Generalizations Freudenthal magic square

1 generalizations

1.1 split composition algebras
1.2 arbitrary fields
1.3 more general jordan algebras
1.4 symmetric spaces

generalizations
split composition algebras

in addition normed division algebras, there other composition algebras on r, namely split-complex numbers, split-quaternions , split-octonions. if 1 uses these instead of complex numbers, quaternions, , octonions, 1 obtains following variant of magic square (where split versions of division algebras denoted dash).

here lie algebras split real form except so3, sign change in definition of lie bracket can used produce split form so2,1. in particular, exceptional lie algebras, maximal compact subalgebras follows:

a non-symmetric version of magic square can obtained combining split algebras usual division algebras. according barton , sudbery, resulting table of lie algebras follows.

the real exceptional lie algebras appearing here can again described maximal compact subalgebras.

arbitrary fields

the split forms of composition algebras , lie algebras can defined on field k. yields following magic square.

there ambiguity here if k not algebraically closed. in case k = c, complexification of freudenthal magic squares r discussed far.

more general jordan algebras

the squares discussed far related jordan algebras j3(a), division algebra. there jordan algebras jn(a), positive integer n, long associative. these yield split forms (over field k) , compact forms (over r) of generalized magic squares.

for n=2, j2(o) jordan algebra. in compact case (over r) yields magic square of orthogonal lie algebras.

the last row , column here orthogonal algebra part of isotropy algebra in symmetric decomposition of exceptional lie algebras mentioned previously.

these constructions closely related hermitian symmetric spaces – cf. prehomogeneous vector spaces.

symmetric spaces

riemannian symmetric spaces, both compact , non-compact, can classified uniformly using magic square construction, in (huang & leung 2011). irreducible compact symmetric spaces are, finite covers, either compact simple lie group, grassmannian, lagrangian grassmannian, or double lagrangian grassmannian of subspaces of



(

a

⊗

b


)

n


,


{\displaystyle (\mathbf {a} \otimes \mathbf {b} )^{n},}

normed division algebras , b. similar construction produces irreducible non-compact symmetric spaces.