Extended affine Lie algebras, groups and representation theory

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Since the 19th century the mathematical concept of a group has been one of the most basic abstract structures used by mathematicians to describe symmetry arising in many different incarnations, e.g. in science and engineering. Mathematicians have developed a sophisticated theory for groups with more and more profound applications. The applications of groups within and outside of mathematics have been a remarkable success story. The most important type of groups originates from the fundamental work of the Norwegian mathematician Sophus Lie (1842--1899). These so-called Lie groups and their algebraic analogues, the algebraic groups, are distinguished by the fact that one can associate another mathematical structure, a Lie algebra, to them. Roughly speaking, Lie algebras are first order approximations of the corresponding groups. My research is concerned with certain types of Lie algebras and some related structures. In the first part of the 20th century, Cartan, Weyl, Jacobson and Chevalley have created a magnificent theory of finite-dimensional semisimple Lie algebras. Their theory was later extended to certain infinite-dimensional Lie algebras by Kac and Moody. In particular, the so-called affine Kac-Moody algebras have been a central topic of research in Lie algebras with many profound applications. Motivated by applications in quantum gauge theory and singularity theory, affine Kac-Moody algebras have been generalized to so-called extended affine Lie algebras. This new class of Lie algebras offers exciting new possibilities. It provides us with many more structures and hence possible applications than what was previously known. The goal of my research is to develop the structure and representation theory of extended affine Lie algebras and some other related Lie algebras. I will also study the groups associated to these Lie algebras and another algebraic structure, so-called Jordan algebras.