Standard Representations of Some Affine Lie Algebras

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.

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Categories of nonstandard highest weight modules for affine Lie algebras

Mathematische Zeitschrift ◽

10.1007/bf02622110 ◽

1996 ◽

Vol 221 (1) ◽

pp. 193-209 ◽

Cited By ~ 2

Author(s):

B. Cox ◽

V. Futorny ◽

D. Melville

Keyword(s):

Lie Algebras ◽

Affine Lie Algebras ◽

Highest Weight ◽

Highest Weight Modules ◽

Weight Modules

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Braided Tensor Categories of Admissible Modules for Affine Lie Algebras

Communications in Mathematical Physics ◽

10.1007/s00220-018-3217-6 ◽

2018 ◽

Vol 362 (3) ◽

pp. 827-854 ◽

Cited By ~ 13

Author(s):

Thomas Creutzig ◽

Yi-Zhi Huang ◽

Jinwei Yang

Keyword(s):

Lie Algebras ◽

Affine Lie Algebras ◽

Tensor Categories

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Affine Lie algebras and vertex operator algebras

Generalized Vertex Algebras and Relative Vertex Operators ◽

10.1007/978-1-4612-0353-7_13 ◽

1993 ◽

pp. 141-160

Author(s):

Chongying Dong ◽

James Lepowsky

Keyword(s):

Lie Algebras ◽

Vertex Operator ◽

Operator Algebras ◽

Vertex Operator Algebras ◽

Affine Lie Algebras

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Isomorphisms of Twisted Hilbert Loop Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-2016-003-x ◽

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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