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

Mariano Santander·Francisco J. Herranz
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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.

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Mariano Santander, Francisco J. Herranz. 1997-02-24. "Cayley-Klein" schemes for real Lie algebras and Freudhental Magic Squares. https://arxiv.org/abs/physics/9702031

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