sl(n,H)-Current Algebra on S^3

We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H otimes gl(n,C)$. Then we have the corresponding central extension $S^3gl(n,H)oplus oplus_k (Ca_k)$. As a subalgebra of $S^3H$ we have the algebra $C[phi]$ of the Laurent polynomial spinors on $S^3$. Then we have a Lie subalgebra $hat{gl}(n, H)=C[phi] otimes gl(n, C)$ of $S^3gl(n,H)$, as well as its central extension by the 2-cocycles ${c_k}$ and the Euler vector field $d$: $hat{gl}=hat{gl}(n, H) oplus oplus_k(Ca_k)oplus Cd$ . The Lie algebra $hat{sl}(n,H)$ is defined as a Lie subalgebra of $hat{gl}(n,H)$ generated by $C[phi]otimes sl(n,C))$. We have the corresponding central extension of $hat{sl}(n,H)$ by the 2-cocycles ${c_k}$ and the derivation $d$, which becomes a Lie subalgebra $hat{sl}$ of $hat{gl}$. Let $h_0$ be a Cartan subalgebra of $sl(n,C)$ and $hat{h}=h_0 oplus oplus_k(Ca_k)oplus Cd$. The root space decomposition of the $ad(hat{h})$-representation of $hat{sl}$ is obtained. The set of roots is $Delta ={ m/2 delta + alpha ; alpha in Delta_0, m in Z} bigcup {m/2 delta ; m in Z }$ . And the root spaces are $hat{g}_{m/2 delta+ alpha}= C[phi ;m] otimes g_{alpha}$, for $alpha eq 0$ , $hat{g}_{m/2 delta}= C[phi ;m] otimes h_0$, for $m eq 0$, and $hat{g}_{0 delta}= hat{h}$, where $C[phi ;m]$ is the subspace with the homogeneous degree m. The Chevalley generators of $hat{sl}$ are given.