Yongchang Zhu

Vertex operator algebras associated to representations of affine and Virasoro algebras

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].

On the set-theoretical Yang-Baxter equation

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.

Global vertex operators on Riemann surfaces

We develop an approach towards construction of conformal field theory starting from the basic axioms of vertex operator algebras.

Self-dual modules of semisimple Hopf algebras

Abstract We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.

On Hopf algebras with positive bases

We show that if a finite dimensional Hopf algebra H over C has a basis with respect to which all the structure constants are nonnegative, then H is isomorphic to the bi-cross-product Hopf algebra constructed by Takeuchi and Majid from a finite group G and a unique factorization G = G+ G− of G into two subgroups. We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in a similar way. Our results can be used to explain some results on Hopf algebras from the set-theoretical point of view.

Theta functions and Weil representations of loop symplectic groups

We introduce Weil representations for loop symplectic groups and prove the convergence and modularity of the related theta functions.

THE DIMENSION OF IRREDUCIBLE MODULES FOR TRANSITIVE MODULE ALGEBRAS

We prove that for a semisimple Hopf algebra H, if A is a transitive H-module algebra and M is an irreducible A-module, then dim(A) divides dim(M)2dim(H). *Supported by RGC Earmark Grant HKUST629/95P

Stabilizer for hopf algebra actions

The definition of stabilizer and orbit for Hopf algebra action is given, and a duality theorem on stabilizer is proved.