A Chevalley–Kostant presentation of basic modules for sl(2)̂ and the associated affine KPRV determinants at q=1
✍ Scribed by Jacob Greenstein; Anthony Joseph
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- French
- Weight
- 180 KB
- Volume
- 125
- Category
- Article
- ISSN
- 0007-4497
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✦ Synopsis
Let g be a semisimple Lie algebra with triangular decomposition g = n -⊕ h ⊕ n + and ρ the half-sum of the roots of n + . It is a remarkable fact that up to a shift by ρ the weights of the exterior algebra of n -are exactly those of the simple highest weight module V (ρ) with highest weight ρ. Moreover a construction of Chevalley and Kostant realizes V (ρ) in the Clifford algebra over n -. With a little care this construction can be generalized to the central extensions of the corresponding loop algebra and in particular for sl(2).
In this paper a Chevalley-Kostant construction is given for the basic modules of sl( 2). This has no analogue for semisimple g and indeed is dependent on the peculiarities of bounded modules. It is used to construct a certain endomorphism of a basic module and to compute its trace on weight subspaces, which relates to the corresponding KPRV determinant at q = 1. 2001 Éditions scientifiques et médicales Elsevier SAS