Classification Of Finite Irreducible Modules Of Simple Lie Pseudoalgebras Of Type W, S, And K

In order to understand the algebraic structures underlying the notion of op-erator product expansion (OPE) of chiral fields in conformal field theory, there has been a great deal of work as well as many significant achievements since the papers of Belavin, Borcherds.etc. For instance, conformal algebra theory has provided us with a very powerful tool for studying the singular parts of OPE. In the meantime, the theory of conformal algebra has played a very important role in the area of vertex algebras as well. Consequently, more and more mathe-maticians have become interested in conformal algebra theory. On the one hand, there have been substantial results in the research of the structure, representa-tion theory and cohomology theory of Lie conformal algebras. On the other hand, people begin to pay attentions to the generalization of conformal algebras. Pseu-doalgebras, which we will introduce in this paper, are exactly multi-dimensional analogues of conformal algebras. The main body of the paper is to summarize the results in the field of pseudoalgebras which were made in recent years.The primary contents include properties of pseudo-tensor categories and pseudoalge-bras, classification of finite simple Lie pseudo algebras, and classification of finite irreducible representations of finite simple Lie pseudoalgebras.