Infinite dimensional lie algebras and their coordinate algebras

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Finite dimensional simple Lie algebras, introduced by Sophus Lie over a hundred years ago, are of fundamental importance in geometry and physics as a measure of symmetry.  About 30 years ago, Victor Kac and Robert Moody independently introduced some infinite dimensional analogs, now called affine Kac-Moody algebras, of finite dimensional simple Lie algebras.  A rich mathematical theory of affine Kac-Moody Lie algebras has been developed since that time.  Furthermore, these algebras have been used in physics to study the symmetries that occur in field theories that model interactions of elementary particles. In 1990 a new class of Lie algebras, called extended affine Lie algebras, was introduced by two mathematical physicists Høegh-Krohn and Torresani.  This new class includes all finite dimensional simple Lie algebras and all affine Kac-Moody Lie algebras.  Remarkable progress has been made in the last decade in the study of the structure of these algebras, but many interesting questions remain to be investigated.  In the proposed research, I will continue my study of the structure of extended affine Lie algebras along with some related classes of algebras. One of my approaches will be to investigate realizations of these Lie algebras. Realizations allow one to make explicit calculations involving the algebras. One type of realization is as matrices with coordinates from an algebra called a coordinate algebra.  Part of my investigation will be devoted to the study of these coordinate algebras.