Research Mathematics

In brief, I am interested in noncommutative structures. My research thus far has focused on quasideterminants and Hopf algebras, but, frankly, if your problem has the word "noncommutative" somewhere in its description, you've peaked my interest. Research Statement

Recent Communications
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Presentations
  • PO-set paths and q-commuting minors, Sèminaire de combinatoire et d'informatique mathématique, LaCIM-UQAM, Montréal, April 2006. [slides]
  • Noncommutative flag varieties and Yangians, AMS Sectional Meeting, Eugene, OR, November, 2005. [slides]
  • A quasideterminantal approach to noncommutative flag varieties, Dissertation Defense, Rutgers University, April, 2005. [slides]
Papers, Notes
  • Quasideterminants and q-commuting minors. [submitted]
  • Flag varieties for the Yangian Y(gl_n). [submitted]
  • Coideals coming from coactions. [note]
  • (with E.J. Taft) A class of left quantum groups modeled after SLq(n). [preprint][to appear, JPAA]
  • NSym → Q∞ is not a Hopf map. [note]
  • Quantum- and quasi-Plücker coordinates. [preprint][J.Algebra `06]
Research Interests, Briefly
Quasideterminants
While following courses as a graduate student, one question I often asked myself was, "but what can we say when things don't commute?" Obviously, I was elated to discover the work of Gelfand & Retakh in this direction. In 1991, they introduced to the world what Cayley (1845), and others had been searching for… a proper determinant-like tool for the noncommutative setting. Their goal has been to provide explict formulas and objects with which to work---bringing the al-jabr back into the world of noncommutative algebra. In this they have been extremely successful. Since 1991, the quasideterminant has appeared as part of the story---if not THE story---in numerous seemingly diverse areas: Casimir operators in Lie theory, quantum determinants for quantum groups, the theory of noncommutative symmetric functions and the factorization of noncommutative polynomials. What's more, there's even a Cramer's rule with which to do noncommutative linear algebra!

In my dissertation, I introduce the notion of "amenable determinant" and use it, together with quasideterminants, to define (flag) varieties for a great many type A (for GLn) noncommutative settings. There is some promise that quasideterminants can also provide flag varieties for other types, and for Schubert subvarieties of these. An open question is whether quasideterminantal constructs alone can "completely describe" these noncommutative varieties; or if specific results in each setting are needed to provide the proper flag analogs there. No doubt, this is a project I will revisit frequently. Check back later.

I. Gel'fand, V. Retakh, Determinants of matrices over noncommutative rings, Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 13--25.

I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon, Noncommutative Symmetric Functions, Advances in Math 112 (1995), no. 2, 218--348.

I. Gel'fand, S. Gel'fand, V. Retakh, and R. Wilson, Quasideterminants, Advances in Mathematics, Volume 193, Issue 1, 1 May 2005, Pages 56-141

Hopf Algebras
The Hopf algebra phenomenon was first explored in algebraic topology (Heinz Hopf introduced them in his study of Sn); I don't know anything about these algebras or their success. The second wave of Hopf algebras were launched from Lie- and Algebraic-group theory. These are very nice, and continue to be a source of interesting mathematics (e.g. quantum groups). The latest wave of Hopf algebras were introduced by G.-C. Rota and his contemporaries in the service of combinatorics (or vice versa?). These are my favorites! Singling out two papers on this subject which I admire would be wrong, but I'll do it anyway (cf. below). One timeless example, so so important in representation theory: the ring of symmetric functions. A newer, equally beautiful example: the ring NSym of noncommutative symmetric functions introduced in the N.C.S.F. paper referenced above. I am interested in finding more uses for the quasideterminant in the study of NSym. Luckily, I'm currently surrounded by several experts in the field who may turn this nebulous interest into a specific question. Check back later.

N. Bergeron, S. Mykytiuk, F. Sottile, S. van Willigenburg, Noncommutative Pieri operators on posets, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 84--110.

M. Aguiar, N. Bergeron, F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, arXiv: math.CO/0310016

Representation Theory
The interplay between combinatorics and Lie theory at work in the representation theory of irreducible representations of Lie groups and Lie algebras is a source of endless amazement for me. Again, it would be an injustice for me to single out two papers on this subject which I admire, but I will do it anyway. While I presently have no projects in this direction, I will surely visit this beautiful subject in the future.

S. Sahi, A new formula for weight multiplicities and characters, Duke Mathematical Journal 101 (2000), no. 1, 77--84.

C. Lenart, A. Postnikov, Affine Weyl groups in K-theory and representation theory, arXiv: math.RT/0309207





Books That I Like
I list these in the hope that those students beginning study in these areas will find the suggestions useful.
(last updated: 09/2003)
Basic Algebra

T. Hungerford, Algebra, Springer-Verlag (reprint), 1997.

M. Artin, Algebra, Prentice-Hall, 1991.

N. Jacobson, Basic Algebra II, W.H. Freeman and Co., 1989.

Ring Theory & Module Theory

T.Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, 1991.

F. Anderson, K. Fuller, Rings and categories of modules, (2nd ed.) Springer-Verlag, 1992.

D.Sharpe, P. Vamos, Injective Modules, Cambridge U.P., 1972.

Lie Theory

N. Jacobson, Lie Algebras, Dover (reprint), 1979.

I. Stewart, Lie Algebras, LNM v. 127, Springer-Verlag, 1970.

J.-P. Serre, Complex Semisimple Lie Algebras, Springer-Verlag (reprint), 1987.

R. Carter Simple Groups of Lie Type, Wiley (reprint), 1989.

Group Theory

J.E. Humphreys, Reflection Groups and Coxeter Groups, CSAM no.29, Cambridge University Press, 1990.

T. Springer, Linear Algebraic Groups, Birkhauser, 1998.

Other Topics

M.E. Sweedler, Hopf Algebras, W.A. Benjamin, Inc., 1969.

J.J. Rotman, An Introduction to Homological Algebra, Academic Press, Inc., 1979.

W. Fulton, Young Tableaux, LMSST no.35, Cambridge University Press, 1997.

Orals Syllabus
Again, I include this primarily for students beginning study in these areas. It may be of interest to others as well: if you use TeX and don't use BibTeX, start using it right away! (Use the documents below to guide you.)
Syllabus
Bibliography

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