Here are some approaches from other points of view:

- Geoffrey Dixon
- John Baez - Division Algebras
- C.H. Barton and A. Sudbery
- J. M. Landsberg - Algebraic Geometry

The E6-E7-E8 structures are based on the Freudenthal-Tits Magic Square, which shows relationships between division algebras and matrix algebras. In particular: Division algebras define the rows of the Magic Square; Jordan algebras define the columns of the Magic Square; and Lie algebras define the entries of the Magic Square. The Jordan algebras are Hermitian matrices with a symmetric product. The Lie algebras are anti-Hermitian matrices with an antisymmetric product. The Magic Square includes ALL the exceptional Lie algebras, but only some of the A, B, C, and D series Lie algebras. There don't seem to be many references in standard textbooks. I first read about it in the article Jordan Algebras and their Applications, by Kevin McCrimmon, Bull. AMS 84 (1978) 612-627, at pp. 620-621 and in the unpublished 1976 Caltech notes of Pierre Ramond, who I think learned about it from Feza Gursey at Yale. Freudenthal wrote about it in Adv. Math. 1 (1964) 143, and Tits in Indag. Math. 28 (1966) 223-237. It is: (here I use Q for quaternion, and I follow McCrimmon except that I correct a misprint of A6 that should (I think) be D6, and I will also say that McCrimmon writes it as 4x5 instead of 4x4, so that it is rectangle instead of square - other authors omit the first column of McCrimmon's figure, and they get a 4x4 square, but they don't have G2 as the derivation algebra of octonions O.) R J3(R) J3(C) J3(Q) J3(O) R 0 A1 A2 C3 F4 C 0 A2 A2+A2 A5 E6 Q A1 C3 A5 D6 E7 O G2 F4 E6 E7 E8 The columns are labelled by Jordan algebras J = R, J3(R), J3(C), J3(Q), J3(O) (where J3(K) is the algebra of 3x3 Hermitian matrices over K) The rows are labelledby composition algebras A = R, C, Q, O It is a 4x5 rectangle, but the right 4 columns make a 4x4 square. The 4x5 Magic Square entries are Lie algebras L formed by the rule: L = Der(A) + (A0xJ0) + Der(J) where Der means derivation, + is direct sum, x is tensor product, A0 are the pure imaginary elements of A, R0=S0, C0=S1, Q0=S3, O0=S7,(here Sn means the algebra of tangent vectors to an n-dim sphere, and S0,S1,S3 are Lie algebras and S7 is a Malcev algebra), and J0 are the elements of trace zero of the Jordan algebra J. (McCrimmon shows what the Lie products of such elements are, on page 620 of his article in Bull. AMS). Further, notice that A1 is SU(2), A2 is SU(3), A5 is SU(6), C3 is Sp(3) (denoted by some people Sp(6)), D6 is SO(12) (same Lie algebra as Spin(12), and G2, F4, E6, E7, and E8 are the exceptional Lie algebras. The corresponding exceptional Lie groups are the subject of the book Lectures on Exceptional Lie Groups by J. F. Adams, published posthumously by Un. of Chicago Press in 1996, edited by Zafer Mahmud and Mamoru Mimura. To try to make the magic squares a little clearer, here is the Magic Square with the dimensions of the Lie algebras as entries: R J3(R) J3(C) J3(Q) J3(O) R 0 3 8 21 52 C 0 8 16 35 78=52+1x26 Q 3 21 35 66 133=52+3x26+3 O 14 52 78 133 248=52+7x26+14 R J3(R) J3(C) J3(Q) R 0 3 8 21 C 0 8 16 35=21+1x14 Q 3 21 35 66=21+3x14+3 O 14 52 78 133=21+7x14+14 R J3(R) J3(C) R 0 3 8 C 0 8 16=8+1x8 Q 3 21 35=8+3x8+3 O 14 52 78=8+7x8+14 R J3(R) R 0 3 C 0 8=3+1x5 Q 3 21=3+3x5+3 O 14 52=3+7x5+14 To help a little more, consider the dimension of3x3 matrices with entries Aij such that Aij = Aji*(where * denotes conjugate) If the entries of Aij are of dimension k then the diagonal Aii are real, and the matrix dimension is 3k + 3,and if trace = sum of diagonals = 0, the dimension is 3k + 2. Therefore:for reals: 3x1 + 2 = 5;for complex: 3x2 + 2 = 8;for quaternion: 3x4 + 2 = 14;for octonion: 3x8 + 2 = 26. As for the first column,which is rectangle part, not really part of the 4x4 square: R R 0 (no imaginary part) C 0 (imaginary part dissipates) Q 3 (imaginary part stable) 3 is dim of SU(2) O 14=7+7 (imaginary part expands) 14 is dim of G2

GeoffreyDixon shows how to construct F4, E6, E7, and E8 from a slightlydifferent point of view:

- F4 = Spin(8) + SO(3) + 3x7 = 28 + 3 + 3x7 = 52
- E6 = Spin(8) + SU(3) + 6x7 = 28 + 8 + 6x7 = 78
- E7 = Spin(8) + Sp(3) + 12x7 = 28 + 21 + 12x7 = 133
- E8 = Spin(8) + F4 + 24x7 = 28 + 52 + 24x7 = 248

To get from Geoffrey Dixon's construction to my construction,

use the fibrations S7 = Spin(8) / Spin(7) and S7 = Spin(7) /G2

to break the 28 of Spin(8) into 14 + 7 + 7 of G2, S7, and S7,

and add the two 7's to his 3x7, 6x7, 12x7, and 24x7

to get my 5x7, 8x7, 14x7, and 26x7.

John Baez writes This Weeks Finds in Math Physics on the WWW. His Week 64 describes E6; his Week 90 describes E8; and his Week 91 describes triality. The Freudenthal-Tits Magic Square can be formulated in the terms of his description of E8: Start with D4 = Spin(8): 28 = 28 + 0 + 0 + 0 + 0 + 0 + 0 Add spinors and vector to get F4: 52 = 28 + 8 + 8 + 8 + 0 + 0 + 0 Now, "complexify" the 8+8+8 part of F4 to get E6: 78 = 28 + 16 + 16 + 16 + 1 + 0 + 1 Then, "quaternionify" the 8+8+8 part of F4 to get E7: 133 = 28 + 32 + 32 + 32 + 3 + 3 + 3 Finally, "octonionify" the 8+8+8 part of F4 to get E8: 248 = 28 + 64 + 64 + 64 + 7 + 14 + 7 This way shows you that the "second" Spin(8) in E8 breaks down as 28 = 7 + 14 + 7 which is globally like two 7-spheres and a G2, one S7 for left-action, one for right-action, and a G2 automorphism group of octonions that is needed to for "compatibility" of the two S7s. the 3+3+3 of E7, the 1+0+1 of E6, and the 0+0+0 of F4 and D4 are the quaternionic, complex, and real analogues of the 7+14+7.

In MAGIC SQUARES OF LIE ALGEBRAS, math.RA/0001083,C.H. Barton and A. Sudbery say: "... This paper is an investigationof the relation between Tit's magic square of Lie algebras andcertain Lie algebras of 3 x 3 and 6 x 6 matrices with entries inalternative algebras. By reformulating Tit's definition in terms oftrialities (a generalisation of derivations), we give a systematicexplanation of the symmetry of the magic square. We show that whenthe columns of the magic square are labelled by the real divisionalgebras and the rows by their split versions, then the rows can beinterpreted as analogues of the matrix Lie algebras su(3), sl(3) andsp(6) defined for each division algebra. We also define another magicsquare based on 2 x 2 and 4 x 4 matrices and prove that it consistsof various orthogonal or (in the split case) pseudo-orthogonal Liealgebras. ...

... Tits ... showed ... the so-called magic square of Lie algebrasof 3 x 3 matrices whose complexifications are

R C H O R A1 A2 C3 F4C A2 A2xA2 A5 E6H C3 A5 B6 E7O F4 E6 E7 E8

The striking properties of this square are (a) its symmetry and(b) the fact that four of the five exceptional Lie algebras occur inits last row. ... The fifth exceptional Lie algebra, G2, can beincluded by adding a extra row corresponding to the Jordan algebra R. ...

... most exceptional Lie algebras are related to the exceptionalJordan algebra of 3 x 3 hermitian matrices with entries from theoctonions, O. ... this relation yields descriptions of certain realforms of the complex Lie algebras

- F4 ... which can be interpreted as octonionic versions of ... the Lie algebra of antihermitian 3 x 3 matrices ...
- E6 ... which can be interpreted as octonionic versions of ... the Lie algebra of .... special linear 3 x 3 matrices and ...
- E7 which can be interpreted as octonionic versions of ... the Lie algebra of ... symplectic 6 x 6 matrices. ...".

In math.AG/9810140,J. M. Landsberg and Laurent Manivel say: "... This is the first paperin a series establishing new relations between the representationtheory of complex simple Lie groups and thealgebraic and differential geometry of their homogeneous varieties.In this paper we determine the varieties of linear spaces on rationalhomogeneous varieties, provide explicit geometric models for thesespaces, and establish basic facts about the local differentialgeometry of rational homogeneous varieties. Let G be a complex simpleLie group, P a maximal parabolic subgroup. The space of lines on G /P in its minimal homogeneous embedding was determined ...[ byCohen and Cooperstein ]... in terms of Lie incidence systems.There is a dichotomy between the cases for which the simple rootassociated to P is short or not: for non-short roots, the space oflines on G / P is G-homogeneous and can be described using ideas ofTits; for short roots, it is not G-homogeneous. ... We present arefinement of their result ... (for a parabolic subgroup P which doesnot need to be maximal), that each connected component of the spaceof lines consists of exactly two G-orbits. ... We explain how todetermine the higher dimensional linear spaces associated tonon-short roots using Tits methods. For short roots, we provideexplicit descriptions of the spaces we study, especially in theexceptional cases where we use Cayley'soctonions. In all cases, each connected component of the varietyof linear spaces on a G / P is quasi-homogeneous; more precisely, itis the union of a finite number of G-orbits. ...

... Here is a table of the G-minuscule varieties: there are fourinfinite series and two exceptional spaces.

... Here E and Q are the tautological and quotient vector bundleson the Grassmannian or their pullbacks to the varieties in question.S+ is the half spin representation of D5, and J3(O) is the space of 3x 3 O-Hermitian matrices, the representation V_w1 for E6 ... G_w(O3,O6 ) may be interpreted as the space of O3 's in O6 that are nullfor an O-Hermitian symplectic form ...

... As an algebraic variety, G2 / P1 is a familiar space, G2 / P1= Q5 in P6 . ... **G2**is not really an exceptional group, because it is defined by ageneric form. ... The ... interpretation can be understood in termsof folding Dynkin diagrams:

This indicates that G2 / P1 should be be understandable in termsof D4 / P1 = Q6 , and in fact it is a generic hyperplane section. ImO in O should be thought of as the traceless elements, where thetrace of an element is its "real" part and we call the hyperplanesection { tr = 0 } . ...

... J3(O) ... a Jordan algebra ... Thereis a well-defined determinant on J3(O ), which is defined by sameexpression as the classical determinant in terms of traces:

... F4 / P4 = OP2_0 . ... The description [of] ...**F4** in GL(J3(O))... [as F4 = { g in E6 | g+ = g- } ... is motivated by folding ofDynkin diagrams:

... Note that F4 is generated by SO3 and Spin8 ... This defines anautomorphism of the Jordan algebra J3( O)_0 because of the trialityprinciple. ...

... **E6** is the subgroup ofGL( J 3(O )) = GL(27, C ) preserving det. The notion of rank onematrices is also well defined and the Cayley plane, E6 / P1 = OP2 inP(J3(O)) is the projectivization of the rank one elements, with idealthe 2 x 2 minors ... Since alpha_1 is not short, all linear spaces onOP2 are described by Tits geometries. In particular, E6/ P3 is thespace of lines on OP2 and E6 / P2 is the space of P5 's on OP2...

In math.AG/9902102,J. M.Landsberg and Laurent Manivel say: "... Complex simple Liealgebras were classified by Cartan and Killing 100 years ago. Theirproof proceeds by reducing the question to a combinatorical problem:the classification of irreducible root systems, and then performingthe classification. We present a new proof of the classification viathe projective geometry of homogeneous varieties. Our proof isconstructive: we build a homogeneous space X in PN from a smallerspace Y in Pn via a rational map Pn -> PN defined using the idealsof the secant and tangential varieties of Y. Our proof has threesteps.

- Among homogeneous varieties, there is a preferred class, the minuscule varieties ...
- We next construct all the fundamental adjoint varieties from certain minuscule varieties. ...
- Finally, we prove that these are all the adjoint varieties except for the two "exceptional" cases of Am and Cm. ...

... Our proof can be translated into a combinatorical argument:the construction consists of two sets of rules for adding new nodesto marked Dynkin diagrams. As a combinatorical algorithm, it is lessefficient than the standard proof ...

... minuscule representations define algebraic structures that arecousins of Clifford algebras ... the raisingand lowering action corresponds to Clifford multiplication. ...

In math.AG/9908039,J. M.Landsberg and L. Manivel say: "... We connect the algebraicgeometry and representation theory associated to Freudenthal's magicsquare. We give unified geometric descriptions of several classes oforbit closures, describing their hyperplane sections anddesingularizations, and interpreting them in terms of compositionalgebras. In particular, we show how a class of invariant quarticpolynomials can be viewed as generalizations of the classicaldiscriminant of a cubic polynomial. ...

... Freudenthal associates to each group in the square a set ofpreferred homogeneous varieties (k-spaces for each group in the k-throw). These spaces have the same incidence relations with thecorresponding varieties for the groups in the same row. He calls thegeometries associated to the groups of the rows respectively,2-dimensional elliptic, 2-dimensional plane projective, 5-dimensionalsymplectic and metasymplectic. The distinguished spaces are calledrespectively, spaces of points, lines, planes and symplecta. To avoidconfusion, we will use the terminology F-points, F-planes etc. ...While Freudenthal was interested in the synthetic/axiomatic geometryof the spaces, we are primarily interested in the spaces assubvarieties of a projective space. ...

The Severi varieties ... the projective planes over thecomposition algebras ... have the unusual property that a generichyperplane section of a Severi variety is still homogeneous. Puttingthe resulting varieties into a chart we have:

... These varieties are homogeneous spaces of groups whoseassociated Lie algebras are:

This chart is called Freudenthal's magic square of semi-simple Liealgebras.

The magic square was constructed by Freudenthal and Tits asfollows: Let A denote a complex composition algebra (i.e. thecomplexification of R , C , the quaternions H or the octonions O ).For a pair ( A ; B ) of such composition algebras, the correspondingLie algebra is

... To deduce the first line of the magic square from the secondone we use the folding of a root system. ... Here is a chartsummarizing the representations arising from folding:

... the notations are explained ...[ in the paper]...".

In math.AG/0107032,J. M.Landsberg and L. Manivel say: "... we thought it might beinteresting to parametrize the exceptional series by a = dim CA ,where A is respectively the complexification of 0, R, C, H, O for thelast five algebras in the exceptional series (so a = 0, 1, 2, 4, 8).... The construction we use highlights the triality principle, sincewe put a natural Lie algebra structure on the direct sum

where t( A ) is a certain triality algebra associated to A . ...If A is a real Cayley algebra, it is a classical fact that T( A ) isan algebraic group of type D4. ... we get the following types for theLie algebra t( A ) of T( A ) ...

t( R ) t( C ) t( H ) t( O ) 0 R2 so3 x so3 x so3 so8

... For B = O , our construction gives the last line ofFreudenthal square. ... Consider the case of e8, i.e., A = O . ...Now we make a few observations on the weightsof Ai. ... The weight structure is as follows:

...".

Here are some more references: the book Nonassociative Algebras in Physics, by the Estonians Jaak Lohmus, Eugene Paal, and Leo Sorgsepp, Hadronic Press (1994) p. 58 and the book Division Algebras by Geoffrey Dixon, Kluwer (1994) chapter 8 and the article Division Algebras ... by A. Sudbery J. Phys. A: Math. Gen. 17 (1984) 939-955

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