Central extensions of graded Lie algebras

Abstract

In this thesis we describe the universal central extension of two important classes of so-called root-graded Lie algebras defined over a commutative associative unital ring k. Root-graded Lie algebras are Lie algebras which are graded by the root lattice of a locally finite root system and contain enough s^-triples to separate the homogeneous spaces of the grading. Examples include the infinite rank analogs of the simple finite-dimensional complex Lie algebras. In the thesis we show that in general the universal central extension of a root-graded Lie algebra L is not root-graded anymore, but that we can measure quite easily how far it is away from being so, using the notion of degenerate sums, introduced by van der Kallen. We then concentrate on root-graded Lie algebras which are graded by the root systems of type A with rank at least 2 and of type C. For them one can use the theory of Jordan algebras. Given a Jordan algebra J, we establish a functorial construction which produces a Lie algebra from J, called the universal Tits-Kantor-Koecher algebra of J. We are led to study the derivation algebras of Jordan algebras and alternative algebras. Under mild assumptions on the base ring k, it is proven that the Albert algebra (a Jordan algebra) and the octonion algebra (an alternative algebra) have derivation algebras which are isomorphic to exceptional Lie algebras of type F4 and G^ respectively. We also show that certain root-graded Lie algebras which are defined by the Albert algebra resp. the octonion algebra are simply connected, i.e., coincide with their central extensions.