Abstract
The closest infinite-dimensional relatives of compact Lie algebras are
Hilbert-Lie algebras, i.e., real Hilbert spaces with a Lie algebra structure for
which the scalar product is invariant. Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie
algebras , also called
affinisations of . They possess a
root space decomposition whose corresponding root system is a locally affine root
system of one of the 7 families
for some infinite set J. To each of these types
corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra
, which we call
standard.
In this paper, we give for each affinisation g of a simple Hilbert-Lie
algebra an explicit
isomorphism from g to one of the standard affinisations of . The existence of
such an isomorphism could also be derived from the classiffication of locally
affine root systems, but for representation theoretic purposes it is crucial to
obtain it explicitly as a deformation between two twists that is compatible with
the root decompositions. We illustrate this by applying our isomorphism theorem to
the study of positive energy highest weight representations of g. In subsequent
work, this paper will be used to obtain a complete classification of the positive
energy highest weight representations of affinisations of .
FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS