Pi-systems of symmetrizable Kac-Moody Algebras[HBNI Th173]

Abstract

In this thesis, we undertake a systematic study of π-systems of symmetrizable Kac-Moody algebras and regular subalgebras of affine Kac-Moody algebras.

A π-system Σ is a finite subset of the real roots of a Kac-Moody algebra g satisfying the property that pairwise differences of elements of Σ are not roots of g. As part of his classification of regular semisimple subalgebras of semisimple Lie algebras, Dynkin introduced and studied the notion of π-systems. These precisely form the simple systems of such subalalgebras. We generalize the definition of π-systems and regular subalgebras and establish their fundamental properties. We show that π-systems, regular subalgebras and closed subroot systems of affine Kac-Moody algebras are in one-to-one correspondence. We completely classify and give explicit descriptions of the maximal closed subroot systems (or maximal π-systems in other words) of affine Kac-Moody algebras. As an application we describe a procedure to get the classification of all regular subalgebras of affine Kac Moody algebras in terms of their root systems. We also study the orbits of the Weyl group action on π-systems of symmetrizable Kac-Moody algebras, showing that for many π-systems of interest in physics, the action is transitive (up to negation).

Finally, we formulate general principles for constructing π-systems and criteria for the non-existence of π-systems of certain types and use these to determine the set of maximal hyperbolic diagrams in ranks 3-10 relative to the partial order of admitting a π-system.