In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z) is manifest. In the study of conformal field theory, physicists arrived at the notion of chiral algebras (see e.g. [MS]). Independently, in the attempt to realize the Monster sporadic group as a symmetry group of certain algebraic structure, an infinite dimensional graded representation of the Monster sporadic group, the so called Moonshine module, was constructed in [FLM1]. This algebraic structure was later found in [Bo] and called the vertex algebra; the first axioms of vertex operator algebras were formulated in that paper. The proof that the Moonshine module is a vertex operator algebra and the Monster group acts as its automorphism group was given in [FLM2]. Notably the character of the Moonshine module is also a modular function, namely j(τ) − 744. It turns out that the vertex operator algebra can be regarded as a rigorous mathematical definition of the chiral algebras in the physical literature. And it is expected that a pair of isomorphic vertex operator algebras and their representations (corresponding to the holomorphic and antiholomorphic sectors) are the basic objects needed to build a conformal field theory of a certain type.

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Modular invariance of characters of vertex operator algebras

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the number of such categories (functors) realizing a given fusion datum is finite. Finally, we develop the theory of Frobenius-Perron dimensions in an arbitrary fusion category. At the end of the paper we generalize some of these results to positive characteristic.

http://annals.math.princeton.edu/wp-content/uploads/annals-v162-n2-p01.pdf

On fusion categories

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

Introduction Definition of vertex algebras Vertex algebras associated to Lie algebras Associativity and operator product expansion Applications of the operator product expansion Modules over vertex algebras and more examples Vertex algebra bundles Action of internal symmetries Vertex algebra bundles: Examples Conformal blocks I Conformal blocks II Free field realization I Free field realization II The Knizhnik-Zamolodchikov equations Solving the KZ equations Quantum Drinfeld-Sokolov reduction and $\mathcal{W}$-algebras Vertex Lie algebras and classical limits Vertex algebras and moduli spaces I Vertex algebras and moduli spaces II Chiral algebras Factorization Appendix Bibliography Index List of frequently used notation.

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Vertex Algebras and Algebraic Curves

We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra
 representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the “face” formulation for any type of Lie algebra
 
$$\mathfrak{g}$$

 and arbitrary finite-dimensional representations of
 . We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq→1 these solutions degenerate again into
 solutions with
 
$$q' = \exp \left( {\frac{{2\pi i}}{{k + g}}} \right)$$

 . We also study the simples examples of solutions of our holonomic difference equations associated to
 
$$U_q (\widehat{\mathfrak{s}\mathfrak{l}(2)})$$

 and find their expressions in terms of basic (orq−)-hypergeometric series. In the special case of spin −1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.

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Quantum affine algebras and holonomic difference equations

The first construction of the integrable highest-weight representations of affine Lie algebras or loop algebras by Kac i-K] was greatly inspired by the generalization of the Weyl denominator formula for affine roots systems discovered earlier by Macdonald [M]. Though the Macdonald identity found its natural context in representation theory, its mysterious modular invariance was not understood until the work of Witten [W-I on the geometric realization of representations of the loop groups corresponding to loop algebras. The work of Witten clearly indicated that the representations of loop groups possess a very rich structure of conformal field theory which appeared in physics literature in the work of Belavin, Polyakov, and Zamolodchikov [BPZ-I. Independently (though two years later), Borcherds, in an attempt to find a conceptual understanding of a certain algebra of vertex operators invariant under the Monster [FLM1], introduced in [B-I a new algebraic structure. We call vertex operator algebras a slightly modified version of Borcherd’s new algebras [FLM2].

Vertex operator algebras associated to representations of affine and Virasoro Algebras

We propose a general way of constructing set-theoretical solutions of the YangBaxter equation. We study the properties of the construction. We also show that our construction includes the earlier ones given by Weinstein-Xu and Etingof-SchedlerSoloviev.

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On the set-theoretical Yang-Baxter equation

Hopf algebras of prime dimension

Discrete series of the Virasoro algebra and the moonshine module, In: Algebraic groups and their generalizations: quantum and infinite-dimensional methods

Yongchang Zhu

Papers

Modular invariance of characters of vertex operator algebras

Vertex operator algebras associated to representations of affine and Virasoro Algebras

On the set-theoretical Yang-Baxter equation

Hopf algebras of prime dimension

Discrete series of the Virasoro algebra and the moonshine module, In: Algebraic groups and their generalizations: quantum and infinite-dimensional methods