The centralizer of $K$ in $U(g)\otimes C(p)$ for the group $SO_e(4,1)$

Author: Ana Prlić

Glasnik Matematički, Vol. 52, No. 2 (2017), 275-288.

DOI: 10.3336/gm.52.2.07

Abstract. Let $G$ be the Lie group $SO_e(4,1)$ , 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 $(U(\mathfrak{g}) \otimes C(\mathfrak{p}))^{K}$ .

2010 Mathematics Subject Classification. 22E47, 22E46.

Key words and phrases. Lie group, Lie algebra, representation, Dirac operator, Dirac cohomology.

Pages

Glasnik Matematički, Vol. 52, No. 2 (2017), 275-288.