Quaternifications and Extensions of Current Algebras on S3

Abstract

Let \(\mathbf{H}\) be the quaternion algebra. Let \(\mathfrak{g}\) be a complex Lie algebra and let \(U(\mathfrak{g})\) be the enveloping algebra of \(\mathfrak{g}\). The quaternification \(\mathfrak{g}^{\mathbf{H}}=\)\(\,(\,\mathbf{H}\otimes U(\mathfrak{g}),\,[\quad,\quad]_{\mathfrak{g}^{\mathbf{H}}}\,)\) of \(\mathfrak{g}\) is defined by the bracket \( \big[\,\mathbf{z}\otimes X\,,\,\mathbf{w}\otimes Y\,\big]_{\mathfrak{g}^{\mathbf{H}}}\,=\)\(\,(\mathbf{z}\cdot \mathbf{w})\otimes\,(XY)\,- \)\(\, (\mathbf{w}\cdot\mathbf{z})\otimes (YX)\,,\nonumber \) for \(\mathbf{z},\,\mathbf{w}\in \mathbf{H}\) and {the basis vectors \(X\) and \(Y\) of \(U(\mathfrak{g})\).} Let \(S^3\mathbf{H}\) be the ( non-commutative) algebra of \(\mathbf{H}\)-valued smooth mappings over \(S^3\) and let \(S^3\mathfrak{g}^{\mathbf{H}}=S^3\mathbf{H}\otimes U(\mathfrak{g})\). The Lie algebra structure on \(S^3\mathfrak{g}^{\mathbf{H}}\) is induced naturally from that of \(\mathfrak{g}^{\mathbf{H}}\). We introduce a 2-cocycle on \(S^3\mathfrak{g}^{\mathbf{H}}\) by the aid of a tangential vector field on \(S^3\subset \mathbf{C}^2\) and have the corresponding central extension \(S^3\mathfrak{g}^{\mathbf{H}} \oplus(\mathbf{C}a)\). As a subalgebra of \(S^3\mathbf{H}\) we have the algebra of Laurent polynomial spinors \(\mathbf{C}[\phi^{\pm}]\) spanned by a complete orthogonal system of eigen spinors \(\{\phi^{\pm(m,l,k)}\}_{m,l,k}\) of the tangential Dirac operator on \(S^3\). Then \(\mathbf{C}[\phi^{\pm}]\otimes U(\mathfrak{g})\) is a Lie subalgebra of \(S^3\mathfrak{g}^{\mathbf{H}}\). We have the central extension \(\widehat{\mathfrak{g}}(a)= (\,\mathbf{C}[\phi^{\pm}] \otimes U(\mathfrak{g}) \,) \oplus(\mathbf{C}a)\) as a Lie-subalgebra of \(S^3\mathfrak{g}^{\mathbf{H}} \oplus(\mathbf{C}a)\). Finally we have a Lie algebra \(\widehat{\mathfrak{g}}\) which is obtained by adding to \(\widehat{\mathfrak{g}}(a)\) a derivation \(d\) which acts on \(\widehat{\mathfrak{g}}(a)\) by the Euler vector field \(d_0\). That is the \(\mathbf{C}\)-vector space \(\widehat{\mathfrak{g}}=\left(\mathbf{C}[\phi^{\pm}]\otimes U(\mathfrak{g})\right)\oplus(\mathbf{C}a)\oplus (\mathbf{C}d)\) endowed with the bracket \( \bigl[\,\phi_1\otimes X_1+ \lambda_1 a + \mu_1d\,,\phi_2\otimes X_2 + \lambda_2 a + \mu_2d\,\,\bigr]_{\widehat{\mathfrak{g}}} \, =\)\( (\phi_1\phi_2)\otimes (X_1\,X_2) \, -\,(\phi_2\phi_1)\otimes (X_2X_1)+\mu_1d_0\phi_2\otimes X_2- \) \(\mu_2d_0\phi_1\otimes X_1 + \) \( (X_1\vert X_2)c(\phi_1,\phi_2)a\,. \) When \(\mathfrak{g}\) is a simple Lie algebra with its Cartan subalgebra \(\mathfrak{h}\) we shall investigate the weight space decomposition of \(\widehat{\mathfrak{g}}\) with respect to the subalgebra \(\widehat{\mathfrak{h}}= (\phi^{+(0,0,1)}\otimes \mathfrak{h} )\oplus(\mathbf{C}a) \oplus(\mathbf{C}d)\).