Spring, 2007
- Speaker Minxian Zhu, Yale University
- Title Vertex operator algebras associated to modified
regular representations of affine Lie algebras
- Time/place Tuesday, 1/23/2007 3:30 pm in Hill 425
- Note Joint Quantum Mathematics/Topology-Geometry Seminar
- Abstract Vertex operator algebras can be regarded as
generalizations of associative algebras,
but have much richer structures. We study a
family of vertex operator algebras which admit two copies of the
affine Lie algebra actions with dual central charges, and whose top
levels are identified with regular functions on the Lie groups. We
discuss two constructions: one is based on the properties of
intertwining operators and Knizhnik-Zamolodchikov equations; the other
is to use the enveloping algebra of the vertex algebroid associated to
the Lie group and a fixed level. We show that the two constructions
yield the same vertex operator algebra. The case of integral central
charges will also be discussed.
- Speaker Bill Cook, Rutgers University
- Title Vertex operator algebras and recurrence relations
- Time/place Friday, 3/30/2007 1:00 pm in Hill 705
- Abstract There are many important classes of examples
of vertex operator
algebras including Heisenberg VOAs, Virasoro VOAs, lattice VOAs, and
the VOAs associated with affine Lie algebras.
We will begin with an introduction to the class of VOAs (along with
their modules) associated with affine Lie algebras. Then in the
latter part of the talk we will discuss an interesting theorem of
Haisheng Li. Applying this theorem to our class of examples, we
will obtain recurrence relations among the characters of these
Vertex Operator Algebras (and VOA modules).
- Speaker Antun Milas, University at Albany, SUNY
- Title On a certain family of W-algebras
- Time/place Friday, 4/6/2007 1:00 pm in Hill 705
- Abstract Rational conformal field theories can be
characterized by
the property that there are, up to equivalence, finitely many
irreducible representations of the vertex operator algebra, and that
every representation is completely reducible.
It is tempting to relax the semisimplicity condition and study more
general classes of conformal field theories. One such class are
rational logarithmic conformal field theories, where not all modules
are completely reducible and not even L(0) diagonalizable (here L(0)
is the degree zero Virasoro generator). The only known examples of
rational LCFT come from certain W-algebras.
In this talk, I will first define several families of W vertex
algebras. We will then focus on their representations, graded
dimensions and various modularity issues.
Some parts of the talk are based on a joint work with D. Adamovic.
- Speaker Vincent Graziano, SUNY at Stony Brook
- Title G-equvariant modular categories and Verlinde formula
- Time/place Friday, 4/13/2007 1:00 pm in Hill 705
- Abstract Many features of a conformal field theory can
be captured in the language
of categories. Modular tensor categories provide the appropriate
framework
and we will start by discussing the properites of such a category. We
will
then introduce the Verlinde algebra associated to such a category, the
action of the S-matrix, and the Verlinde formula.
Our goal will be to generalize this setup to the case of theories with
additional symmetries, such as a vertex operator algebra with a finite
group of symmetries. We discuss the extended Verlinde algebra, the
S-matrix, and the 'extended' Verlinde formulas.
- Speaker Corina Calinescu, Ohio State University
- Title Vertex-algebraic structure of certain modules for
affine Lie algebras underlying recursions
- Time/place Friday, 4/20/2007 1:00 pm in Hill 705
- Abstract Many combinatorial identities and recursions
have been proved or conjectured via vertex operator constructions
of representations of
affine Lie algebras.
In this talk we discuss vertex-algebraic structure of the principal
subspaces of all the standard A_1^(1)-modules and we prove suitable
presentations for these subspaces. These presentations were used by
Capparelli, Lepowsky and Milas for the purpose of obtaining the
classical Rogers-Ramanujan and Rogers-Selberg recursions. This is
joint work with Jim Lepowsky and Antun Milas.
- Speaker Tom Robinson, Rutgers University
- Title A Formal Variable Approach to Special
Hyperbinomial Sequences
- Time/place Friday, 4/27/2007 1:00 pm in Hill 705
- AbstractIn a nearly self-contained and elementary
treatment, we develop the formal calculus used in the theory of
vertex algebras to describe certain formal changes of variable.
In particular, we extend the logarithmic formal Taylor theorem as
found in the work of Y.Z. Huang, J. Lepowsky, and L. Zhang. We
apply our results to obtain combinatorial identities concerning
generalizations of the Stirling numbers and find that our
development leads naturally to a combinatorial definition of the
exponential Riordan group which was studied by L.W. Shapiro,
S. Getu, W.J. Woan, and L.C. Woodson.
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