Let L be the space of spinors on the 3-sphere that are the restrictions 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 the tensor product of L and g is called the g-current algebra. The real part K of L becomes a commutative subalgebra of L. For a Cartan subalgebra h of g, h tensored by K becomes a Cartan subalgebra Kh of Lg. The set of non-zero weights of the adjoint representation of Kh corresponds bijectively to the root space of g. Let g=h+e+ f be the standard triangular decomposition of g, and let Lh, Le and Lf respectively be the Lie subalgebras of Lg generated by the tensor products of L with h, e and f respectively . 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 the 3-shpere that arise from the infinitesimal representation of SO(3) we introduce a triple of 2-cocycles {c_k; k=0,1,2} on Lg. Then we have the central extension: Lg+ sum Ca_k associated to the 2-cocycles {c_k; k=0,1,2}. Adjoining a derivation coming from the radial vector field on S^3 we obtain the second central extension g^=Lg+ sum Ca_k + Cn. The root space decomposition of g^ as welll as the Chevalley generators of g^ will be given.
تحميل البحث