"... a semisimple GLA [Graded Lie Algebra] has the form
g = SUM(-v < k < +v) g(k)
with g(-v) =/= 0. Such a GLA is called a GLA of the v-th kind. ... the pair (Z,t) is the associated pair, where Z is the characteristic element and t is a grade-reversing Cartan involution. ... Let g ... be a real simple GLA of the v-th kind, and (Z,t) be the associated pair. Let Go be the group of grade-preserving automorphisms of G. ... Let U = Go exp(g(1) + ... + g(v)), which is a parabolic subgroup of G. The real flag manifold M = G/U is called a flag manifold of the v-th kind. ...
... The Shilov boundary of an irreducible bounded symmetric domain is a flag manifold of the 1st kind or of the 2nd kind, according as the domain is of tube type or not. ... For an irreducible symmetric domain of tube type with dimension greater than 1, we show the coincidence of the causal automorphism group of the Shilov boundary and the full holomorphic automorphism group of the domain. ...
... The class of symmetric R-spaces contains the Shilov boundaries of bounded symmetric domains of tube type. ... By a symmetric R-space we mean a compact irreducible Hermitian symmetric space or a real form (i.e., the set of fixed points by an involutive anti-holomorphic isometry) of it. ... The purpose of these notes is to give an introduction and survey of recent results on semisimple pseudo-Hermitian symmetric spaces. ...
... In the following tables we use the following notation: H the quaternion algebra over R, O (resp. O') the Cayley (resp. split Cayley) algebra over R, and OC = O (x)R C. Mp,q(F) the vector space of p x q matrices with entries in F, where F = R, C, H, O, O' or OC; SHn(H) the vector space of skew-hermitian quaternion matrices of degree n; SYMn(C) the vector space of complex symmetric matrices of degree n.; ALTn(F) the vector space of alternating F-matrices of degree n; Hn(F) the vector space of F-hermitian matrices of degree n. ...[ II a restricted fundamental root system of g; II1 the part of II corresponding to grade g(1); a used for alpha and y used for gamma ]...
... Table 1 TABLE OF SIMPLE GLA'S OF THE FIRST KIND
(I1) g = sl(n,R) , n >= 3, 1 <= p <= [n/2],
II = A(n-1),
II1 = {ap},
g(0) = sl(p,R) + sl((n-p),R) + R ,
g(-1) = Mp,(n-p)(R).
(I2) g = sl(n,H) , n >= 3, 1 <= p <= [n/2],
II = A(n-1),
II1 = {ap},
g(0) = sl(p,H) + sl((n-p),H) + R ,
g(-1) = Mp,(n-p)(H).
(I3) g = su(n,n) , n >= 3,
II = Cn,
II1 = {an},
g(0) = sl(n,C) + R ,
g(-1) = Hn(C).
(I4) g = sp(n,R) , n >= 3,
II = Cn,
II1 = {an},
g(0) = sl(n,R) + R ,
g(-1) = Hn(R).
(I5) g = sp(n,n) , n >= 2,
II = Cn,
II1 = {an},
g(0) = sl(n,H) + R ,
g(-1) = SHn(H).
(I6) g = so(p+1,q+1) , 0 <= p < q or 3 <= p = q,
II = B(p+1) for p < q or D(p+1) for p = q,
II1 = {a1},
g(0) = so(p,q) + R ,
g(-1) = M1,(p+q)(R).
(I7) g = so*(4n) , n >= 3,
II = Cn,
II1 = {an},
g(0) = sl(n,H) + R ,
g(-1) = Hn(H).
(I8) g = so(n,n) , n >= 4,
II = Dn,
II1 = {an},
g(0) = sl(n,R) + R ,
g(-1) = ALTn(R).
(I9) g = E6(6) ,
II = E6 ,
II1 = {a1},
g(0) = so(5,5) + R ,
g(-1) = M1,2(O').
(I10) g = E6(-26) ,
II = A2 ,
II1 = {a1},
g(0) = so(1,9) + R ,
g(-1) = M1,2(O).
(I11) g = E7(7) ,
II = E7 ,
II1 = {a7},
g(0) = E6(6) + R ,
g(-1) = H3(O').
(I12) g = E7(-25) ,
II = C3 ,
II1 = {a3},
g(0) = E6(-26) + R ,
g(-1) = H3(O).
(I13) g = sl(n,C) , n >= 3, 1 <= p <= [n/2],
II = A(n-1),
II1 = {ap},
g(0) = sl(p,C) + sl((n-p),C) + C ,
g(-1) = Mp,(n-p)(C).
(I14) g = sp(n,C) , n >= 3,
II = Cn,
II1 = {an},
g(0) = sl(n,C) + C ,
g(-1) = SYMn(C).
(I15) g = so(n+2)C ,
II = B[(n+2)/2] or D[(n+2)/2] ,
II1 = {a1},
g(0) = so(n)C + C ,
g(-1) = M1,n(C).
(I16) g = so(2n,C) , n >= 4,
II = Dn,
II1 = {an},
g(0) = sl(n,C) + C ,
g(-1) = ALTn(C).
(I17) g = E6C ,
II = E6 ,
II1 = {a1},
g(0) = so(10)C + C ,
g(-1) = M1,2(O)C.
(I18) g = E7C ,
II = E7 ,
II1 = {a7},
g(0) = E6C + C ,
g(-1) = H3(O)C.
Table 2 TABLE OF CLASSICAL SIMPLE GLA'S OF THE SECOND KIND
(c1) g = sl(n,F) , N >= 3 , F = R or C ,
II = {y1, ... , y(n-1)} of type A(n-1),
II1 = {yp, y(p+q)}, 1 <= p <= [n/2], 1 <= q <= (n-2p),
g(0) = sl(p,F) + sl(q,F) + sl((n-p-q),F) + F + F,
g(-1) = Mp,q(F) x Mq,(n-p-q)(F),
g(-2) = Mp,(n-p-q)(F).
(c2) g = sl(n,H) , N >= 3 ,
II, II1 the same as in (c1) with the same conditions,
g(0) = sl(p,H) + sl(q,H) + sl((n-p-q),H) + R + R,
g(-1) = Mp,q(F) x Mq,(n-p-q)(F),
g(-2) = Mp,(n-p-q)(F).
(c3) g = su(p,q) , 1 <= p < q or 3 <= p=q ,
II = {y1, ... , yp} of type BCp (p<q),
or type Cp (p=q),
II1 = {yk}, 1 <= k <= p, if p<q,
or 1 <= k <= (p-1), if p=q,
g(0) = sl(k,C) + su((p-k),(q-k)) + R + iR,
g(-1) = Mp,(p+q-2k)(C),
g(-2) = Hk(C).
(c4) g = so(p,q) , 2 <= p < q, or 4 <= p=q ,
II = {y1, ... , yp} of type Bp (p<q),
or type Dp (p=q),
II1 = {yk}, 2 <= k <= p, if p<q,
or 2 <= k <= (p-2), if p=q,
g(0) = sl(k,R) + so((p-k),(q-k)) + R,
g(-1) = Mp,(p+q-2k)(R),
g(-2) = ALTk(R).
(c5) g = sp(n,F) , N >= 3 , F = R or C ,
II = {y1, ... , y(n)} of type Cn,
II1 = {yk}, 1 <= k <= (n-1),
g(0) = sl(k,F) + sp((n-k),F) + F,
g(-1) = Mk,(2n-2k)(F),
g(-2) = SYM,(C) if F=C, or Hk(R) if F=R.
(c6) g = sp(p,q) , 1 <= p < q, or 2 <= p=q ,
II = {y1, ... , yp} of type BCp (p<q),
or type Cp (p=q),
II1 = {yk}, 1 <= k <= p, if p<q,
or 1 <= k <= (p-1), if p=q,
g(0) = sl(k,H) + sp((p-k),(q-k)) + R,
g(-1) = Mp,(p+q-2k)(H),
g(-2) = SHk(H).
(c7) g = so*(2n) , n even >= 6, or n odd >= 5,
II = {y1, ... , y[n/2]} of type C[n/2] (n even),
or BC[n/2] (n odd),
II1 = {yk}, 1 <= k <= ([n/2]-1) if n even,
or 1 <= k <= [n/2] if n odd,
g(0) = sl(k,H) + so*(2n-4k) + R,
g(-1) = Mk,(n-2k)(H),
g(-2) = Hk(H).
(c8) g = so(n,n;F) , F = R or C,
II = {y1, ... , yn} of type Dn,
a) II1 = {y(n-1),yn} (N >= 4),
g(0) = sl((n-1),F) + F + F,
g(-1) = M(n-1),2(F),
g(-2) = ALT(n-1)(F).
b) II1 = {y1, yn}, (N >= 5),
g(0) = sl((n-1),F) + F + F,
g(-1) = M1,(n-1)(F) x ALT(n-1)(F),
g(-2) = F^(n-1).
(c9) g = so(n,C) , n odd >= 5, or n even >= 8,
II = {y1, ... , y[n/2]} of type B[n/2] (n odd),
or type D[n/2] (n even),
II1 = {yk}, 2 <= k <= [n/2] if n odd,
or 2 <= k <= ((n/2)-2) if n even,
g(0) = sl(k,C) + so((n-2k),C) + C,
g(-1) = Mk,(n-2k)(C),
g(-2) = ALTk(C).
Table 3 TABLE OF EXCEPTIONAL SIMPLE GLA'S OF THE SECOND KIND
(e1) g = E6(6),
II = E6,
II1 = {y3},
g(0) = sl(5,R) + sl(2,R) + R,
dimR g(-1) = 20,
dimR g(-2) = 5.
(e2) g = E6(6),
II = E6,
II1 = {y2},
g(0) = sl(6,R) + R,
dimR g(-1) = 20,
dimR g(-2) = 1.
(e3) g = E6(6),
II = E6,
II1 = {y1, y6},
g(0) = so(4,4) + R + R,
dimR g(-1) = 16,
dimR g(-2) = 8.
(e4) g = E6(2),
II = F4,
II1 = {y1},
g(0) = su(3,3) + R,
dimR g(-1) = 20,
dimR g(-2) = 1.
(e5) g = E6(2),
II = F4,
II1 = {y4},
g(0) = so(3,5) + R + iR,
dimR g(-1) = 16,
dimR g(-2) = 8.
(e6) g = E6(-14),
II = BC2,
II1 = {y1},
g(0) = su(1,5) + R,
dimR g(-1) = 20,
dimR g(-2) = 1.
(e7) g = E6(-14),
II = BC2,
II1 = {y2},
g(0) = so(1,7) + R + iR,
dimR g(-1) = 16,
dimR g(-2) = 8.
(e8) g = E6(-26),
II = A2,
II1 = {y1, y2},
g(0) = so(8) + R + R,
dimR g(-1) = 16,
dimR g(-2) = 8.
(e9) g = E7(7),
II = E7,
II1 = {y6},
g(0) = so(5,5) + sl(2,R) + R,
dimR g(-1) = 32,
dimR g(-2) = 10.
(e10) g = E7(7),
II = E7,
II1 = {y1},
g(0) = so(6,6) + R,
dimR g(-1) = 32,
dimR g(-2) = 1.
(e11) g = E7(7),
II = E7,
II1 = {y2},
g(0) = sl(7,R) + R,
dimR g(-1) = 35,
dimR g(-2) = 7.
(e12) g = E7(-5),
II = F4,
II1 = {y1},
g(0) = so*(12) + R,
dimR g(-1) = 32,
dimR g(-2) = 1.
(e13) g = E7(-5),
II = F4,
II1 = {y4},
g(0) = so(3,7) + su(2) + R,
dimR g(-1) = 32,
dimR g(-2) = 10.
(e14) g = E7(-25),
II = C3,
II1 = {y1},
g(0) = so(2,10) + R,
dimR g(-1) = 32,
dimR g(-2) = 1.
(e15) g = E7(-25),
II = E7,
II1 = {y2},
g(0) = so(1,9) + sl(2,R) + R,
dimR g(-1) = 32,
dimR g(-2) = 10.
(e16) g = E8(8),
II = E8,
II1 = {y8},
g(0) = E7(7) + R,
dimR g(-1) = 56,
dimR g(-2) = 1.
(e17) g = E8(8),
II = E8,
II1 = {y1},
g(0) = so(7,7) + R,
dimR g(-1) = 64,
dimR g(-2) = 14.
(e18) g = E8(-24),
II = F4,
II1 = {y1},
g(0) = E7(-25) + R,
dimR g(-1) = 56,
dimR g(-2) = 1.
(e19) g = E8(-24),
II = F4,
II1 = {y4},
g(0) = so(3,11) + R,
dimR g(-1) = 64,
dimR g(-2) = 14.
(e20) g = F4(4),
II = F4,
II1 = {y1},
g(0) = sp(3,R) + R,
dimR g(-1) = 14,
dimR g(-2) = 1.
(e21) g = F4(4),
II = F4,
II1 = {y4},
g(0) = so(3,4) + R,
dimR g(-1) = 8,
dimR g(-2) = 7.
(e22) g = F4(-20),
II = BC1,
II1 = {y1},
g(0) = so(7) + R,
dimR g(-1) = 8,
dimR g(-2) = 7.
(e23) g = G2(2),
II = G2,
II1 = {y2},
g(0) = sl(2,R) + R,
dimR g(-1) = 4,
dimR g(-2) = 1.
(e24) g = E6C,
II = E6,
II1 = {y3},
g(0) = sl(5,C) + sl(2,C) + C,
dimC g(-1) = 20,
dimC g(-2) = 5.
(e25) g = E6C,
II = E6,
II1 = {y2},
g(0) = sl(6,C) + C,
dimC g(-1) = 20,
dimC g(-2) = 1.
(e26) g = E6C,
II = E6,
II1 = {y1, y6},
g(0) = so(8,C) + C + C,
dimC g(-1) = 16,
dimC g(-2) = 8.
(e27) g = E7C,
II = E7,
II1 = {y6},
g(0) = so(10,C) + sl(2,C) + C,
dimC g(-1) = 32,
dimC g(-2) = 10.
(e28) g = E7C,
II = E7,
II1 = {y1},
g(0) = so(12,C) + C,
dimC g(-1) = 32,
dimC g(-2) = 1.
(e29) g = E7C,
II = E7,
II1 = {y2},
g(0) = sl(7,C) + C,
dimC g(-1) = 35,
dimC g(-2) = 7.
(e30) g = E8C,
II = E8,
II1 = {y8},
g(0) = E7C + C,
dimC g(-1) = 56,
dimC g(-2) = 1.
(e31) g = E8C,
II = E8,
II1 = {y1},
g(0) = so(14,C) + C,
dimC g(-1) = 64,
dimC g(-2) = 14.
(e32) g = F4C,
II = F4,
II1 = {y1},
g(0) = sp(3,C) + C,
dimC g(-1) = 14,
dimC g(-2) = 1.
(e33) g = F4C,
II = F4,
II1 = {y4},
g(0) = so(7,C) + C,
dimC g(-1) = 8,
diC g(-2) = 7.
(e34) g = G2C,
II = G2,
II1 = {y2},
g(0) = sl(2,C) + C,
dimC g(-1) = 4,
dimC g(-2) = 1.
... consider ... g(ev) = g(-2) + g(0) + g(2) ...
( g, g(ev) )
(c1) ( sl(n,F) , sl((n-q),F) + sl(q,F) + F ), F = R, C
(c2) ( sl(n,H) , sl((n-q),H) + sl(q,H) + R )
(c3) ( su(p,q) , su(k,k) + su((p-k),(q-k)) + iR )
(c4) ( so(p,q) , so(k,k) + so((p-k),(q-k)) )
(c5) ( sp(n,F) , sp(k,F) + sp((n-k),F) ), F = R, C
(c6) ( sp(p,q) , sp(k,k) + sp((p-k),(q-k)) )
(c7) ( so*(2n) , so*(4k) + so*(2n-4k) )
(c8) a. ( so(n,n) , so(1,1) + so((n-1),(n-1)) )
b. ( so(n,n) , so(n,R) + R )
aC. ( so(2n,C) , so(2,C) + so((2n-2),C) )
bC. ( so(2n,C) , sl(n,C) + C )
(c9) ( so(n,C) , so(2k,C) + so((n-2k),C) )
(e1) ( E6(6), sl(6,R) + sl(2,R) )
(e2) ( E6(6), sl(6,R) + sl(2,R) )
(e3) ( E6(6), so(5,5) + R )
(e4) ( E6(2), su(3,3) + sl(2,R) )
(e5) ( E6(2), so(4,6) + iR )
(e6) ( E6(-14), su(1,5) + sl(2,R) )
(e7) ( E6(-14), so(2,8) + iR )
(e8) ( E6(-26), so(1,9) + R )
(e9) ( E7(7), so(6,6) + sl(2,R) )
(e10) ( E7(7), so(6,6) + sl(2,R) )
(e11) ( E7(7), sl(8,R) )
(e12) ( E7(-5), so*(12) + sl(2,R) )
(e13) ( E7(-5), so(4,8) + su(2) )
(e14) ( E7(-25), so(2,10) + sl(2,R) )
(e15) ( E7(-25), so(2,10) + sl(2,R) )
(e16) ( E8(8), E7(7) + sl(2,R) )
(e17) ( E8(8), so(8,8) )
(e18) ( E8(-24), E7(-25) + sl(2,R) )
(e19) ( E8(-24), so(4,12) )
(e20) ( F4(4), sp(3,R) + sl(2,R) )
(e21) ( F4(4), so(4,5) )
(e22) ( F4(-20), so(1,8) )
(e23) ( G2(2), sl(2,R) + sl(2,R) )
(e24) ( E6C, sl(6,C) + sl(2,C) )
(e25) ( E6C, sl(6,C) + sl(2,C) )
(e26) ( E6C, so(10,C) + C )
(e27) ( E7C, so(12,C) + sl(2,C) )
(e28) ( E7C, so(12,C) + sl(2,C) )
(e29) ( E7C, sl(8,C) )
(e30) ( E8C, E7C + sl(2,C) )
(e31) ( E8C, so(16,C) )
(e32) ( F4C, sp(3,C) + sl(2,C) )
(e33) ( F4C, so(9,C) )
(e34) ( G2C, sl(2,C) + sl(2,C) ) ...".
Consider a 5-level Graded Lie Algebra
g =
g(-2)
+ g(-1)
+ g(0)
+ g(1)
+ g(2)
and its corresponding 3-level structure in which g(-2) g(0) and g(2) are combined to form g(ev)
g =
+ g(-1)
+ g(ev)
+ g(1)
E6(-14) =
8-dim
+ 16-dim
+ R + so(1,7) + iR
+ 16-dim
+ 8-dim
with physical interpretation in the Lagrangian of the D4-D5-E6-E7-E8 VoDou Physics Model as:
By using only the Shilov boundary for representing 8-dimensional spacetime and the 8+8 = 16 first-generation fermions, we have not yet given physical interpretation to these parts of E6:
+ R + iR
+ 16-dim
+ 8-dim
Physically, they correspond to another copy of 8-dimensional spacetime, another copy of the 8 fermion particles, another copy of the 8 fermion antiparticles, and 2 U(1)-type symmetry dimensions. Mathematically, they correspond to the symmetric space E6 / F4 which can be represented by three Octonions Ov, O+, and O-, and two Real numbers a and b, which, in turn, can be represented by 8+8+8+1+1 = 26-dimensional traceless 3x3 Hermitian Octonion matrices H3(O)o:
a O+ Ov O+* - O- Ov* O-* b
A point in that 26-dimensional space corresponds (up to the two U(1)-type symmetries) to a configuration of fermion particle and/or antiparticle at a point in spacetime. A line in that 26-dimensional space looks like a string and corresponds to a world-line describing history of a particle (closed because globally time of S1 x S7 spacetime is cyclic S1, although the time scale may be so large that we don't readily see its cyclic nature). The physics of strings in that 26-dimensional space describes a Bohm Quantum Potential.
As an aside, note that the 3-level structure corresponding to the GLA of type e7 is
E6(-14) =
+ 16-dim
+ so(2,8) + iR
+ 16-dim
where so(2,8) is the double cover of the Conformal group over the Octonions.
E7(-26) =
+ H3(O)
+ E6(-26) + R
+ H3(O)
where
the E6 of g(0) corresponds to a configuration of fermions and gauge bosons at a given point of 8-dimensional spacetime; and
the H3(O) of g(-1) plus the R of g(0) plus the H3(O) of g(1) correspond to a 27-Complex-dimensional Complexified H3(O) Jordan algebra ( also denoted by J3(O) ) plus a U(1) Complex symmetry generated by the R, which complex domain has as its Shilov boundary (1+26)=27-dimensional S1 x H3(O)o which in turn represents a 27-dimensional bosonic timelike brane M-theory of the MacroSpace of the Many-Worlds.
E8(8) =
1-dim
+ 56-dim
+ R + E7(7)
+ 56-dim
+ 1-dim
The corresponding 3-level structure is
E8(8) =
+ 56-dim
+ E7(7) + sl(2,R)
+ 56-dim
where
the E7 of g(ev) = sl(2,R) + E7(7) corresponds to a timelike brane M-theory; and
the 56-dim of g(-1) plus the sl(2,R) of g(ev) plus the 56-dim of g(1) correspond to a 28-Quaternion-dimensional Quaternified J4(Q) Jordan algebra plus a SU(2) = Sp(1) Quaternionic symmetry generated by the sl(2,R). Although this is not a Bounded Complex Domain corresponding to a Hermitian Symmetric Space, and therefore does not technically have a Shilov boundary, the 28-dim Jordan algebra J4(Q) can be regarded as effectively a Shilov-like boundary which in turn represents a 28-dimensional bosonic spacelike brane F-theory of the MacroSpace of the Many-Worlds.
Note that:
In a post to the spr thread Re: Structures preserved by e_8 Thomas Larsson says:
"... e_8 also seems to admit a 7-grading,g = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3,of the form
e_8 = 8 + 28* + 56 + (sl(8) + 1) + 56* + 28 + 8*.Kaneyuki does not mention anything about this, because from his point of view 3- and 5-gradings are more interesting. Incidentally, this grading refutes my claim that mb(3|8) is deeper than anything seen in string theory, since now e_8 also admits a grading of depth 3 and I learned about it in an M theory paper: P West, E_11 and M theory, hep-th/0104081, eqs (3.2) - (3.8). OTOH, the above god-given 7-grading of e_8 is not really useful in M theory, because g_-3 is identified with spacetime translations and one would therefore get that spacetime has 8 dimensions rather than 11. ...".
That structure shows the relationship between 248-dim E8 and the 256-dim graded exterior algebra /\(8) with graded structure
1 8 28 56 70 56* 28* 8* 1* (the 70 is self-dual)
That is,
E8 = 8 + 28 + 56 + 64 + 56* + 28* + 8*
and you get E8 by dropping the 1 and 1* and 70, and adding a 64.
If you look at 64 in Lie algebra terms, it is natural to think of it as 8 x 8* which is in compact terms the adjoint rep of U(8) = SU(8) x U(1).
However, because of connections with the D4-D5-E6-E7-E8 VoDou Physics model, I like to think of the 64 as the space of the Clifford algebra Cl(2,4) as 4x4 quaternionic matrices. From that point of view the 7-grading of E8 looks like
E8 = 8 + 28 + 56 + Cl(2,4) + 56* + 28* + 8*
If you do the natural thing and let the 28 be the D4 Lie algebra, you have
E8 = 8 + D4 + 56 + Cl(2,4) + 56* + D4* + 8*
and if you regard the 8 as the root vector space of E8 you have
Vector Lie alg Clif alg Lie alg* vect* E8 = 8 + D4 + 56 + Cl(2,4) + 56* + D4* + 8*
As to the 56, recall that the Lie Group E6 is the Automorphism Group of the 56-dimensional Freudenthal Algebra Fr3(O) of 2x2 Zorn-type vector-matrices
a X Y b
where a and b are real numbers and X and Y are elements of the 27-dimensional Jordan Algebra J3(O), so we have:
Vector Lie alg Fr alg Clif alg Fr alg* Lie alg* vect* E8 = 8 + D4 + Fr3(O) + Cl(2,4) + Fr3(O)* + D4* + 8*
I am NOT saying that the E8 Lie multiplication is the same as the Lie, Freudenthal, and Clifford multiplications, but it does seem that E8 has graded structure such that the vector spaces of the graded elements are the same as vector spaces of interesting Lie, Freudenthal, and Clifford algebras. Soji Kaneyuki makes that clear when he says "... Hn(F) [is] the vector space of F-hermitian matrices of degree n ...", so that in his notation H3(O) is NOT the Jordan algebra J3(O), but is only the vector space on which you can put the Jordan product if you want to make J3(O) from it.
Since the Freudenthal algebra Fr3(O) is closely related to the Jordan algebra J3(O), you can say that the 7-grading structure shows that E8 looks like a combination of Vector, Lie, Jordan, and Clifford things.
I very much like that 7-grading because the D4-D5-E6-E7-E8 VoDou Physics model uses:
g(1) + g(-1) is given the structure
of 4x4 Complex antisymmtric matrices of the form
0 a x d
-a 0 b y
-x -b 0 c
-d -y -c 0
and 8-real-dimensional {a,b,c,d} corresponds to su(3)
and 4-real-dimensional {x,y} corresponds to u(2) = su(2)+u(1).
The 4-dimensional space on which those matrices act
corresponds to the post-dimensional-reduction
internal symmetry space with CP2 global structure.
In other words, unlike West, I am happy with the 7-grading as is, because the D4-D5-E6-E7-E8 VoDou Physics model (unlike M-theory) has a physical interpretation for each term in the 7-grading.
In a 19 February 2003 post to the sci.physics.research thread Re: Structures preserved by e_8, John Baez says:
"... I would like to understand all the [N-]gradings of E8 ...[by emulating]... a jeweler, searching for mathematics pretty enough to be a theory of everything, examin[ing] an 8-dimensional gemstone with 240 vertices, turning it until ...[the jeweler]... finds that the vertices line up to form [N] parallel planes.The E8 lattice consists of all 8-tuples (x_1,...,x_8) of real numbers such that the x_i are either all integers or all half-integers (a half-integer being an integer plus 1/2), and they satisfy x1 + ... + x8 is even.
The nearest neighbors of the origin are called the "roots" of E8. They all have length equal to sqrt(2), and here they are:
- (1,1, 0,0,0,0,0,0) and all permutations: there are 8 choose 2 = 28 of these
- (-1,-1, 0,0,0,0,0,0) and all permutations: there are 8 choose 2 = 28 of these
- (1,-1, 0,0,0,0,0,0) and all permutations: there are twice 8 choose 2 = 56 of these
- (1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2): there is 1 of these
- (-1/2,-1/2, 1/2,1/2,1/2,1/2,1/2,1/2): there are 8 choose 2 = 28 of these
- (-1/2,-1/2,-1/2,-1/2, 1/2,1/2,1/2,1/2): there are 8 choose 4 = 70 of these
- (-1/2,-1/2,-1/2,-1/2,-1/2,-1/2, 1/2,1/2) there are 8 choose 2 = 28
- (-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2) there is 1 of these
So, there are a total of 28 x 6 + 70 + 2 = 168 + 72 = 240 roots. I wrote out this calculation because it's one of the strangest ways I've seen so far of counting the 240 roots of E8; for example, the number 168 is the size of the symmetry group of the Fano plane! One can construct the octonions starting with the Fano plane, and E8 from the octonions... hmm....
Anyway ... it seems evident that we get gradings of E8 by finding integer-valued linear functionals L on the E8 lattice; the value of L on a given root is its "grade". We say we have an "n-grading" if L takes exactly n values on the roots and the origin. I could be wrong, but that's how I think it works. So, let's find some gradings.
This sounds very technical, but you should really imagine it like this: you're a jeweler holding up a precious 8-dimensional gemstone cut in a shape with 240 vertices, and you want to rotate it around and see how the vertices (and the center of the gem) can line up to lie on parallel planes. ...
I'll just try a simple method: I'll define L to equal the first component of our vector. This can be 1, 1/2, 0, -1/2, or -1, so we get a 5-grading. Let's figure out the dimension of each grade,to guess if it's one of the 5-gradings we've already seen above.
- The number of roots with a "1" as the first component is 7 + 7 = 14
- The number of roots with a "1/2" as the first component is 1 + (7 choose 5) + (7 choose 3) + (7 choose 1) = 1 + 21 + 35 + 7 = 64
- The number of roots with a "-1/2" as the first component is 1 + (7 choose 5) + (7 choose 3) + (7 choose 1) = 1 + 21 + 35 + 7 = 64
- The number of roots with a "-1" as the first component is 7 + 7 = 14
- The rest of the roots have a "0" as the first component; there are 240 - 14 - 64 - 64 - 14 = 84 We also have to count the Cartan subalgebra (corresponding to the origin); this gives 8 more dimensions in this grade, for a total of 92.
So we get a 5-grading of this sort:
14 + 64 + 92 + 64 + 14
Unless the gods are playing pranks, this must be Larsson's
14 + 64 + (so(14) + 1) + 64 + 14
... I should mention that while I was doing these calculations, I noticed all sorts of strange things. For example, 64 + 14 = 78, the dimension of e_6. ..."
You can use these E8 lattice coordinates
±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke, (±1 ±ie ±je ±ke)/2 (±e ±i ±j ±k)/2 (±1 ±ke ±e ±k)/2 (±i ±j ±ie ±je)/2 (±1 ±k ±i ±je)/2 (±j ±ie ±ke ±e)/2 (±1 ±je ±j ±e)/2 (±ie ±ke ±k ±i)/2 (±1 ±e ±ie ±i)/2 (±ke ±k ±je ±j)/2 (±1 ±i ±ke ±j)/2 (±k ±je ±e ±ie)/2 (±1 ±j ±k ±ie)/2 (±je ±e ±i ±ke)/2
to find the 5-grading of e-8 constructed by John Baez, by ordering the grades by the sum of the coefficients of all 8 basis vectors {1,i,j,k,e,ie,je,ke} of the root vector space of the e_8 lattice:
grade(3) +2 14 = 14 grade(2) +1 8 + 14x4 = 8+56 = 64 grade(0) 0 14x6 + 8cartan = 92 grade(-1) -1 8 + 14x4 = 8+56 = 64 grade(-2) -2 14 = 14
Unless the gods are playing pranks, this must be the e_8 grading e31 (or a related real form), which is Larsson's
Note that 14 + (so(14) + 1) + 14 = so(16) and 64+64 = 128 = dim(so(16)half-spinor).
You can find another 5-grading of e_8 using these E8 lattice coordinates
±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke, (±1 ±ie ±je ±ke)/2 (±e ±i ±j ±k)/2 (±1 ±ke ±e ±k)/2 (±i ±j ±ie ±je)/2 (±1 ±k ±i ±je)/2 (±j ±ie ±ke ±e)/2 (±1 ±je ±j ±e)/2 (±ie ±ke ±k ±i)/2 (±1 ±e ±ie ±i)/2 (±ke ±k ±je ±j)/2 (±1 ±i ±ke ±j)/2 (±k ±je ±e ±ie)/2 (±1 ±j ±k ±ie)/2 (±je ±e ±i ±ke)/2
and ordering the grades by the coefficient of {1}:
grade(2) +1 1 grade(1) +1/2 8x7 = 56 grade(0) 0 16x7 = 112 + 14 + 8cartan = 134 grade(-1) -1/2 8x7 = 56 grade(-2) -1 1
Unless the gods are playing pranks, this must be the e30 5-grading of e_8 (or a related real form):
Now, look for a 7-grading of e_8 by using the E8 lattice coordinates
±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke, (±1 ±ie ±je ±ke)/2 (±e ±i ±j ±k)/2 (±1 ±ke ±e ±k)/2 (±i ±j ±ie ±je)/2 (±1 ±k ±i ±je)/2 (±j ±ie ±ke ±e)/2 (±1 ±je ±j ±e)/2 (±ie ±ke ±k ±i)/2 (±1 ±e ±ie ±i)/2 (±ke ±k ±je ±j)/2 (±1 ±i ±ke ±j)/2 (±k ±je ±e ±ie)/2 (±1 ±j ±k ±ie)/2 (±je ±e ±i ±ke)/2
and ordering the grades by the sum of the coefficients of {1,i,j,e}:
grade(3) +3/2 2+2+2 + 2 = 8 grade(2) +1 4 + 4+4+4 + 4+4+4 = 28 grade(1) +1/2 8+6+6+6 + 6+8+8+8 = 26 + 30 = 56 grade(0) 0 8 + 8+8+8 + 8+8+8 = 56 + 8cartan = 64 grade(-1) -1/2 8+6+6+6 + 6+8+8+8 = 26 + 30 = 56 grade(-2) -1 4 + 4+4+4 + 4+4+4 = 28 grade(-3) -3/2 2+2+2 + 2 = 8
Unless the gods are playing pranks, this must be Larsson's 7-grading of e_8
grade(4) 57 (vector of Fr3(O) plus 1) -----------------\
grade(3) 27 (vector of J3(O)) -------------\ \
grade(2) 16 (half of 5-dim hypercube) ---------\ \ \
grade(1) 8 (4-dim cross polytope) -----\ \ \ \
grade(0) 24 (D4 24-cell) + cartan* ---D4 |-D5 |-E6 |-E7 |-E8
grade(-1) 8 (4-dim cross polytope) -----/ / / /
grade(-2) 16 (half of 5-dim hypercube) ---------/ / /
grade(-3) 27 (vector of J3(O)) -------------/ /
grade(-4) 57 (vector of Fr3(O) plus 1) -----------------/
There are 4 Cartan elements for D4, 4+1=5 for D5,
4+2=6 for E6, 4+3=7 for E7, and 4+4=8 for E8.
To see whether such a 9-grading of e-8 really exists as a Lie algebra grading, consider these E8 lattice coordinates
±1, ±i, ±j, ±k, ±e, ±ie, ±je, ±ke, (±1 ±ie ±je ±ke)/2 (±e ±i ±j ±k)/2 (±1 ±ke ±e ±k)/2 (±i ±j ±ie ±je)/2 (±1 ±k ±i ±je)/2 (±j ±ie ±ke ±e)/2 (±1 ±je ±j ±e)/2 (±ie ±ke ±k ±i)/2 (±1 ±e ±ie ±i)/2 (±ke ±k ±je ±j)/2 (±1 ±i ±ke ±j)/2 (±k ±je ±e ±ie)/2 (±1 ±j ±k ±ie)/2 (±je ±e ±i ±ke)/2
and ordering the grades by the sum of the coefficients of basis vectors.
Since the maximum sum value is 2, the highest possible grade is 4.
Since the maximum possible number of highest-grade elements is 14, you cannot get 57 grade-4 elements, and such a 9-grading of e_8 does not exist.
The reason that 9-level structure of E8 is not a Lie algebra grading is:
Each level of that 9-level structure of E8 is seen from a different perspective, while a Lie algebra grading requires that, as John Baez says, ".. the vertices line up to form [9] parallel planes ..." from the same perspective.
The different perspectives of that 9-level structure of E8 are:
grade(4) 57 in the 8th dimension grade(3) 27 in the 7th dimension grade(2) 16 in the 6th dimension grade(1) 8 in the 5th dimension grade(0) 24 first 4 dimensions grade(-1) 8 in the 5th dimension grade(-2) 16 in the 6th dimension grade(-3) 27 in the 7th dimension grade(-4) 57 in the 8th dimension
In order to vary perspective from grade to grade, you would have to be able to change perspective from grade level to grade level, sort of like shifting the spheres in this
Chinese Nested Sphere carving.
...