June 26, 2016

Email: alex@math.binghamton.edu

- 1973-77 Ph.D. Yale University, New Haven, CT, Department of Mathematics
- 1971-73 Health Services Officer (Computer Programming) Health Services Research, U.S. Public Health Service Hospital, 3100 Wyman Park Drive, Baltimore, MD 21211
- 1967-71 B.A., M.A.Johns Hopkins University, Baltimore, MD, Major: Mathematics

Director: Professor George B. Seligman

- 1987-88 Member of the Institute for Advanced Study, Princeton, NJ
- 1984 Member of the Mathematical Sciences Research Institute, Berkeley, California, for part of the summer.
- 1982-83 Member of the Institute for Advanced Study, Princeton, NJ
- 1973-77 Yale University Fellowship
- 1971 Gilman Fellowship, Johns Hopkins University
- 1971 Junior Instructorship
- 1969-70 Phi Beta Kappa
- 1967-70 U.S. Senatorial Scholarship

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

- American Mathematical Society
- Phi Beta Kappa (President of Local Chapter at Binghamton University)
- Pi Mu Epsilon (Permanent Faculty Correspondent and Charter
Member
of the New York Alpha Omicron Chapter at Binghamton University)

- 1998- Professor of Mathematics, SUNY, Binghamton, NY,
- 1987-97 Associate Professor of Mathematics, SUNY, Binghamton, NY,
- 1987-88 Member of The Institute for Advanced Study, Princeton, NJ
- 1982-83 Member of The Institute for Advanced Study, Princeton, NJ
- 1979-87 Assistant Professor of Mathematics, SUNY, Binghamton, NY,
- 1977-79 Visiting Assistant Professor of Mathematics, Drexel University, Philadelphia, PA
- 1975-77 Graduate Teaching Assistant, Mathematics, Yale University, New Haven, CT
- 1970-71 Junior Instructor, Mathematics, Johns Hopkins University, Baltimore, MD

- Reviewer for the Mathematical Reviews since 1980 (over 56 papers reviewed)
- Refereed papers for:

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

- Reviewed Manuscript of a proposed Linear Algebra textbook for John Wiley & Sons, Inc. Publishers (consulting activity).
- Reviewer of Grant proposals for the National Science Foundation.
- Reviewer of Grant proposals for the National Security Agency.

- Wrote a book review published in the Bulletin of the American Mathematical Society, Vol. 25, No. 2, Oct. 1991, 432-440: ``Group Theory in Physics, Volume III, Super-symmetries and Infinite Dimensional Algebras'', by J. F. Cornwell, Techniques of Physics: 10, ed. N. H. Marsh, Academic Press, 1989.
- Served as a mentor during 2009-2010 for a high school student
(Elizabeth Dwornik, Union-Endicott High School) doing a
mathematics
research project for the Intel Science Talent Search. This
culminated
in the joint writing of the expository paper ``Matrix
Realizations of
Hyperbolic Triangle Groups''.

- Cornell Algebra Seminar
- Temple Univesity Colloquium
- Yale University Colloquium
- Yale University Lie Groups Seminar
- Rutgers University Lie Algebras and Lie Groups Seminar
- University of Maryland Number Theory Seminar
- University of Rochester, Department of Physics
- Baruch College, CUNY, Department of Mathematics
- University of New Hampshire, Durham, Department of Mathematics
- SUNY at Geneseo, Department of Mathematics
- CUNY, Graduate Center, Department of Mathematics
- Cornell Lie Groups Seminar
- Various American Math Society Meetings
- Mathematical Research Institute, Oberwolfach, Germany
- Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India
- Max Planck Institute for Gravitational Physics, Potsdam, Germany.
- College of Charleston, Charleston, South Carolina.
- Illinois State University, Normal, Illinois.
- University of Alberta, Edmonton, Alberta, Canada.
- Centre de recherches mathématiques (CRM), Montreal, Quebec, Canada.
- University of Connecticut, Storrs, Connecticut.
- ``Representation Theory XIII", Inter-University Centre, Dubrovnik, Croatia, June 20-27, 2013.
- 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.
- Institut des Hautes Etudes Scientifique (IHES),
Bures-sur-Yvette, France, May 2015.

- Joint Summer Research Conference in the Mathematical Sciences,
*Conformal Field Theory, Topological Field Theory, and Quantum Groups*, Mount Holyoke College, South Hadley, MA, June 13-19, 1992. - NSF-CBMS Conference on ``Applications of the Representation Theory of Quantum Affine Lie Algebras to Solvable Lattice Models'', North Carolina State University, Raleigh, North Carolina, June 1-5, 1993.
- Joint Summer Research Conference in the Mathematical Sciences,
*Moonshine, The Monster, and Related Topics*, Mount Holyoke College, South Hadley, MA, June 18-24, 1994. - Conference on Infinite-Dimensional Lie Theory and Conformal Field Theory, University of Virginia, Charlottesville, Virginia, May 23-27, 2000.
- Workshop
*Conformal Field Theory and Supersymmetry,*April 15--19, 2002, Mathematical Sciences Research Institute, Berkeley, California. - Attended the workshop ``Moonshine -- the First Quarter Century
and Beyond. A Workshop on the Moonshine Conjectures and Vertex
Algebras'', July 5--13, 2004, Heriot-Watt Univeristy,
International
Centre for Mathematical Sciences, Edinburgh, Scotland.

- Organized the Special Session on ``Lie Algebras, Conformal Field Theory, and Related Topics", at the 990th Sectional Meeting of the American Mathematical Society, Oct. 11--12, 2004, Binghamton University, Binghamton, NY (with co-organizers Chongying Dong and Gaywalee Yamskulna).
- Co-organizer of a Special Session on ``Theory of
Infinite-Dimensional Lie Algebras, Vertex Operator Algebras, and
Related Topics", at the 1009
^{th}Sectional Meeting of the American Mathematical Society, Oct. 8--9, 2005, Annandale-on-Hudson, NY, Bard College (with co-organizers Antun Milas and Yi-Zhi Huang). - Attended and presented an invited talk in the Special Session
on
Geometric and Combinatorial Methods in Representation Theory at
the 1024
^{th}American Mathematical Society meeting at Davidson College, Davidson, North Carolina, March 3-4, 2007. - Visited the Albert Einstein Institute, Max Planck Institute for Gravitational Physics, in Potsdam, Germany, July 30 - August 10, 2007, for collaboration with Hermann Nicolai and Axel Kleinschmidt on Weyl groups of hyperbolic Kac-Moody algebras. Also gave a talk on ``A New Perspective on the Frenkel-Zhu Fusion Rule Theorem".
- Visited the Max Planck Institute for Mathematics in Bonn, Germany, August 10 - 12, for consultations with Don Zagier, on the way to Paris.
- Attended part of the XXXVIIth Paris Summer Institute on Black Holes, Black Rings and Modular Forms, at the Ecole Normale Superieure, August 13 - 15, 2007.
- Attended the conference ``Representation Theory and
Mathematical
Physics" in honor of the 60
^{th}birthday of Gregg Zuckerman, Yale University, New Haven, CT, October 24-27, 2009. - Co-organizer of a Special Session on ``Kac-Moody Algebras, Vertex Algebras, and Related Topics,", at the 1072$^{nd}$ Sectional Meeting of the American Mathematical Society, September 10-11, 2011, Cornell University, Ithaca, NY, (with co-organizer Antun Milas).
- Attended the conference in honor of Igor Frenkel's 60
^{th}birthday, ``Perspectives in Representation Theory", May 12-17, 2012, Yale University, New Haven, Connecticut. - Attended and presented a talk at a workshop on
*Infinite dimensional Lie theory: Algebra, Geometry and Combinatorics*, Centre de recherches mathématiques (CRM), Montreal, Quebec, Canada, August 21-24, 2012, organized by Joel Kamnitzer (University of Toronto) and Michael Lau (Université Laval). - Attended and presented a talk at ``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.
- Invited visitor at the Institut des Hautes Etudes Scientifique (IHES), Bures-sur-Yvette, France, May 20-31, 2013.
- Invited visitor at the Albert Einstein Institute, Max-Planck Institute for Gravitational Physics, Potsdam, Germany, June 1-19, 2013.
- Invited to attend and present a talk at the special session on vertex operator algebras, Kac-Moody Lie algebras and related topics, at the conference ``Representation Theory XIII", Inter-University Centre, Dubrovnik, Croatia, June 20-27, 2013.
- Invited visitor at the Institut des Hautes Etudes Scientifique (IHES), Bures-sur-Yvette, France, May 18-31, 2014.
- Invited visitor at the Albert Einstein Institute, Max-Planck Institute for Gravitational Physics, Potsdam, Germany, June 1-22, 2014.
- Invited speaker at the 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.
- Invited visitor at the Institut des Hautes Etudes Scientifique (IHES), Bures-sur-Yvette, France, May 18-31, 2015.
- Invited visitor at the Albert Einstein Institute, Max-Planck Institute for Gravitational Physics, Potsdam, Germany, June 1-17, 2014.
- Invited speaker at the conference on ``Generalizations of Symmetric Spaces", Nahsholim Sea Resort, Israel, June 17-24, 2015.
- Invited visitor at the Universitaet zu Koeln, Mathematisches Institut, Koeln, Germany, June 25-29, 2015.
- Invited to present a talk in the Special Session on “Representation Theory, Vertex Operator Algebras, and Related Topics” at the AMS meeting at Rutgers University, New Brunswick, November 14-15, 2015.
- Invited speaker at the conference on ``Lie and Jordan algebras, Their Representations and Applications-VI", Bento Goncalves,Brazil, December 13-19, 2015. Unable to attend.
- Presented an invited talk in the Special Session on
``Algebraic structures in mathematical physics: Lie
algebras, vertex algebras, quantum algebra'' at the
1117
^{th}AMS meeting at the University of Georgia, Athens, GA, March 4-6, 2016.

Modular Forms, Siegel Modular Forms, Conformal Field Theory, Fusion Algebras.

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 E

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 E

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 A

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 D

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.

- 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.
- 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.
- 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.
- 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.
- The 1991 American Mathematical Society Summer Research Institute, on Algebraic Groups and Their Generalizations, Pennsylvania State University, University Park, PA, July 8-26, 1991.
- 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.
- The Structure and Representation Theory of Lie Algebras conference in honor of George Seligman, April 10-12, 1992, Yale University, New Haven, CT.
- The 884th Meeting of the AMS, Special Session on Lie Theoretical Methods in Mathematical Physics, Sept. 18-19, 1993, Syracuse University, Syracuse, NY.
- 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. - The 906th Meeting of the AMS, Special Session on Quantum Kac-Moody Lie Algebras and Related Topics, Nov. 17-18, 1995, Greensboro, North Carolina.
- The 922nd Meeting of the AMS, Special Session on VOA's, Monstrous Moonshine and Related Topics, May 2-4, 1997, Detroit, Michigan.
- 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.
- 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.
- 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.
- 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.
- The 1024
^{th}Meeting of the AMS, Special Session on Geometric and Combinatorial Methods in Representation Theory, March 3-4, 2007, Davidson College, Davidson, NC. - 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.
- The 1048
^{th}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. - The 1062
^{nd}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). - The 1065
^{th}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 G_{2}^{(1)}and D_{4}^{(3)}(joint with Quincy Loney). - 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 D
_{4}^{(1)}with respect to its subalgebra G_{2}^{(1)}in the spinor construction (joint with Quincy Loney). - A workshop on
*Infinite dimensional Lie theory: Algebra, Geometry and Combinatorics*, Centre de recherches mathématiques (CRM), Montreal, Quebec, Canada, August 21-24, 2012, organized by Joel Kamnitzer (University of Toronto) and Michael Lau (Université Laval). - ``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.
- ``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.
- 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.
- Conference on ``Generalizations of Symmetric Spaces", Nahsholim Sea Resort, Israel, June 17-24, 2015.
- Special Session on ``Representation Theory, Vertex Operator Algebras, and Related Topics”, AMS meeting at Rutgers University, New Brunswick, November 14-15, 2015.
- Conference on "Lie and Jordan algebras, Their Representations and Applications-VI", Bento Goncalves, Brazil, December 13-19, 2015. Unable to attend.
- Special Session on ``Algebraic structures in mathematical
physics: Lie algebras, vertex algebras, quantum algebra'' at the
1117
^{th}AMS meeting at the University of Georgia, Athens, GA, March 4-6, 2016.

- ``Zones of uniform decomposition in tensor products'', Proceedings of the American Mathematical Society, Vol. 70, No. 2, July 1978, 109-113.
- ``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.
- ``The Weyl-Kac character formula and power series identities'', Advances in Mathematics 29, No. 3, September 1978, 271-309 (with J. Lepowsky).
- ``A hyperbolic GCM Lie algebra and the Fibonacci numbers'', Proceedings of the American Mathematical Society, Vol. 80, No. 3, November 1980, 379-385.
- ``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.
- ``A hyperbolic Kac-Moody algebra and the theory of Siegel modular forms of genus 2'', Mathematische Annalen 263, 1983, 87-144 (with I. Frenkel).
- ``Classical affine algebras'', Advances in Mathematics, Vol. 56, No. 2, May 1985, 117-172 (with I. Frenkel).
- ``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.
- ``The exceptional affine algebra E
_{8}^{(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). - ``Spinor Construction of Vertex Operator Algebras, Triality
and E
_{8}^{(1)}'', Contemporary Mathematics, Vol. 121, American Mathematical Society, Providence, RI, 1991, 146 pp. monograph (with Igor Frenkel and John F. X. Ries). - ``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).
- ``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.
- ``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. - ``Minimal model fusion rules from 2-groups'', Letters in Mathematical Physics, Vol. 40, No. 2 (1997), 159-169, (with Fusun Akman and Michael Weiner).
- ``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.
- ``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.
- ``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).
- ``A New Perspective on the Frenkel-Zhu Fusion Rule Theorem'', Journal of Algebra 320 (2008), 2079--2100, (with Stefan Fredenhagen).
- ``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).
- ``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.
- ``Matrix realizations of hyperbolic triangle groups'', completed July 2010, joint with Elizabeth Dwornik, unpublished.
- ``The 3-State Potts model and Rogers-Ramanujan series'', Central European Journal of Mathematics, 11(1), 2013, pp. 1--16 (with Antun Milas).
- ``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
- ``A Lightcone embedding of the twin building of a hyperbolic
Kac-Moody group'', (with Lisa Carbone and Walter Freyn), 2017,
to be submitted.

- ``Structure of Cartan subalgebras in hyperbolic Kac-Moody algebras'', (with Walter Freyn).
- ``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).

- ``On principal subspaces of certain admissible C
_{n}^{(1)}-modules'', (with C. Calinescu and Antun Milas).

- ``Further research on hyperbolic Kac-Moody algebras'', (with Hermann Nicolai and Axel Kleinschmidt).
- ``Vertex Operator Algebras, Triality and E
_{8}^{(1)}'', book, (with the late John F. X. Ries).

- 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.)
- 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.)

- Quincy Loney, Ph.D. July 2012, Thesis: Decomposition of
level-1 representations of D
_{4}^{(1)}with respect to its subalgebra G_{2}^{(1)}in the spinor construction. - Christopher Mauriello, Ph.D. May 2013, Thesis: Branching rule
decomposition of irreducible level-1 E
_{6}^{(1)}-modules with respect to the affine subalgebra F_{4}^{(1)}.

- 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.