Freudenthal magic square

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. It associates a Lie group (or corresponding Lie algebra) to a pair of division algebras "A", "B". The resulting Lie algebras have Dynkin diagrams according to the following table.

Barton and Sudbery then identify the magic square Lie algebra corresponding to ("A","B") with a Lie algebra structure on the vector space:$mathfrak\{tri\}(A)oplusmathfrak\{tri\}(B)oplus\; (A\_1otimes\; B\_1)oplus\; (A\_2otimes\; B\_2)oplus\; (A\_3otimes\; B\_3).$The Lie bracket is compatible with a Z₂×Z₂ grading, with tri("A") and tri("B") in degree (0,0), and the three copies of $Aotimes\; B$ in degrees (0,1), (1,0) and (1,1). The bracket preserves tri("A") and tri("B") and these act naturally on the three copies of $Aotimes\; B$, as in the other constructions, but the brackets between these three copies are more constrained.

For instance when "A" and "B" are the octonions, the triality is that of Spin(8), the double cover of SO₈, and the Barton-Sudbery description yields:$mathfrak\; e\_8cong\; mathfrak\{so\}\_8oplushat\{mathfrak\{so\_8oplus(Votimes\; hat\; V)oplus\; (S\_+otimeshat\; S\_+)oplus\; (S\_-otimes\; hat\; S\_-)$where V, S₊ and S_- are the three 8 dimensional representations of $mathfrak\{so\}\_8$ (the fundamental representation and the two spin representations), and the hatted objects are an isomorphic copy.

With respect to one of the Z₂ gradings, the first three summands combine to give $mathfrak\{so\}\_\{16\}$ and the last two together form one of its spin representations Δ₊¹²⁸ (the superscript denotes the dimension). This is a well known symmetric decomposition of E8.

The Barton-Sudbery construction extends this to the other Lie algebras in the magic square. In particular, for the exceptional Lie algebras in the last row (or column), the symmetric decompositions are::$mathfrak\; f\_4cong\; mathfrak\{so\}\_9oplus\; Delta^\{16\}$:$mathfrak\; e\_6cong\; (mathfrak\{so\}\_\{10\}oplus\; mathfrak\; u\_1)oplus\; Delta^\{32\}$:$mathfrak\; e\_7cong\; (mathfrak\{so\}\_\{12\}oplus\; mathfrak\{sp\}\_1)oplus\; Delta\_+^\{64\}$:$mathfrak\; e\_8cong\; mathfrak\{so\}\_\{16\}oplus\; Delta\_+^\{128\}.$

Generalizations

plit composition algebras

In addition to the normed division algebras, there are other composition algebras over R, namely the split-complex numbers, the split-quaternions and the split-octonions. 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 dash).

The real exceptional Lie algebras appearing here can again be described by their maximal compact subalgebras.

For "n"=2, J_"2"("O") is also a Jordan algebra. In the compact case (over R) this yields a magic square of orthogonal Lie algebras.

The last row and column here are the orthogonal algebra part of the isotropy algebra in the symmetric decomposition of the exceptional Lie algebras mentioned previously.

These constructions are closely related to hermitian symmetric spaces – cf. prehomogeneous vector spaces.

ee also

* E8 (mathematics)
* E7 (mathematics)
* E6 (mathematics)
* F4 (mathematics)
* G2 (mathematics)
* Jordan triple system

References

* John Frank Adams (1996), "Lectures on Exceptional Lie Groups" (Chicago Lectures in Mathematics), edited by Zafer Mahmud and Mamora Mimura, University of Chicago Press, ISBN 0-226-00527-5.
*; also available [http://math.ucr.edu/home/baez/octonions/node16.html here] , [http://www.arxiv.org/abs/math/0105155v1 arXiv:math.AG/0105155v1] .
* C. H. Barton and A. Sudbery (2003), "Magic squares and matrix models of Lie algebras", Adv. in Math. 180 (2003), 596-647, [http://arxiv.org/abs/math.RA/0203010 arXiv:math.RA/0203010] .
*J.M. Landsberg and L. Manivel (2001), "The projective geometry of Freudenthal's magic square", Journal of Algebra, Volume 239, Issue 2, pages 477-512, doi|10.1006/jabr.2000.8697, [http://www.arxiv.org/abs/math/9908039 arXiv:math.AG/9908039v1] .
* Pierre Ramond (1976), " [http://www.slac.stanford.edu/spires/find/hep/www?r=CALT-68-577 Introduction to Exceptional Lie Groups and Algebras] ", CALT-68-577, California Institute of Technology, Pasadena.