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Affine Lie algebras are one of the most important classes of infinite-dimensional Lie algebras. Suitable module categories for affine Lie algebras have structures of modular tensor categories. Moreover, these modular tensor categories are equivalent to the modular tensor categories constructed from suitable module categories for the corresponding quantum groups at the corresponding roots of unity. These modular tensor categories give the quantum invariants of knots and three-manifolds and play an important role in topological quantum computing.
In this course, I will discuss the construction of these modular tensor categories. Below are the detailed topics to be covered in this course:
1. Finite-dimensional Lie algebras and modules.
2. Affine Lie algebras and modules.
3. The categories generated by integrable highest weight modules for affine Lie algebras.
4. Vertex operator algebras and modules associated to affine Lie algebras.
5. Intertwining operators and operator product expansion.
6. Ribbon braided tensor category structures.
7. Modular invariance of intertwining operators.
8. Moore-Seiberg equations and the Verlinde formula
9. Rigidity and nondegeneracy properties.
Prerequisites: First year graduate algebra and analysis courses. Basic knowledge in Lie algebras will be very helpful but is not required.
Text: No text book. Lectures are based on papers and books that are available online. I will give links below to the papers relevant to the lectures.
Papers relevant to the lectures:
- Y.-Z. Huang, Section 3, Lecture notes for my course ``Introduction to representation
theory and tensor categories,'' March to June, 2011, Beijing
International Center for Mathematical Research, Peking University,
Beijing, China.
pdf
file
- I. B. Frenkel and Y. Zhu, Vertex operator algebras associated to representations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.
https://projecteuclid.org/euclid.dmj/1077294666
- Two constructions of grading-restricted vertex (super)algebras, J. Pure
Appl. Alg. 220 (2016), 3628-3649.
arXiv:1507.06098
- Robert McRae, Non-negative integral level affine Lie algebra tensor categories and their associativity isomorphisms, Comm. Math. Phys 346 (2016), 349-395.
arXiv:1506.00113
- Y.-Z. Huang and J. Lepowsky, Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke Math. J. 99 (1999), 113-134.
arXiv:q-alg/9706028
- G. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Comm. Math. Phys. 123 (1989),
177-254.
Scanned pdf file (KEK scanned document)
- Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, I, Selecta Mathematica 1 (1995), 699-756.
arXiv:hep-th/9309076
- Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, III,
J. Pure Appl. Alg. 100 (1995), 141--171.
arXiv:q-alg/9505018
- Y.-Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Alg. 100 (1995), 173--216.
arXiv:q-alg/9505019
- Y.-Z. Huang, Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories, in: Operads: Proceedings of Renaissance Conferences, ed. J.-L. Loday, J. Stasheff, and A. A. Voronov, Contemporary Math., Vol. 202, Amer. Math. Soc., Providence, 1997, 335--355.
arXiv:q-alg/9512024
- Y.-Z. Huang, Differential equations, duality and modular invariance, Comm. Contemp. Math. 7 (2005), 649--706.
arXiv:math/0303049
pdf file
- Y.-Z. Huang, Vertex operator algebras and the Verlinde conjecture, Comm. Contemp. Math. 10 (2008), 103--154.
arXiv:math/0406291
pdf file
- Y.-Z. Huang, Rigidity and modularity of vertex tensor categories, Comm. Contemp. Math. 10 (2008), 871--911.
arXiv:math/0502533
pdf file
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