In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups).
It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently.
The resulting Lie algebras have Dynkin diagrams according to the table at the right.
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": G2 itself is the automorphism group of the octonions (also, it 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).
These were originally constructed circa 1958 by Freudenthal and Tits, with more elegant formulations following in later years.
(which is not a subalgebra) is not obvious, but Tits showed how it could be defined, and that it produced the following table of compact Lie algebras.
In the presence of derivations, these form a subalgebra acting naturally on
A more recent construction, due to Pierre Ramond (Ramond 1976) and Bruce Allison (Allison 1978) and developed by Chris Barton and Anthony Sudbery, uses triality in the form developed by John Frank Adams; this was presented in (Barton & Sudbery 2000), and in streamlined form in (Barton & Sudbery 2003).
Whereas Vinberg's construction is based on the automorphism groups of a division algebra A (or rather their Lie algebras of derivations), Barton and Sudbery use the group of automorphisms of the corresponding triality.
The triality is the trilinear map obtained by taking three copies of the division algebra A, and using the inner product on A to dualize the multiplication.
The automorphism group is the subgroup of SO(A1) × SO(A2) × SO(A3) preserving this trilinear map.
For instance when A and B are the octonions, the triality is that of Spin(8), the double cover of SO(8), and the Barton-Sudbery description yields where V, S+ and S− are the three 8-dimensional representations of
and the last two together form one of its spin representations Δ+128 (the superscript denotes the dimension).
The Barton–Sudbery construction extends this to the other Lie algebras in the magic square.
If one uses these instead of the complex numbers, quaternions, and octonions, one obtains the following variant of the magic square (where the split versions of the division algebras are denoted by a prime).
Here all the Lie algebras are the split real form except for so3, but a sign change in the definition of the Lie bracket can be used to produce the split form so2,1.
According to Barton and Sudbery, the resulting table of Lie algebras is as follows.
The real exceptional Lie algebras appearing here can again be described by their maximal compact subalgebras.
In the case K = C, this is the complexification of the Freudenthal magic squares for R discussed so far.
There are also Jordan algebras Jn(A), for any positive integer n, as long as A is associative.
In the compact case (over R) this yields a magic square of orthogonal Lie algebras.
These constructions are closely related to hermitian symmetric spaces – cf.
Riemannian symmetric spaces, both compact and non-compact, can be classified uniformly using a magic square construction, in (Huang & Leung 2010).
A similar construction produces the irreducible non-compact symmetric spaces.
Following Ruth Moufang's discovery in 1933 of the Cayley projective plane or "octonionic projective plane" P2(O), whose symmetry group is the exceptional Lie group F4, and with the knowledge that G2 is the automorphism group of the octonions, it was proposed by Rozenfeld (1956) that the remaining exceptional Lie groups E6, E7, and E8 are isomorphism groups of projective planes over certain algebras over the octonions:[1] This proposal is appealing, as there are certain exceptional compact Riemannian symmetric spaces with the desired symmetry groups and whose dimension agree with that of the putative projective planes (dim(P2(K ⊗ K′)) = 2 dim(K)dim(K′)), and this would give a uniform construction of the exceptional Lie groups as symmetries of naturally occurring objects (i.e., without an a priori knowledge of the exceptional Lie groups).
The Riemannian symmetric spaces were classified by Cartan in 1926 (Cartan's labels are used in sequel); see classification for details, and the relevant spaces are: The difficulty with this proposal is that while the octonions are a division algebra, and thus a projective plane is defined over them, the bioctonions, quateroctonions and octooctonions are not division algebras, and thus the usual definition of a projective plane does not work.
More broadly, these compact forms are the Rosenfeld elliptic projective planes, while the dual non-compact forms are the Rosenfeld hyperbolic projective planes.
[1] While at the level of manifolds and Lie groups, the construction of the projective plane P2(K ⊗ K′) of two normed division algebras does not work, the corresponding construction at the level of Lie algebras does work.
That is, if one decomposes the Lie algebra of infinitesimal isometries of the projective plane P2(K) and applies the same analysis to P2(K ⊗ K′), one can use this decomposition, which holds when P2(K ⊗ K′) can actually be defined as a projective plane, as a definition of a "magic square Lie algebra" M(K,K′).
This definition is purely algebraic, and holds even without assuming the existence of the corresponding geometric space.