Aldaya, Victor; Navarro-Salas, José Kac-Moody group representations and generalization of the Sugawara construction of the Virasoro algebra. (English) Zbl 0657.17014 Lett. Math. Phys. 16, No. 2, 117-124 (1988). 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 Cited in 3 Documents 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 Keywords:Sugawara construction; perturbative technique; group laws; infinite- dimensional Lie group; Virasoro algebra; affine Kac-Moody algebra; commutation relations; central extensions; subalgebras; polarizations; vacuum symmetry; wave functions of representations; highest weight representations; strings PDFBibTeX XMLCite \textit{V. Aldaya} and \textit{J. Navarro-Salas}, Lett. Math. Phys. 16, No. 2, 117--124 (1988; Zbl 0657.17014) Full Text: DOI References: [1] Aldaya, V. and Navarro-Salas, J., Commun. Math. Phys. 113, 375 (1987). · Zbl 0637.22011 · doi:10.1007/BF01221252 [2] Aldaya, V. and de Azcarraga, J. A., J. Math. Phys. 23, 1297 (1982). · Zbl 0502.58018 · doi:10.1063/1.525513 [3] Aldaya, V. and de Azcarraga, J. A., Proceeding Conf. ?Constraints Theory and Relativistic Dynamics?, Firence, 1986. [4] Mickelsson, J., CPT preprint # 1448. [5] Jacobsen, H. P. and Kac, V. G., A new class of unitarizable highest weight representations of infinite dimensional Lie algebras, in N. Sanchez (ed.), Nonlinear Equations in Classical and Quantum Field Theory, pp. 1-20, Lecture Notes in Physics 226, Springer-Verlag, 1985. [6] Gepner, D. and Witten, E., Nucl. Phys. B278, 493 (1986). · doi:10.1016/0550-3213(86)90051-9 [7] Aldaya, V. and de Azcarraga, J. A., Ann. Phys. 165, 484 (1985). · Zbl 0588.58027 · doi:10.1016/0003-4916(85)90306-9 [8] Witten, E., Princeton preprint (1987). [9] Sugawara, H., Phys. Rev. 170, 1659 (1986). · doi:10.1103/PhysRev.170.1659 [10] Knizhnik, V. G. and Zamolodchikov, A. B., Nucl. Phys. B247, 83 (1984). · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2 [11] Goddard, P., Kent, A., and Olive, D. I., Phys. Lett. B152, 88 (1985); Goddard, P. and Olive, D. I. Int. J. Mod. Phys. A1, n2 (1986). [12] Kac, V. G., Highest weight representations of conformal current algebras, in Topological and Geometrical Methods in Field Theory, Helsinki, World Scientific (1986). [13] Aldaya, V. and Navarro-Salas, J., Imperial preprint TP/87-88/4. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.