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 = (H ⊗ U(g), [,]_gH) of g is defined by the bracket [z ⊗ X, w ⊗ Y]_gH = (z · w) ⊗ (XY) - (w · z) ⊗ (YX), for z, w e H and the basis vectors X and Y of U(g). Let S³H be the (non-commutative) algebra of H-valued smooth mappings over S³ and let S³g^H = S³H ⊗ U(g). The Lie algebra structure on S³g^H is induced naturally from that of g^H. We introduce a 2-cocycle on S³g^H by the aid of a tangential vector field on S³ ⊂ C² and have the corresponding central extension S³g^H⊕(Ca). As a subalgebra of S³H we have the algebra of Laurent polynomial spinors C[φ^±] spanned by a complete orthogonal system of eigen spinors (φ^{± (m, l,k)})_{m, l,k} of the tangential Dirac operator on S³. Then C[φ^±] ⊗ U(g) is a Lie subalgebra of S³g^H. We have the central extension g^(a) = (C[φ^±] ⊗ U(g)) ⊕ (Ca) as a Lie-subalgebra of S³g^H ⊕ (Ca). Finally we have a Lie algebra g^ which is obtained by adding to g^(a) a derivation d which acts on g^(a) by the Euler vector field d₀. That is the C-vector space g^ = (C[φ^±] ⊗ U(g)) ⊕ (Ca)⊕(Cd) endowed with the bracket [φ₁ ⊗ X₁ +λ₁a+μ₁d, φ₂ ⊗ X₂ +λ₂a+μ₂d]_g^ = (φ₁φ₂) ⊗ (X₁ X₂) - (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 g^ with respect to the subalgebra h^ = (φ+^(0,0,1) ⊗ h)⊕(Ca)⊕(Cd).