Representations of Spin(4), Spin(2,2) and Spin(3,1)

TL;DR

This paper introduces a new outer product on R^6 that enables the construction of Lie algebra structures corresponding to Spin(4), Spin(2,2), and Spin(3,1), and explores a related almost complex structure.

Contribution

It presents a novel outer product on R^6 that generates multiple Lie algebra structures and identifies a 4D submanifold with an almost complex structure.

Findings

R^6 with the new outer product can form Lie brackets for so(4), so(2,2), and so(3,1)

Existence of a 4D submanifold with an almost complex structure

Unified framework for different Spin groups via R^6 outer product

Abstract

In this essay we present an outer product on R^6, and we show that R^6 with this outer product can take three kinds of Lie brackets. We prove that R^6 with these Lie brackets take the structures of so(4), so(2,2) and so(3,1). We also deduce a 4-dimensional submanifold of R^6 with an almost complex structure.