Lie and Jordan algebras, and related groups

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Symmetry is omnipresent in nature and the source of much enjoyment for all of us. But it is not only in nature that we find symmetry. It is everywhere in our lives. One just has to think about music (Bach) or paintings (Escher). Even more, symmetry is basic for our understanding of the physical world. 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. 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 19th-century Norwegian mathematician Sophus Lie. 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. In essence, Lie algebras are first order approximations of the corresponding groups. My research is concerned with Lie algebras, algebraic groups and the interplay between them. The main goals in any theory of Lie algebras are to understand their internal structure and their representations. These are also my long-term goals for the Lie algebras I plan to study: extended affine Lie algebras, root-graded Lie algebras and equivariant map Lie algebras. These Lie algebras are important for present-day Lie algebra theory. For example, extended affine Lie algebras generalize finite-dimensional semisimple and affine Kac-Moody Lie algebras. A fundamental result for these two examples of extended affine Lie algebras is conjugacy of their Cartan subalgebras (celebrated results of Chevalley and Peterson-Kac). My research aims to extend conjugacy to all extended affine Lie algebras. This will have important consequences for their structure theory. In the area of groups, I will be working on two types of groups, Steinberg groups and exceptional groups. The former have been studied in many different types. The theory of Steinberg groups that I have developed jointly with Ottmar Loos will unify all of them, thus allowing substantial simplifications of the theory. It introduces new methods by systematically employing root systems and Jordan pairs. Exceptional groups are the most fascinating, but also most complicated of all algebraic groups. They have found applications in many areas of mathematics and in physics. So far, they have mostly been studied over base fields. My goal is to describe exceptional groups over rings, using a combination of advanced methods from nonassociative algebras, group schemes and descent theory. The extension from base fields to base rings will shed new light on the construction of these groups.