Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras

10.1142/0476 ◽  
1988 ◽  
Author(s):  
V Kac ◽  
A Raina
Keyword(s):  
Lie Algebras ◽  
Infinite Dimensional ◽  
Highest Weight ◽  
Highest Weight Representations

scholarly journals Isomorphisms of Twisted Hilbert Loop Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 453-480
Author(s):  
Timothée Marquis ◽  
Karl-Hermann Neeb
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Energy ◽  
Algebra Structure ◽  
Affine Lie Algebras ◽  
Subsequent Work ◽  
Highest Weight ◽  
Loop Algebras ◽  
Highest Weight Representations

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 .

scholarly journals HIGHEST WEIGHT REPRESENTATIONS AND KAC DETERMINANTS FOR A CLASS OF CONFORMAL GALILEI ALGEBRAS WITH CENTRAL EXTENSION

2012 ◽  
Vol 23 (11) ◽  
pp. 1250118 ◽  
Author(s):  
NARUHIKO AIZAWA ◽  
PHILLIP S. ISAAC ◽  
YUTA KIMURA
Keyword(s):  
Central Extension ◽  
Spatial Dimension ◽  
Verma Modules ◽  
Infinite Dimensional ◽  
Singular Vectors ◽  
Highest Weight ◽  
Irreducible Modules ◽  
Highest Weight Representations ◽  
Quotient Modules

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrödinger algebra.

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