# Symmetry, Vol. 7, Pages 2150-2180: Quaternifications and Extensions of Current Algebras on S3

Research paper by Tosiaki Kori, Yuto Imai

Indexed on: 16 May '16Published on: 27 Nov '15Published in: Symmetry

#### 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 $$\widehat{\mathfrak{h}}= (\phi^{+(0,0,1)}\otimes \mathfrak{h} )\oplus(\mathbf{C}a) \oplus(\mathbf{C}d)$$.