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

## TL;DR

This paper explicitly describes the generators of the algebra of $K$-invariants within the tensor product of the universal enveloping algebra of $rak{g}$ and the Clifford algebra of $rak{p}$ for the group $SO_e(4,1)$, advancing understanding of its symmetry structure.

## Contribution

The paper provides explicit generators for the algebra of $K$-invariants in $U(rak{g}) ensor C(rak{p})$, a novel detailed algebraic description for the specific group $SO_e(4,1)$.

## Key findings

- Explicit generators of $(U(rak{g}) ensor C(rak{p}))^{K}$ are obtained.
- The structure of the algebra of invariants is clarified for the group $SO_e(4,1)$.
- The results facilitate further study of representations and symmetry properties.

## 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 $(U(\mathfrak{g}) \otimes C(\mathfrak{p}))^{K}$.