Monday, June 24, 2013

1306.5030 (Tosiaki Kori et al.)

On the quarternifiation of the Lie algebra $Map(S^3,g)$ and its

Tosiaki Kori, Yuto Imai
Let $H$ be the quarternion 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 quarternification $g^H$ of $g$. Let $S^3g^H$ be the set of $g^H$-valued smooth mappings over $S^3$. The Lie algebra structure on $S^3g^H$ is induced naturally from that of $g^H$. We introduce a 2-cocycle on $ S^3g^H$ by the aid of Clifford multiplication of the radial vector on $C^2$. Then we have the corresponding central extension $S^3g^H + Ca$. On S^3 exists the space of Laurent polynomial spinors spanned by a complete orthogonal system of eigen spinors of the tangential Dirac operator on $S^3$. Tensoring $U(g)$ we have the space of $U(g)$-valued Laurent polynomial spinors, which is a Lie subalgebra of $S^3g^H$, hence its central extension $g^(a)$ is obtained. 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. $g^$ is not a central extension of $g^(a)$. When $g$ is a simple Lie algebra with its Cartan subalgebra $h$, $g^$ is an infiite dimensional simple Lie algebra. We shall investigate the root space decomposition of $g^$ .
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