"Cayley-Klein" schemes for real Lie algebras and Freudhental Magic Squares

TL;DR

This paper introduces Cayley-Klein families of Lie algebras using matrix realizations over various number systems, leading to infinite Freudenthal-like magic squares that connect different algebraic structures.

Contribution

It develops new Cayley-Klein schemes for Lie algebras and constructs extended Freudenthal-like magic squares relating these algebras.

Findings

Infinite family of 3x3 Freudenthal-like magic squares derived

Extensions involving octonions and 4x4 squares for classical cases

Connections between real, complex, and quaternionic Lie algebras

Abstract

We introduce three "Cayley-Klein" families of Lie algebras through realizations in terms of either real, complex or quaternionic matrices. Each family includes simple as well as some limiting quasi-simple real Lie algebras. Their relationships naturally lead to an infinite family of $3\times 3$ Freudenthal-like magic squares, which relate algebras in the three CK families. In the lowest dimensional cases suitable extensions involving octonions are possible, and for $N=1, 2$, the "classical" $3\times 3$ Freudenthal-like squares admit a $4\times 4$ extension, which gives the original Freudenthal square and the Sudbery square.