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Kac-Moody group representations and generalization of the Sugawara construction of the Virasoro algebra. (English) Zbl 0657.17014

In the present paper the authors use a previously developed perturbative technique for calculating group laws. The group they apply their techniques to is the infinite-dimensional Lie group with as Lie algebra the semidirect product of the Virasoro algebra V with the affine Kac- Moody algebra of sl(2,\({\mathbb{R}})\) (i.e. the Kac-Moody algebra \(A_ 1^{(1)}).\)
They give the commutation relations of \(\tilde V\times A_ 1^{(1)}\), parametrized so as to include all nontrivial central extensions. They classify all possible characteristic subalgebras, leading to different polarizations and then to different representations with a determined vacuum symmetry. The corresponding group law is given as a power series in the parameters (up to fourth order). Making use of this they can give perturbative expressions for the wave functions of representations of the group, and then of the actions of the Lie algebra generators on the wave functions. The highest weight representations considered here are associated with strings on SU(2).
Reviewer: J.Van der Jeugt

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
17B15 Representations of Lie algebras and Lie superalgebras, analytic theory
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
81T99 Quantum field theory; related classical field theories
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