Quaternifiations and extensions of current algebras on S³

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Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. We define a Lie algebra structure on the tensor product space of H and U(g), and obtain the quaternification gʰ of g. Let S³gʰ be the set of gʰ-valued smooth mappings over S³. The Lie algebra structure on S³gʰ is induced naturally from that of gʰ. On S³ exists the space of Laurent polynomial spinors spanned by a complete orthogonal system of eigen spinors of the tangential Dirac operator on S³. Tensoring U(g) we have the space of U(g)-valued Laurent polynomial spinors, which is a Lie subalgebra of S³gʰ. We introduce a 2-cocycle on the space of U(g)-valued Laurent polynomial spinors by the aid of a tangential vector field on S³. Then we have the corresponding central extension g(a) of the Lie algebra of U(g)-valued Laurent polynomial spinors. Finally we have the a Lie algebra g= g(a)+Cd which is obtained by adding to g(a) a derivation d which acts on g(a) as the radial derivation. When g is a simple Lie algebra with its Cartan subalgebra h, We shall investigate the weight space decomposition of (g, ad(h)), where h=h+Ca+Cd. The previous versions (v1-v7) of this article contained several incorrect assertions and here we have corrected them.