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

Abstract

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

arXiv admin note: text overlap with arXiv:1306.5030