Centralizers of spin subalgebras

TL;DR

This paper characterizes the centralizers of specific spin subalgebras within orthogonal Lie algebras, expanding understanding of their algebraic structure and symmetries.

Contribution

It explicitly determines the centralizers of isomorphic copies of spin subalgebras in orthogonal Lie algebras, using Clifford algebra representations.

Findings

Explicit descriptions of centralizers for various spin subalgebras

Connections between Clifford algebra representations and Lie algebra centralizers

Enhanced understanding of symmetries in orthogonal Lie algebras

Abstract

We determine the centralizers of certain isomorphic copies of spin subalgebras $\mathfrak{spin}(r)$ in $\mathfrak{so}(d_rm)$, where $d_r$ is the dimension of a real irreducible representation of $Cl_r^0$, the even Clifford algebra determined by the positive definite inner product on $\mathbb{R}^r$, where $r, m\in\mathbb{N}$.