A Kac-Moody group is a group attached to a Kac-Moody algebra, acting on it adjointly with small central kernel. The minimal such group over a ring is built by the constructive Tits functor: root groups indexed by the real roots, together with a torus, presented by commutator relations whose structure constants are read off exponentials in a completed integral form of the universal enveloping algebra; only real root spaces exponentiate, because the adjoint operator is locally nilpotent exactly there. Students learn how those root groups form an RGD system giving twinned BN-pairs and an action on a twin building, how the axioms of a Tits functor pin the group down up to a central kernel, and how completing in the positive direction yields the algebraic, representation-theoretic, geometric and scheme-theoretic maximal Kac-Moody groups.
Lecture notes from a three-hour minicourse condensing the textbook 'An Introduction to Kac-Moody Groups over Fields'. A Kac-Moody group is a group attached to a Kac-Moody algebra g(A) of generalised …