Alladi Ramakrishnan Hall

On symmetric regular subalgebras of Kac-Moody algebras

R Venkatesh

IISc

A symmetrizible Kac-Moody algebra g is generated (as a Lie algebra)
by its root spaces corresponding to real roots. In this paper, we address
the following natural question: given any subset of real root vectors, is the
Lie subalgebra of g generated by these again a Kac-Moody algebra? We
call such Lie subalgebras root generated and show that there is a one-to-one
correspondence between them, real closed subroot systems and π-systems
contained in the positive system of g. Finally, we apply these identifica-
tions to classify symmetric regular subalgebras (introduced first by Dynkin
in finite types) in the untwised affine case. We show that any root gener-
ated subalgebra associated to a maximal real closed subroot system can be
embedded into a unique maximal symmetric regular subalgebra.