Current algebras on S^3 of complex Lie algebras

Abstract

Let L be the space of spinors on S 3 that are the restrictions to S 3 of the Laurent polynomial type harmonic spinors on C 2 . L becomes an associative algebra. For a simple Lie algebra g the real Lie algebra Lg generated by L ⊗ C g is called g-current algebra. The real part K of L becomes a commutative subalgebra of L. For the Cartan subalgebra h of g , Kh = K ⊗ R h becomes a Cartan subalgebra of Lg. We investigate the adjoint representation of Kh and find that the set of non-zero weights corresponds bijectively to the root space of g. Let g = h + e + f be the standard triangular decomposition of g, and let L⊗ C h, L⊗ C e and L⊗ C f generate respectively the Lie subalgebras Lh, Le and Lf of Lg. Then we have the triangular decomposition Lg = Lh + Le + Lf , that is also associated with the weight space decomposition of Lg. With the aid of the basic vector fields on S 3 that arise from the infinitesimal representation of SO(3) we introduce a triple of 2-cocycles {c k ; k = 0, 1, 2 } on the Lie algebra Lg. Then we have the central extenstion Lg ⊕ ⊕ 2 k=0 Ca k associated to the 2-cocycles {c k } k=0,1,2 . Adjoining a derivation coming from the radial vector field n on S 3 we obtain the second central extension g = Lg ⊕ ⊕ 2 k=0 Ca k ⊕ Cn. The root space decomposition and the Chevalley generators of g will be given.