Graduate Student Seminar, 4:00pm January 20, 2003, Hayes-Healy 231



Operads: The Geometric Road to Vertex Operator Algebras


Operads, or "systems with compositions", were first formally defined in the 1970's by topologists to unify such problems as looking at iterated loop spaces, compositions of maps and commutativity in simplicial chain complexes [1]. By taking the geometric structure of a particular operad, one can induce classical algebraic structures such as monoids, (associative) algebras, coalgebras and Lie algebras. One can also induce the structure of a vertex operator algebra (VOA)[2 & 3], which is a non-associative algebra first used to describe the largest order sporadic finite simple group (the Monster) and also found to have applications in Conformal Field Theory. We will first discuss what these operads are as well as examples of them and how they can be used to induce classical algebraic structures, then look at the operad that induces the VOA structures along with the basic parts of that structure.

[1] J. Peter May's ``The geometry of iterated loop spaces", Lecture Notes in Math., No. 271, Springer-Verlag, Berlin, 1972.

[2] Y.-Z. Huang and J. Lepowsky, ``Vertex operator algebras and operads", The Gelfand Mathematical Seminar, 1990--1992, ed. L. Corwin, I. Gelfand and J. Lepowsky, Birkhauser, Boston, 1993, 145--161.

[3] Y.-Z. Huang and J. Lepowsky, ``Operadic formulation of the notion of vertex operator algebra", Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, ed. P. Sally, M. Flato, J. Lepowsky, N. Reshetikhin and G. Zuckerman, Contemp. Math. 175, Amer. Math. Soc., Providence, 1994, 131--148.

To volunteer to give a talk, or for any other questions regarding this schedule, contact Wesley Calvert