Cartan-preserving automorphisms of untwisted and twisted Kac–Moody algebras

Abstract

The group Ñ defined as the normalizer of the Cartan subalgebra in the group of all (inner and outer) automorphisms of affine Kac–Moody (KM) algebras is a natural extension of the Weyl group and is shown to play a fundamental role in the structure of these algebras (both untwisted and twisted). It is a (discrete) Galilean group which incorporates the affine and Weyl group structure of the KM algebra and the space-time structure of the string. It links the Virasoro and KM algebras in a nontrivial way and plays a key role in the ‘‘vertex’’ construction. Moreover, the elements of its covering group can be used to uniquely parametrize (and untwist) the pseudotwisted KM algebras.