A perspective on the Magic Square and the 'special unitary' realizations of simple Lie algebras

TL;DR

This paper explores the classification of simple real Lie groups through an extended Magic Square framework, incorporating split and tensor product algebras to realize all real forms as 'special unitary' groups.

Contribution

It introduces a 'Grand Magic Square' that includes split and tensor product algebras, providing a unified realization of all simple real Lie algebras as 'special unitary' over these algebras.

Findings

Complete list of realizations of simple real Lie algebras as 'special unitary' over tensor products.

Extension of the Magic Square to include split and tensor product algebras.

Unified framework for understanding all real forms of simple Lie algebras.

Abstract

This article contains the last part of the mini-course `Spaces: a perspective view' delivered at the IFWGP2012. Here I deal with the part of the mini-course which centers on the classification questions associated to the simple real Lie groups. I review the original introduction of the Magic Square `a la Freudenthal', putting the emphasis in the role played in this construction by the four normed division algebras ${\mathbb R}, {\mathbb C}, {\mathbb H}, {\mathbb O}$. I then explore the possibility of understanding some simple real Lie algebras as `special unitary' over some algebras ${\mathbb K}$ or tensor products ${\mathbb K}_1\otimes {\mathbb K}_2$, and I argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions of complex, quaternions and octonions as well. This way we get a `Grand Magic Square' and we fill in the details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as `special unitary' (or only `unitary' when $n=2$) over some tensor product of two $*$-algebras ${\mathbb K}_1, {\mathbb K}_2$, which in all cases are obtained from ${\mathbb R}, {\mathbb C}, {\mathbb H}, {\mathbb O}$ and their split versions as sets, endowing them with a $*$-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation.