## On the quarternifiation of the Lie algebra \$Map(S^3,g)\$ and its extensions    [PDF]

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^\$ .
View original: http://arxiv.org/abs/1306.5030