The representation theory of Lie algebras is equivalent to the representation theory of universal enveloping algebras of Lie algebras. In particular, the universal enveloping algebras of Lie algebras and their quotients play an important role in the representation theory of Lie algebras.
For vertex operator algebras, though we also have universal enveloping algebras constructed by Frenkel and Zhu, they are not very useful since we consider only lower-bounded generalized modules. It is therefore important to find an associative algebra associated to a vertex operator algebra such that the category of modules for the associative algebra is equivalent to the category of lower-bounded generalized modules for the vertex operator algebra. In the case all lower-bounded generalized modules are completely reducible, such an algebra was found by Zhu. In the general case, such an algebra was found by Huang. In this course, we will study these algebras and their applications in the representation theory of vertex operator algebras. We will also study other related associative algebras.
Here is a list of topics to be covered in the course:
- Vertex operator algebras and modules.
- Universal enveloping algebras of vertex operator algebras.
- Zhu algebra and zero-mode algebra.
- The higher level generalizations of Zhu algebra by Dong-Li-Mason.
- Huang's associative algebras and lower-bounded generalized modules.
- Twisted generalizations of the associative algebras above.
- Applications of the associative algebras above.
Prerequisites:
Some knowledge of first year algebra and complex analysis for graduate students
Text:
No textbook. Papers will be discussed in the classes.
Papers and lecture notes to be used in the lectures:
- Y.-Z. Huang, Lecture notes on vertex algebras and quantum vertex algebras, 96 pages on April 26, 2020, for the graduate course "Math 555: Selected Topics in Algebra: Vertex algebras and quantum vertex algebras," Spring, 2020.
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- I. B. Frenkel and Y. Zhu, Vertex operator
algebras associated to representations of affine and Virasoro
algebras, Duke Math. J. 66 (1992), 123--168.
Online version at Project Euclid
- Y. Zhu, Modular invariance of characters of vertex operator algebras, J.
Amer. Math. Soc. 9 (1996), 237--307.
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- D. Brungs and W. Nahm, The associative algebras of conformal field theory,
Lett. Math. Phys. 47 (1999), 379--383.
Online version at Srpinger Nature Link
- Y.-Z. Huang, Differential equations, duality and modular invariance, Comm. Contemp. Math. 7 (2005), 649--706.
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- C. Dong, H. Li and G. Mason, Vertex operator algebras and
associative algebras, J. Algebra 206 (1998), 67-96..
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- Y.-Z. Huang, Associative algebras and the representation theory of
grading-restricted vertex algebras, Comm. Comtemp. Math. 26 (2024), paper no. 2350036 .
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- Y.-Z. Huang, Associative algebras and intertwining operators, Comm. Math. Phys. 396 (2022), 1--44.
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- Y.-Z. Huang, Modular invariance of (logarithmic) intertwining operators, Comm. Math. Phys. 405 (2024), article number 131,1-- 82.
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- C. Dong, H. Li and G. Mason,
Twisted representations of vertex operator algebras,
Math. Ann. 310 (1998), 571-600.
Online version at Springer Nature Link
- Y.-Z. Huang and J. Yang, Associative algebras for (logarithmic) twisted modules for a vertex operator algebra,
Trans. Amer. Math. Soc. 371 (2019), 3747--3786.
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