1. Isomorphisms of Twisted Hilbert Loop Algebras
- Author
-
Timothée Marquis and Karl-Hermann Neeb
- Subjects
17B65, 17B70, 17B22, 17B10 ,General Mathematics ,010102 general mathematics ,Hilbert space ,Mathematics - Rings and Algebras ,01 natural sciences ,Combinatorics ,Loop (topology) ,symbols.namesake ,Isomorphism theorem ,Rings and Algebras (math.RA) ,Affine root system ,Product (mathematics) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Isomorphism ,Representation Theory (math.RT) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Representation Theory ,Mathematics - 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 $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections
- Published
- 2017