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.