Infinite-dimensional Lie algebras and their associated groups

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The principal goal of research in pure mathematics is to study structures. Because of the abstract approach the mathematical results can then be aplied in different areas, like finance or physics or the human genome. The structure I am interested in is part of algebra. It arises from the study of solutions of equations. These solutions often have some symmetry, leading to a structure that mathematicians call a group. It satisfies some nice internal laws, like a group of friends in ordinary life. The groups I am studying not only have infinitely members, but also members which are close-by look more or less the same. In the late 19th century the Norwegian mathematician Sophus Lie (pronouced Lee) discovered that one can study lots of properties of these groups with the means of an even nicer structure, a sort of approximationby approximation, which nowadays is called a Lie algebra. The study of Lie algebras has been at the centre of 20th century mathematics. Lie algebras have found many applications within and outside of mathematics, for example in particle physics. While most of the studies in the first part of the last century were concerned with finite-dimensional Lie algebras, i.e., those allowing only a finite number of freedoms, in recent decades the interest has shifted to infinite-dimensional Lie algebras. The proposed research is devoted to a special class of infinite-dimensional Lie algebras, the so-called extended affine Lie algebras which have been invented by physicists. I will determine the precise structure of these Lie algebras. The second part of the proposal deals with groups one can associate to these Lie algebras. The main point will be in constructing the groups, using the knowledge obtained in the first part.