A remark on the connection between affine Lie algebras and soliton equations

1988 ◽  
Vol 16 (2) ◽  
pp. 125-132 ◽  
Author(s):  
Etsuro Date
Keyword(s):  
Lie Algebras ◽  
Affine Lie Algebras ◽  
Soliton Equations

Representations of affine Lie algebras and soliton equations

1985 ◽  
Vol 315 (1533) ◽  
pp. 391-391
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Gordon Equation ◽  
Vries Equation ◽  
Affine Lie Algebras ◽  
Loop Groups ◽  
Sine Gordon Equation ◽  
Kyoto School ◽  
Hidden Symmetries ◽  
Soliton Equations

A few years ago the 'hidden symmetries’ of the soliton equations had been identified as affine Lie groups, also known as loop groups. The first extensive use of the representation theory of affine Lie algebras for the soliton equations have been developed in a series of works by mathematicians of the Kyoto school. We will review some of their results and develop them further on the basis of the representation theory. Thus an orbit of the simplest affine Lie group SL(2, C)^ in the fundamental representation V will provide the solutions of the Korteweg-de Vries equation, and similarly the solutions of the sine-Gordon equation will come from an orbit of the group (SL(2, C) x SL(2, C)) ^ in V x V*.

A class of non-standard modules for affine Lie algebras

1987 ◽  
Vol 196 (3) ◽  
pp. 303-313 ◽  
Author(s):  
Nolan R. Wallach
Keyword(s):  
Lie Algebras ◽  
Affine Lie Algebras

QUANTUM HAMILTONIAN REDUCTION AND WB ALGEBRA

1992 ◽  
Vol 07 (20) ◽  
pp. 4885-4898 ◽  
Author(s):  
KATSUSHI ITO
Keyword(s):  
Lie Algebras ◽  
Free Field ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Hamiltonian Reduction ◽  
Affine Lie Algebras ◽  
Quantum Hamiltonian Reduction ◽  
Free Field Realization ◽  
Coset Construction ◽  
Quantum Hamiltonian

We study the quantum Hamiltonian reduction of affine Lie algebras and the free field realization of the associated W algebra. For the nonsimply laced case this reduction does not agree with the usual coset construction of the W minimal model. In particular, we find that the coset model [Formula: see text] can be obtained through the quantum Hamiltonian reduction of the affine Lie superalgebra B(0, n)(1). To show this we also construct the Feigin-Fuchs representation of affine Lie superalgebras.

scholarly journals Braided Tensor Categories of Admissible Modules for Affine Lie Algebras

2018 ◽  
Vol 362 (3) ◽  
pp. 827-854 ◽  
Author(s):  
Thomas Creutzig ◽  
Yi-Zhi Huang ◽  
Jinwei Yang
Keyword(s):  
Lie Algebras ◽  
Affine Lie Algebras ◽  
Tensor Categories

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 .

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