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

TL;DR

This paper constructs and analyzes a new class of affine Lie algebras based on the 3-sphere, introducing 2-cocycles, central extensions, and root space decompositions for these algebraic structures.

Contribution

It introduces three non-trivial 2-cocycles on the Lie algebra of maps from S^3, extends them to larger algebras, and develops the root space decomposition for the resulting affine Lie algebras.

Findings

Defined new 2-cocycles on S^3H

Constructed central extensions of these algebras

Derived root space decomposition and Chevalley generators

Abstract

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 $\alpha\neq 0$ , $\hat{g}_{m/2 \delta}= C[\phi ;m] \otimes h_0$, for $m \neq 0$, 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.