Current algebras on S^3 of complex Lie algebras
Abstract
Let L be the space of spinors on the 3sphere 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 gcurrent 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 nonzero 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 3shpere that arise from the infinitesimal representation of SO(3) we introduce a triple of 2cocycles {c_k; k=0,1,2} on Lg. Then we have the central extension: Lg+ \sum Ca_k associated to the 2cocycles {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.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.08389
 Bibcode:
 2021arXiv210108389K
 Keywords:

 Mathematics  Representation Theory;
 Mathematical Physics;
 Mathematics  Differential Geometry;
 17B67(Primary);
 81R10(Secondary);
 F.2.2
 EPrint:
 42 pages