Quaternifications and extensions of current algebras on S3

Abstract

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 gH = (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 S3gH = S3H ⊗ U(g). The Lie algebra structure on S3gH is induced naturally from that of gH. We introduce a 2-cocycle on S3gH by the aid of a tangential vector field on S3 ⊂ C2 and have the corresponding central extension S3gH⊕(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 S3gH. We have the central extension g^(a) = (C[φ±] ⊗ U(g)) ⊕ (Ca) as a Lie-subalgebra of S3gH ⊕ (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).