On Representations of Toroidal Lie Algebras

I gave two talks on some of my results on toroidal Lie algebras in the conference Functional Analysis VIII held in Dubrovnik, Croatia in June 2003. But these results have been submitted to research journals and will appear soon. So I decided to write an expository article on Toroidal Lie Algebras covering my results for the proceedings of Functional Analysis VIII. I will introduce the definitions of toroidal Lie algebras and also the generalized Virasoro algebra in the introduction below. In the main body of the article I will state most of the recent results on representations of toroidal Lie algebras with finite dimensional weight spaces. Toroidal Lie algebras are n-variable generalizations of the well known affine Kac-Moody Lie algebras. The affine Kac-Moody Lie algebra, which is the universal central extension of Loop algebra, has a very rich theory of highest weight modules and some of their characters admit modular properties. The level one highest weight integrable modules has been constructed explicitly on the Fock space through the use of vertex operators (see [FK];