Freudenthal magic square

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In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie groups. It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently.

The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the octonions". In mitigation of the omission, it can be argued that G2 is in many ways like a classical Lie group, because it is the stabilizer of a generic 3-form on a 7-dimensional vector space (see prehomogeneous vector space). In any case, G2 is the automorphism group of the octonions.

Associated with any real division algebra A (i.e., R, C, H or O) is a Jordan algebra

J3(A)

of 3×3 A-Hermitian matrices. For any pair

(A,B)

of such division algebras, one can define a Lie algebra

L=\mathfrak{der}(A)\oplus(A_0\oplus J_3(B)_0)\oplus\mathfrak{der}(J_3(B))

where der denotes the Lie algebra of derivations of an algebra, and the subscript 0 denotes the trace-free part. The Lie algebra structure is not completely obvious, but Freudenthal showed how it could be defined, and that it produced the following table of Lie algebras.

A \ B R C H O
R A1 A2 C3 F4
C A2 A2 × A2 A5 E6
H C3 A5 D6 E7
O F4 E6 E7 E8

[edit] References

  • The projective geometry of Freudenthal’s magic square, J.M. Landsberg and L. Manivel, 1999.