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Course Descriptions

16:640:555 - Selected Topics in Algebra

Yi-Zhi Huang


Modular tensor categories for affine Lie algebras


No textbook. The lectures will be based on research papers.


First year graduate algebra and analysis courses. Basic knowledge in Lie algebras will be very helpful but is not required.


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.

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Department of Mathematics

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Rutgers University
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