On angular momentum algebras and their relations

TL;DR

This paper investigates the structure of the total angular momentum algebra (TAMA), a centraliser in Weyl Clifford algebra, extending known angular momentum algebra relations and providing new presentations for specific ranks.

Contribution

It introduces an analogue relation for the even subalgebra of TAMA and proves it generates a presentation for ranks 4 and 5.

Findings

Constructed an analogue relation for the even subalgebra of TAMA.

Proved the relations generate a presentation for ranks 4 and 5.

Extended the diagrammatic presentation to TAMA's even subalgebra.

Abstract

In this paper, we study the centraliser of $\mathfrak{osp}(1|2)$, denoted the total angular momentum algebra (TAMA), in the Weyl Clifford algebra. The TAMA extends the angular momentum algebra (AMA), which arises as the centraliser of $\mathfrak{sl}(2)$ and admits a diagrammatic presentation via the crossing relation described by Feigin and Hakobyan. Using Young symmetrisers we construct an analogue relation for the even subalgebra of the TAMA. We prove that for rank $4$ and $5$ these relations generate a presentation for the even subalgebra of the TAMA.