Explicit Decompositions of Weyl Reflections in Affine Lie Algebras.pdf

NBI-HE-97-31 July 1997 Explicit Decompositions of Weyl Reflections in Affine Lie Algebras 7 9 Jørgen Rasmussen1 9 1 The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark l u J 1 2 Abstract 1 In this paper explicit decompositions are provided of the Weyl reflections in affine Lie v algebras, in terms of fundamental Weyl reflections. 6 2 0 PACS: 02.20.Sv; 11.25.Hf 7 Keywords: Affine Lie algebra; Lie algebra; Weyl group 0 7 9 / g l a - q : v i X r a 1e-mail address: jrasmussen@nbi.dk 1 Introduction An understanding of the Weyl group of an affine Lie algebra resides in the basis of affine Lie algebra theory [1]. Just as in the case of the usual finite dimensional Lie algebras, the (affine) Weyl group is fundamental in discussions on e.g. characters. In applications of the Weyl group it is sometimes of importance to be able to decompose the Weyl reflections into products of fundamental reflections which are reflections with respect to simple roots. This is the case when considering singular vectors along the lines of Malikov, Feigin and Fuks [2]. The main result in this paper is the presentation of explicit decompositions into fundamental reflections of all Weyl reflections in all affine Lie algebras based on simple finite dimensional Lie algebras of the types A , B , C , D , E , E , E , F and G . The r r r r 6 7 8 4 2 decompositions we present rely on a new universal lemma and on well known expli