Quaternifications and Extensions of Current Algebras on S3

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification g = (H ⊗ U(g), [ , ]gH ) of g is defined by the bracket [ z ⊗ X , w ⊗ Y ] gH = (z · w) ⊗ (XY ) − (w · z) ⊗ (Y X) , for z, w ∈ H and the basis vectors X and Y of U(g). Let SH be the ( non-commutative) algebra of H-valued smooth mappings over S and let Sg = SH ⊗ U(g). The Lie algebra structure on Sg is induced naturally from that of g. We introduce a 2-cocycle on Sg by the aid of a tangential vector field on S ⊂ C and have the corresponding central extension Sg⊕ (Ca). As a subalgebra of SH we have the algebra of Laurent polynomial spinors C[φ±] spanned by a complete orthogonal system of eigen spinors {φ}m,l,k of the tangential Dirac operator on S. Then C[φ±]⊗U(g) is a Lie subalgebra of Sg. We have the central extension ĝ(a) = (C[φ±]⊗U(g) )⊕ (Ca) as a Lie-subalgebra of Sg ⊕ (Ca). Finally we have a Lie algebra ĝ which is obtained by adding to ĝ(a) a derivation d which acts on ĝ(a) by the Euler vector field d0. That is the C-vector space ĝ = (C[φ±]⊗ U(g))⊕ (Ca)⊕ (Cd) endowed with the bracket [ φ1⊗X1 + λ1a+ μ1d , φ2⊗X2 + λ2a+ μ2d ] ĝ = (φ1φ2)⊗ (X1X2) − (φ2φ1)⊗ (X2X1)+μ1d0φ2⊗X2−μ2d0φ1⊗X1 +(X1|X2)c(φ1, φ2)a . When g is a simple Lie algebra with its Cartan subalgebra h we shall investigate the weight space decomposition of ĝ with respect to the subalgebra ĥ = (φ⊗h)⊕ (Ca)⊕ (Cd).