Singular unitary highest weight representations via geometric quantization

1997 ◽  
Vol 40 (2) ◽  
pp. 209-215
Author(s):  
Joachim Hilgert
Keyword(s):  
Geometric Quantization ◽  
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 .

AN EXPLICIT FORMULA FOR SOME SINGULAR VECTORS OF THE NEVEU–SCHWARZ ALGEBRA

1992 ◽  
Vol 07 (13) ◽  
pp. 3023-3033 ◽  
Author(s):  
LOUIS BENOIT ◽  
YVAN SAINT-AUBIN
Keyword(s):  
Explicit Expression ◽  
Explicit Formula ◽  
Singular Vector ◽  
Central Charge ◽  
Discrete Series ◽  
Virasoro Algebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Highest Weight ◽  
Highest Weight Representations

Similarly to the Virasoro algebra, the Neveu–Schwarz algebra has a discrete series of unitary irreducible highest weight representations. These are labeled by the values of [Formula: see text] (the central charge) and of the highest weight hpq = [(p (m + 2) − qm)2 − 4]/(8m (m + 2)) where m, p, q are some integers. The Verma modules constructed with these values (c, h) are not irreducible, however, as they contain two Verma submodules, each generated by a singular vector ψp,q (of weight hpq + pq/2) and ψm−p, m+2−q (of weight hpq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.

Highest weight representations for gl(m/n) and gl(m+n)

1989 ◽  
Vol 30 (7) ◽  
pp. 1415-1432 ◽  
Author(s):  
R. Le Blanc ◽  
D. J. Rowe
Keyword(s):  
Highest Weight ◽  
Highest Weight Representations

scholarly journals Multivariableq-Hahn polynomials as coupling coefficients for quantum algebra representations

2001 ◽  
Vol 28 (6) ◽  
pp. 331-358 ◽  
Author(s):  
Hjalmar Rosengren
Keyword(s):  
Tensor Product ◽  
Quantum Group ◽  
Quantum Algebra ◽  
Coupling Coefficients ◽  
Highest Weight ◽  
Hahn Polynomials ◽  
Highest Weight Representations

We study coupling coefficients for a multiple tensor product of highest weight representations of theSU(1,1)quantum group. These are multivariable generalizations of theq-Hahn polynomials.

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