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Tensor categories are generalizations of monoids and groups.
They appear naturally in representation theory and in quantum
physics. Many mathematical structures and physical phenomena
can be studied using the theory of tensor categories. One
particularly interesting class of tensor categories is the
class from conformal field theories. These tensor categories
have applications in algebra, topology, geometry, string
theory, condensed matter physics and quantum computation.
The first part of the course will be an introduction to tensor
categories using representations of groups, associative
algebras and Lie algebras as motivating examples. The second
part of the course discusses tensor category structures on
suitable module categories for suitable vertex operator
algebras, assuming that the students are familiar with the
material presented in Lepowsky's course in Fall, 2011.
Prerequisites: Algebra and complex analysis at the level of first year graduate course. The second part of the course also requires the students to be familiar with the material presented in Lepowsky's course in Fall, 2011. Text: Y.-Z. Huang, Introduction to representation theory and tensor categories, Lecture notes, 2011. Here is the pdf file: Lecture Notes, 2011. and Y.-Z. Huang, J. Lepowsky and Lin Zhang, Logarithmic tensor category theory, I-VIII, to appear; These are already in the archive. The numbers are: arXiv:1012.4193, arXiv:1012.4196, arXiv:1012.4197, arXiv:1012.4198, arXiv:1012.4199, arXiv:1012.4202, arXiv:1110.1929, arXiv:1110.1931. |