"Cayley-Klein" schemes for real Lie algebras and Freudhental Magic Squares
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
3×3
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"
3×3
Freudenthal-like squares admit a
4×4
extension, which gives the original Freudenthal square and the Sudbery square.