S^3上当前代数的四元化与扩张
Quaternifiations and extensions of current algebras on S^3
摘要：
设$H$为四元数代数。设$g$是复李代数，$U(g)$是$g$的包络代数.在$H$和$U(g)$的张量积空间上定义了一种李代数结构，得到了$g$的四元化$g^H$.设$S^3g^H$是$S^3$上的$g^H$值光滑映射的集合. $S^3g^H$上的李代数结构是由$g^H$的李代数结构自然导出的。在$S^3$上存在由$S^3$上切向Dirac算子的本征旋量的完备正交系所张成的Laurent多项式旋量空间。对$U(g)$进行张量处理，我们得到$U(g)$值Laurent多项式旋量空间，它是$S^3g^H$的一个李子代数.借助于$S^3$上的切向量场，在$U(g)$值Laurent多项式旋量空间上引入了一个2-上循环.然后我们得到了$U(g)$值Laurent多项式旋量的李代数的相应的中心扩张$\hat g(a)$。最后，我们得到了一个李代数$\hat g=\hat g(a)+Cd$，它是通过在$\hat g(a)$上加上一个导子$d$而得到的，导子$d$作为径向导子作用在$\hat g(a)$上。当$g$是一个单李代数，其Cartan子代数为$h$时，我们将研究$(\hat g, ad(\hat h))$的权空间分解，其中$\hat h=h+Ca+Cd$ .本文的以前版本（v1-v7）包含了几个不正确的断言，在这里我们已经纠正了它们。
英文摘要：
Let $H$ be the quaternion 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 quaternification $g^H$ 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 $g^H$. 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 $S^3g^H$. We introduce a 2-cocycle on the space of $U(g)$-valued Laurent polynomial spinors by the aid of a tangential vector field on $S^3$. Then we have the corresponding central extension $\hat g(a)$ of the Lie algebra of $U(g)$-valued Laurent polynomial spinors. Finally we have the a Lie algebra $\hat g=\hat g(a)+Cd$ which is obtained by adding to $\hat g(a)$ a derivation $d$ which acts on $\hat g(a)$ as the radial derivation. When $g$ is a simple Lie algebra with its Cartan subalgebra $h$, We shall investigate the weight space decomposition of $(\hat g, ad(\hat h))$, where $\hat h=h+Ca+Cd$ . The previous versions (v1-v7) of this article contained several incorrect assertions and here we have corrected them.
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