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Isomorphisms of Twisted Hilbert Loop Algebras
- Source :
- Canadian Journal of Mathematics. 69:453-480
- Publication Year :
- 2017
- Publisher :
- Canadian Mathematical Society, 2017.
-
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 $\mathfrak{k}$.<br />Comment: 22 pages; Minor corrections
- 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
Subjects
Details
- ISSN :
- 14964279 and 0008414X
- Volume :
- 69
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics
- Accession number :
- edsair.doi.dedup.....af6fa40ee56a6b0986f0809109ac045e