quadric system in the spirit of the non-commutative algebraic geometry of Artin, Tate and Van den Bergh. We prove that a graded skew Clifford algebra is quadratic and regular if and only if its associated quadric system is normalizing and base-point free. In the process, we extend the notions of complete intersection, base-point free, quadratic form and symmetric matrix to the non-commutative setting. We then use our results to produce several families of quadratic regular skew Clifford algebras of global dimension four.
9. The Yoneda algebra of a K2 algebra need not be another K2 algebra, (with C. Phan and B. Shelton), to appear in Communications in Algebra.
The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K_2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.
10. Noncommutative Koszul algebras from combinatorial topology, (with C. Phan and B. Shelton), under review.
Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups H_X(n,k), generalizing the usual cohomology groups H^n(X). Along with several other results, our methods give a new and primarily topological proof of a result of Serconek and Wilson and of Piontkovski.
11. Quadratic algebras with Ext algebras generated in two degrees, under review.
Green and Marcos call a graded k-algebra delta-Koszul if the corresponding Yoneda algebra Ext (k,k) is finitely generated and Ext^{i,j} (k,k) is zero unless j=delta(i) for some function delta:N -> N. For any integer m>2 we exhibit a non-commutative quadratic delta-Koszul algebra for which the Yoneda algebra is generated in degrees (1,1) and (m,m+1). These examples answer a question of Green and Marcos.
These algebras are not Koszul but are m-Koszul (in the sense of Backelin).
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