Curriculum Vitae

 Alex Jay Feingold

Department of Mathematical Sciences, SUNY-Binghamton, Binghamton, NY 13902-6000
June 26, 2016

Personal:

Born: April 1, 1950, Baltimore, Maryland, USA
Permanent Address: 45 Matthews Street, Binghamton, NY 13905-4038
Phone Numbers: (607) 729-3637 (home), (607) 777-2465 (office)
Email: alex@math.binghamton.edu
Spouse: Nancy Tittler
Married: December 18, 1977
Children: Emily Ruth (Born Aug. 27, 1985), Judith Marian (Born Mar. 10, 1988)

Education:

Dissertation:

``Tensor products of modules for Lie algebras''
Director:  Professor George B. Seligman

Academic Honors:

Grants:

  1. 1994-96    National Security Agency Grant, Mathematical Sciences Program (2 years), for ``Vertex Operator Algebras and Representation Theory''
  2. 1987-88    Grant-in-Aid from the Institute for Advanced Study, Princeton, NJ
  3. 1987        National Science Foundation Grant (2 years) for ``Affine and Hyperbolic Kac-Moody Algebras''
  4. 1985        National Science Foundation Grant (2 years) for ``Affine and Hyperbolic Kac-Moody Algebras''
  5. 1985        National Science Foundation Grant, Mathematical Sciences Research Equipment, (1 year), jointly with other Department members
  6. 1985        Dean's Research Semester Award
  7. 1984        SUNY Faculty Research Fellowship Award
  8. 1982        SUNY Faculty Research Fellowship Award
  9. 1980-81    National Science Foundation Grant (2 years) for ``Generalized Cartan Matrix Lie Algebras''
  10. 1980        SUNY Faculty Research Fellowship Award (declined because of NSF grant)
  11. 1978-79    National Science Foundation Grant for ``Generalized Cartan Matrix Lie Algebras and Power Series Identities''

Professional and Honor Societies:

Professional Experience:

Professional Activities:

    Transactions of the American Mathematical Society,
    Journal of Algebra,
    Journal fur die Reine und Angewandte Mathematik,
    Journal of Number Theory,
    Conference Proceedings on Lie Algebras and Related Topics at Madison, Wisconsin, 1988,
    Duke Mathematical Journal,
    Journal of Pure and Applied Algebra, Advances in Mathematics,
    Communications in Mathematical Physics,
    Journal of Physics A,
    Proceedings of the Conference Moonshine, The Monster,and Related Topics, 1994.

Invited Lectures:

Conferences and Workshops Attended or Organized:

Teaching Interests:

Algebra (Groups, Rings, Fields, etc.), Linear Algebra, Lie Algebras, Vertex Operators,
Modular Forms, Siegel Modular Forms, Conformal Field Theory, Fusion Algebras.

Summary of Research Interests:

My area of special interest is the theory of Lie algebras, their representations, connections to other parts of mathematics and applications to physics. My thesis concerned the decomposition of tensor products of finite-dimensional modules for complex semisimple Lie algebras. While still a graduate student at Yale, strongly influenced by my teacher, Jim Lepowsky, I extended my research into the infinite-dimensional Kac-Moody Lie algebras, independently introduced in 1968 by V.G. Kac (M.I.T.) and R.V. Moody (University of Saskatchewan). This has been an exciting and fruitful area of research because of its remarkable connections with physics (e.g., solitons, quantum field theory, string theory) and other areas of mathematics (e.g., combinatorics, group theory, modular forms, singularities, differential equations, knot theory).

During the period from 1981 to 1991 I had several collaborations with Igor Frenkel (Yale University). Our first paper studied hyperbolic Kac-Moody algebras, showing one such algebra to be closely connected with the theory of Siegel modular forms of genus two and with the related problem of lifting elliptic modular forms (the Saito-Kurokawa conjecture). We also gave a construction which provided closed formulas for an infinite number of root multiplicities (on levels 0, 1 and 2). Our second paper studied affine Kac-Moody algebras, providing a unified approach to constructing certain representations of all the classical affine algebras. These were based on underlying associative algebras of commutation or anticommutation relations whose bosonic or fermionic representations are important in quantum field theory. Another paper, with J. F. X. Ries, studied representations of hyperbolic Kac-Moody algebras, constructing all irreducible highest weight standard modules and providing closed formulas for an infinite number of weight multiplicities (on levels 0, 1 and 2). Other collaborations, also with Ries, studied the vertex operator algebras known in physics as chiral algebras. These algebras play a central role in string theory, conformal field theory, and in the Frenkel-Lepowsky-Meurman construction of the ``Monster'' group. Our main objectives were to obtain independent vertex and spinor constructions of chiral algebras, the isomorphism between the two viewpoints, known as a ``boson-fermion correspondence'', and constructions of the exceptional affine algebra E8(1) based on D4(1) spinor constructions and the principle of triality. Such representations of E8(1) are essential in the anomaly-free heterotic superstring theory of particle physics. Some of these results were announced at the 1988 Conference on Lie Algebras and Related Topics, Madison, Wisconsin. Those results which only involve the spinor constructions are in our Contemporary Mathematics monograph ([10]). A sequel (with Ries only) was planned to give the vertex picture and the boson-fermion correspondence, but the untimely death of Ries has prevented the completion of that project up until the present.

In [13] Weiner (my first Ph.D. student) and I completed a detailed study started by Ries and myself, of the vertex operator superalgebra, modules and intertwining operators constructed from the c = 1/2 Virasoro modules. There are a number of constructions known [12] of vertex operator algebras and their representations, and many of these are quite difficult to give rigorously. Of particular interest are the constructions coming from Virasoro representations where c < 1, known in physics as the discrete series of minimal models, and those coming from representations of affine Kac-Moody Lie algebras, also known as Wess-Zumino-Witten models. These are deeply connected with braid groups and quantum groups, topics of great current interest. The structure of the intertwining operators is governed by the fusion rules which I have studied in [14,15,16,18] with various collaborators, and which were studied in the Ph.D. thesis of my student, Omar Saldarriaga. The results found in [18] brought my research back in a full circle to where I started, showing a remarkable relationship between affine fusion rules and the tensor product multiplicities I studied in my dissertation.

More recently, I returned to the study of hyperbolic Kac-Moody Lie algebras in a collaboration with Hermann Nicolai [17] where we studied subalgebras of hyperbolic algebras, and found, for example, that all the rank 2 hyperbolic algebras with symmetric Cartan matrices are subalgebras of the rank 3 hyperbolic algebra studied in [6]. In collaborations [19,20] with Hermann Nicolai and Axel Kleinschmidt, I  returned to the study of Weyl groups of hyperbolic algebras. It was the key observation of [6] that the Weyl group of the rank 3 hyperbolic algebra studied there is isomorphic to PGL(2,Z), an index 2 extension of the modular group, PSL(2,Z). In [20] we found a generalization to all hyperbolic Weyl groups, which we showed can be realized as certain matrix groups of 2x2 matrices with entries from one of the four normed division algebras, the reals, the complex numbers, the quaternions, or the octonions. This observation could be the starting point for much deeper structural studies of all hyperbolic Kac-Moody algebras, generalizing those found in [6]. Of particular interest is the case of the hyperbolic algebra known in physics as E10, believed to be relevant to string theory. With Terry Gannon (University of Alberta, Edmonton, Alberta, Canada) I started pursuing research into the possibility of applying these methods the 24-dimensional commutative non-associative Chevalley algebra in place of the octonians to understand automorphisms of the 26-dimensional even Lorentzian lattice II{25,1} and groups of automorphisms of the Leech lattice and other lattices.

In collaborations, started during my 2009-2010 sabbatical, with Lisa Carbone (Rutgers University) and Walter Freyn (University of Dortmond, Germany), I am studying Tits buildings associated with hyperbolic Kac-Moody Lie algebras. We have shown how to embed these buildings inside the light-cone of the compact form of the Lie algebra by using the structure of the family of all Cartan subalgebras inside a hyperbolic KM algebra. This is the first time I am working with the Kac-Moody groups, rather than with the Lie algebras.

I also began a new collaboration with Antun Milas (SUNY, Albany), investigating the representation theory behind the appearance of the Rogers-Ramanujan series in the tensor product decomposition of two level-1 modules for the twisted affine Kac-Moody Lie algebra A2(2). In particular, we wanted to explain how the sum of irreducible Virasoro characters with one (positive) central charge could equal an irreducible Virasoro character with a different (negative) central charge.

I started a research project on the decomposition of rank 2 symmetric hyperbolic algebras with respect to the Nicolai-Olive principal so(2,1) Lie subalgebra with Elizabeth Jurisich (College of Charleston, SC).

I supervised the Ph.D. research of three graduate students, Quincy Loney (finished July 2012), Christopher Mauriello (finished May 2013) and Diego Penta (finished May 2016). The dissertation of Quincy Loney concerns the decomposition of level-1 irreducible representations of the affine Kac-Moody algebra D4(1) with respect to its subalgebra G2(1) in the spinor construction. This work uses the Goddard-Kent-Olive coset Virasoro construction to find two commuting Virasoro algebras, one with central charge 1/2 and another with central charge 7/10, which commute with the G2(1) subalgebra, and generate the space of highest weight vectors giving the branching rules. Loney's dissertation was completed and defended on July 30, 2012. Christopher Mauriello investigated a similar branching rule problem for how the level-1 irreducible representations of the affine Kac-Moody algebra  E6(1) decompose with respect to its subalgebra F4(1) . Both projects involved the character theory of the relevant modules, and an important role was played by certain identities for the Roger-Ramanujan functions discovered by Ramanujan. Diego Penta's project involved the decomposition of the rank 3 hyperbolic KM algebra, F, with respect to its rank 2 hyperbolic ``Fibonacci" subalgebra, Fib. Penta's dissertation was defended on May 20, 2016.

I have completed a project with postdoctoral visitor Daniel Valli\'eres studying the Weyl groups of certain hyperbolic Kac-Moody Lie algebras, related to the earlier work I did with Kleinschmidt and Nicolai. This project uses matrices over a Clifford algebra to define Vahlen groups which contain certain hyperbolic Weyl groups.

Invited Addresses:

  1. The Special Session on Lie algebras, organized by Maria Wonenberger at the 775th Meeting of the American Mathematical Society at Bloomington, IN, April 11-12, 1980.
  2. The Special Session on Kac-Moody Lie Theory, organized by Howard Garland and James Hurley at the 789th Meeting of the American Mathematical Society at the University of Massachusetts, Amherst, October 16-18, 1981.
  3. The workshop on Vertex Operators in Mathematics and Physics, organized by James Lepowsky at the Mathematical Sciences Research Institute in Berkeley, CA, November 10-17, 1983.
  4. The Lie Algebras and Related Topics Conference at the University of Wisconsin, Madison, organized by J. Marshall Osborn and Georgia Benkart, May 22-June 1, 1988.
  5. The 1991 American Mathematical Society Summer Research Institute, on Algebraic Groups and Their Generalizations, Pennsylvania State University, University Park, PA, July 8-26, 1991.
  6. The Special Session on Rings and Representations, organized by Martin Lorenz and Shari A. Prevost at the 868th Meeting of the American Mathematical Society, Temple University, Philadelphia, PA, October 12-13, 1991.
  7. The Structure and Representation Theory of Lie Algebras conference in honor of George Seligman, April 10-12, 1992, Yale University, New Haven, CT.
  8. The 884th Meeting of the AMS, Special Session on Lie Theoretical Methods in Mathematical Physics, Sept. 18-19, 1993, Syracuse University, Syracuse, NY.
  9. The Joint Summer Research Conference in the Mathematical Sciences, Moonshine, The Monster, and Related Topics, Mount Holyoke College, South Hadley, MA, June 18-24, 1994.
  10. The 906th Meeting of the AMS, Special Session on Quantum Kac-Moody Lie Algebras and Related Topics, Nov. 17-18, 1995, Greensboro, North Carolina.
  11. The 922nd Meeting of the AMS, Special Session on  VOA's, Monstrous Moonshine and Related Topics, May 2-4, 1997, Detroit, Michigan.
  12. The Conference on Generalized Kac-Moody Algebras at the Mathematical Research Institute at Oberwolfach, Germany, organized by Richard Borcherds and Peter Slodowy, July 19-25, 1998.
  13. The 943rd Meeting of the AMS, Special Session on Representations of Lie Algebras, April 24-25, 1999, State University of New York at Buffalo, NY.
  14. Infinite Dimensional Lie Theory and It's Applications Program, Workshop on Vertex Operator Algebras in Mathematics and Physics, Oct. 23 - 27, 2000, The Fields Institute, Toronto, Ontario, Canada.
  15. Ramanujan International Symposium on Kac-Moody Lie Algebras and Applications January 28--31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India.
  16. The 1024th Meeting of the AMS, Special Session on Geometric and Combinatorial Methods in Representation Theory, March 3-4, 2007, Davidson College, Davidson, NC.
  17. International Conference on Vertex Operator Algebras and Related Areas, A conference to mark the occasion of Geoffrey Mason's 60th Birthday, July 7-11, 2008, Mathematics Department, Illinois State University.
  18. The 1048th Sectional Meeting of the AMS, Special Session on Kac-Moody Algebras, Vertex Algebras, Quantum Groups, and Applications, North Carolina State University, April 5, 2009. Title of Talk: Hyperbolic Weyl Groups and Related Coxeter Groups.
  19. The 1062nd Sectional Meeting of the AMS, Special Session on Lie Algebras and, Representation Theory, Syracuse University, October 2, 2010. Title of Talk: Decomposition of a rank 2 hyperbolic Kac-Moody Lie algebra with respect to the Nicolai-Olive principal so(1,2) subalgebra (joint with Elizabeth Jurisich).
  20. The 1065th Sectional Meeting of the AMS, Special Session on Kac-Moody Algebras, Vertex (Operator) algebras and Applications, University of Richmond, Richmond, VA, November 6, 2010. Title of Talk: Spinor construction of representations of affine Kac-Moody algebras of types G2(1) and D4(3) (joint with Quincy Loney).
  21. The Canadian Mathematical Society Summer Meeting, Special Session on Lie Theory, University of Alberta, Edmonton, Alberta, Canada, June 3, 2011. Title of Talk: Decomposition of level-1 representations of D4(1) with respect to its subalgebra G2(1) in the spinor construction (joint with Quincy Loney).
  22. A workshop on Infinite dimensional Lie theory: Algebra, Geometry and Combinatorics, Centre de recherches math�matiques (CRM), Montreal, Quebec, Canada, August 21-24, 2012, organized by Joel Kamnitzer (University of Toronto) and Michael Lau (Universit� Laval).
  23. ``Symmetries, unification and the search for quantum gravity", A conference on the occasion of Hermann Nicolai's 60th birthday, Sept. 6 - 8, 2012, Albert Einstein Institute, Max-Planck Institute for Gravitational Physics, Potsdam, Germany.
  24. ``Representation Theory XIII", A conference at the Inter-University Centre, Dubrovnik, Croatia, June 20-27, 2013, special session on vertex operator algebras, Kac-Moody Lie algebras and related topics.
  25. International Congress of Mathematicians 2014, Satellite Conference on "Representation Theory and Related Topics", at the Exco Convention Center, Daegu, South Korea, Aug 6-9, 2014.
  26. Conference on ``Generalizations of Symmetric Spaces", Nahsholim Sea Resort, Israel, June 17-24, 2015.
  27. Special Session on ``Representation Theory, Vertex Operator Algebras, and Related Topics�, AMS meeting at Rutgers University, New Brunswick, November 14-15, 2015.
  28. Conference on "Lie and Jordan algebras, Their Representations and Applications-VI", Bento Goncalves, Brazil, December 13-19, 2015. Unable to attend.
  29. Special Session on ``Algebraic structures in mathematical physics: Lie algebras, vertex algebras, quantum algebra'' at the 1117th AMS meeting at the University of Georgia, Athens, GA, March 4-6, 2016.

Publications:

  1. ``Zones of uniform decomposition in tensor products'', Proceedings of the American Mathematical Society, Vol. 70, No. 2, July 1978, 109-113.
  2. ``Tensor products of finite dimensional modules for complex semisimple Lie algebras'', Lie Theories and Their Applications, Proceedings of the 1977 Annual Seminar of the Canadian Mathematical Congress, Queen's Papers in Pure and Applied Mathematics No. 48, Editors: A. J. Coleman and P. Ribenboim, Queen's University, Kingston, Ontario, 1978, 394-397.
  3. ``The Weyl-Kac character formula and power series identities'', Advances in Mathematics 29, No. 3, September 1978, 271-309 (with J. Lepowsky).
  4. ``A hyperbolic GCM Lie algebra and the Fibonacci numbers'', Proceedings of the American Mathematical Society, Vol. 80, No. 3, November 1980, 379-385.
  5. ``Tensor products of certain modules for the Generalized Cartan Matrix Lie Algebra A_1^(1)'', Communications in Algebra, Vol. 9, No. 12, 1981, 1323-1341.
  6. ``A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2'', Mathematische Annalen 263, 1983, 87-144 (with I. Frenkel).
  7. ``Classical affine algebras'', Advances in Mathematics, Vol. 56, No. 2, May 1985, 117-172 (with I. Frenkel).
  8. ``Some applications of vertex operators to Kac-Moody algebras'', Vertex Operators in Mathematics and Physics. Proceedings of a conference Nov. 10-17, 1983. Edited by J. Lepowsky, S. Mandelstam, I. M. Singer. Publications of the Mathematical Sciences Research Institute #3, Springer-Verlag, 1985, 185-206.
  9. ``The exceptional affine algebra E8(1), triality and chiral algebras'', Lie Algebras and Related Topics, Proceedings of a Conference held in Madison, Wisconsin, May 22 to June 1, 1988; Editors: G. Benkart and J. M. Osborn; Contemporary Mathematics, Vol. 110, American Mathematical Society, Providence, RI, 1989, (with Igor Frenkel and John F. X. Ries).
  10. ``Spinor Construction of Vertex Operator Algebras, Triality and E8(1)'', Contemporary Mathematics, Vol. 121, American Mathematical Society, Providence, RI, 1991, 146 pp. monograph (with Igor Frenkel and John F. X. Ries).
  11. ``Representations of hyperbolic Kac-Moody algebras'', Journal of Algebra, Vol. 156, No. 2, April 1993, 433-453 (with Igor Frenkel and John F. X. Ries).
  12. ``Constructions of vertex operator algebras'', Proceedings of Symposia in Pure Mathematics, Vol. 56, Algebraic Groups and Their Generalizations, William J. Haboush and Brian J. Parshall, Editors, American Mathematical Society, Providence, RI, Part 2, 317-336, April 1994.
  13. ``Spinor construction of the c = 1/2 minimal model'', Moonshine, The Monster, and Related Topics, Contemporary Mathematics, Vol. 193, Chongying Dong and Geoffrey Mason, editors, American Mathematical Society, Providence, RI, 1995 (with John F. X. Ries and Michael Weiner), 45-92.
  14. ``Minimal model fusion rules from 2-groups'', Letters in Mathematical Physics, Vol. 40, No. 2 (1997), 159-169, (with Fusun Akman and Michael Weiner).
  15. ``Type A Fusion Rules From Elementary Group Theory'', Comtemporary Mathematics, Vol. 297, Proceedings of the Conference on Infinite-Dimensional Lie Theory and Conformal Field Theory, Charlottesville, VA, American Mathematical Society, Providence, RI, 2002 (with Michael Weiner), 97--115.
  16. ``Fusion Rules for Affine Kac-Moody Algebras'', Kac-Moody Lie Algebras and Related Topics, Ramanujan International Symposium on Kac-Moody Algebras and Applications, Jan. 28-31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, N. Sthanumoorthy, Kailash Misra, Editors, Contemporary Mathematics, Vol. 343, American Mathematical Society, Providence, RI, 2004, 53--96.
  17. ``Subalgebras of hyperbolic Kac-Moody Algebras'', Kac-Moody Lie Algebras and Related Topics, Ramanujan International Symposium on Kac-Moody Algebras and Applications, Jan. 28-31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, N. Sthanumoorthy, Kailash Misra, Editors, Contemporary Mathematics, Vol. 343, American Mathematical Society, Providence, RI, 2004, 97--114, (with Hermann Nicolai).
  18. ``A New Perspective on the Frenkel-Zhu Fusion Rule Theorem'', Journal of Algebra 320 (2008), 2079--2100, (with Stefan Fredenhagen).
  19. ``Hyperbolic Weyl groups and the four normed division algebras'', in Vertex Operator Algebras and Related Areas, Contemporary Mathematics, Vol. 497, (12 pages), Amer. Math. Soc., Providence, RI, in press. (with Hermann Nicolai and Axel Kleinschmidt).
  20. ``Hyperbolic Weyl groups and the four normed division algebras'', Journal of Algebra 322 (2009), 1295-1339 (with Hermann Nicolai and Axel Kleinschmidt). A Corrigendum (author's corrections) to this paper is available through the following link: Corrigendum to ``Hyperbolic Weyl Groups and the Four Normed Division Algebras'' [J. Algebra 322 (2009) 1295-1339], J. Algebra 489 (2017), 586-587. The arXiv copy of the 2009 paper has been updated to contain the corrections. Another paper with the same title and authors as the 2009 paper was published in the proceedings of an international conference on vertex operators and related areas in honor of Geoffrey Mason. That conference took place at Illinois State University in July, 2008. That shorter paper is an introduction to and announcement of the results in this paper, and appeared in Vertex operator algebras and related areas, Contemporary Mathematics, vol. 497, Amer. Math. Soc., Providence, RI (2009), 53-64.
  21. ``Matrix realizations of hyperbolic triangle groups'', completed July 2010, joint with Elizabeth Dwornik, unpublished.
  22. ``The 3-State Potts model and Rogers-Ramanujan series'', Central European Journal of  Mathematics, 11(1), 2013, pp. 1--16 (with Antun Milas).
  23. ``Weyl groups of some hyperbolic Kac-Moody algebras'', Journal of Algebra 500 (2018), 457-497, (with Daniel Valli\`eres). DOI: 10.1016/j.jalgebra.2017.05.003
  24. ``A Lightcone embedding of the twin building of a hyperbolic Kac-Moody group'', (with Lisa Carbone and Walter Freyn), 2017, to be submitted.

Work in Progress:

  1. ``Structure of Cartan subalgebras in hyperbolic Kac-Moody algebras'', (with Walter Freyn).
  2. ``Decomposition of a rank 2 hyperbolic Kac-Moody Lie algebra with respect to the Nicolai-Olive principal so(1,2) subalgebra'' (joint with Elizabeth Jurisich).
  3. ``On principal subspaces of certain admissible Cn(1)-modules'', (with C. Calinescu and Antun Milas).
  4. ``Further research on hyperbolic Kac-Moody algebras'', (with Hermann Nicolai and Axel Kleinschmidt).
  5. ``Vertex Operator Algebras, Triality and E8(1)'', book, (with the late John F. X. Ries).

Graduate Students:

  1. Michael D. Weiner, Ph.D. 1994, Thesis: ``Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras'', published: Memoirs of the American Mathematical Society, Vol. 135, 1998. (Currently Associate Professor at the Altoona Campus of Penn State University.)
  2. Omar Saldarriaga, Ph.D. 2004, Thesis: Fusion Algebras, Symmetric Polynomials, Orbits of N-Groups, and Rank-Level Duality. Published as an article in the Journal of Algebra 312 (2007), 257--293. (Currently Professor at the Universidad de Antioquia, Medellin, Colombia, South America.)
  3. Quincy Loney, Ph.D. July 2012, Thesis: Decomposition of level-1 representations of D4(1) with respect to its subalgebra G2(1) in the spinor construction.
  4. Christopher Mauriello, Ph.D. May 2013, Thesis: Branching rule decomposition of irreducible level-1 E6(1) -modules with respect to the affine subalgebra F4(1) .
  5. Diego Penta, Ph.D. May 2016: Thesis: Decomposition of the rank 3 hyperbolic Kac-Moody Lie algebra, F, with respect to its rank 2 hyperbolic subalgebra, Fib.

Links back to:
Webpage of Alex Feingold
Department of Mathematical Sciences
Binghamton University

This page last modified on 2/19/2018.