The centralizer of $K$ in $U(\mathfrak{g})\otimes C(\mathfrak{p})$ for the group $S{O}_e(4,1)$

Abstract

Let $G$ be the Lie group $S{O}_e(4,1)$, with maximal compact subgroup $K=S(O(4)\times O(1){)}_e\cong SO(4)$. Let $\mathfrak{g}=\mathfrak{so}(5,\mathbb{C})$ be the complexification of the Lie algebra ${\mathfrak{g}}_0=\mathfrak{so}(4,1)$ of $G$, and let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$. Let $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ be the Cartan decomposition of $\mathfrak{g}$, and $C(\mathfrak{p})$ the Clifford algebra of $\mathfrak{p}$ with respect to the trace form $B(X,Y)=\text{tr}(XY)$ on $\mathfrak{p}$. In this paper we give explicit generators of the algebra $(U(\mathfrak{g})\otimes C(\mathfrak{p}){)}^K$.

arXiv comment14 pages

References in corpus (1)

• $K-$structure of $\mathcal{U}(\mathfrak{g})$ for $\mathfrak{su}(n,1)$ and $\mathfrak{so}(n,1)$