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Graded Lie Algebras

 

Lie Algebra Gradings are to SymmetricSpaces as LieAlgebras are to LieGroups. In other words, Graded Lie Algebras are sort oflike the linear tangent spaces of symmetric space manifolds.

 

"... a semisimple GLA [Graded Lie Algebra] has theform

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 characteristicelement 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 associatedpair. Let Go be the group of grade-preserving automorphisms of G. ...Let U = Go exp(g(1) + ... + g(v)), which is a parabolic subgroup ofG. The real flag manifold M = G/U is called a flag manifold of thev-th kind. ...

... The Shilovboundary of an irreducible boundedsymmetric domain is a flag manifold of the 1st kind or ofthe 2nd kind, according as the domain is of tube type or not. ...For an irreducible symmetric domain of tube type with dimensiongreater than 1, we show the coincidence of the causal automorphismgroup of the Shilov boundary and the full holomorphic automorphismgroup of the domain. ...

... The class of symmetric R-spaces contains the Shilovboundaries of bounded symmetric domains of tube type. ... By asymmetric R-space we mean a compact irreducible Hermitiansymmetric space or areal form (i.e., the set of fixed points by an involutiveanti-holomorphic isometry) of it. ... The purpose of these notesis to give an introduction and survey of recent results on semisimplepseudo-Hermitian symmetric spaces. ...

... In the following tables we use the following notation: H thequaternion algebra over R, O (resp. O') the Cayley (resp. splitCayley) algebra over R, and OC = O (x)R C. Mp,q(F) the vector spaceof 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 ofdegree n; SYMn(C) the vector space of complex symmetric matrices ofdegree n.; ALTn(F) the vector space of alternating F-matrices ofdegree n; Hn(F) the vector space of F-hermitian matrices of degree n....[ II a restricted fundamental root system of g; II1 the partof II corresponding to grade g(1); a used for alpha and y used forgamma ]...

 

... Table 1TABLE 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 2TABLE 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 3TABLE 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) ...

... Table 8 of g(ev) for each simple GLA of the 2nd kind...
( 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) )             ...". 

 

Graded Lie Algebra structure of theD4-D5-E6-E7-E8 VoDou Physics Model 

E6 F4E7 E8 

 

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) andg(2) are combined to form g(ev)

g =                                           + g(-1)                 + g(ev)                                        + g(1)

For E6, we have a 5-level GLA oftype e7

E6(-14) =              8-dim                                       + 16-dim            +  R        + so(1,7)      +  iR                                       + 16-dim            + 8-dim

with physical interpretation in the Lagrangianof the D4-D5-E6-E7-E8 VoDou PhysicsModel as:

 

By using only the Shilov boundary for representing 8-dimensionalspacetime and the 8+8 = 16 first-generation fermions, we have not yetgiven physical interpretation to these parts of E6:

            +  R                       +  iR                                       + 16-dim            + 8-dim

Physically, they correspond to another copy of 8-dimensionalspacetime, another copy of the 8 fermion particles, another copy ofthe 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 by8+8+8+1+1 = 26-dimensionaltraceless 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 aconfiguration of fermion particle and/or antiparticle at a point inspacetime. A line in that 26-dimensional space looks like a stringand corresponds to a world-line describing history of a particle(closed because globally time of S1 x S7 spacetime iscyclic S1, although the time scale may be so large that we don'treadily see its cyclic nature). Thephysics of strings in that 26-dimensional space describes a BohmQuantum Potential.

 

As an aside, note that the 3-levelstructure corresponding to the GLA of typee7 is

E6(-14) =                                       + 16-dim            +  so(2,8)                 +  iR                                       + 16-dim

where so(2,8)is the double cover of the Conformal group over theOctonions.

 

 

 

For E7, we have a 3-level GLA oftype I12

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 agiven 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) Jordanalgebra ( also denoted by J3(O)) plus a U(1) Complex symmetry generated by the R, which complexdomain has as its Shilov boundary (1+26)=27-dimensional S1x H3(O)o which in turn represents a27-dimensional bosonic timelike brane M-theory of the MacroSpace ofthe Many-Worlds.

 

 

For E8, we have a 5-level GLA oftype e16

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 ofg(1) correspond to a 28-Quaternion-dimensional Quaternified J4(Q)Jordan algebra plus a SU(2) = Sp(1) Quaternionic symmetry generatedby the sl(2,R). Although this is not a BoundedComplex Domain corresponding to a Hermitian Symmetric Space, andtherefore does not technically have a Shilov boundary, the 28-dimJordan algebra J4(Q) can be regarded as effectively aShilov-like boundary which in turn represents a28-dimensional bosonic spacelike brane F-theory of the MacroSpace ofthe Many-Worlds.

Note that:

 

 

In apost to the spr thread Re: Structures preserved by e_8 ThomasLarsson 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 the256-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 a64.

If you look at 64 in Lie algebra terms, it is natural to think ofit 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 theD4-D5-E6-E7-E8 VoDou Physics model, I like to think of the 64 asthe space of the Clifford algebra Cl(2,4) as 4x4 quaternionicmatrices. 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 youhave

       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 AutomorphismGroup of the 56-dimensional Freudenthal Algebra Fr3(O) of 2x2Zorn-type vector-matrices

a    XY    b

where a and b are real numbers and X and Y are elements of the27-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 theLie, Freudenthal, and Clifford multiplications, but it does seem thatE8 has graded structure such that the vector spaces of the gradedelements are the same as vector spaces of interesting Lie,Freudenthal, and Clifford algebras. Soji Kaneyuki makes that clearwhen he says "... Hn(F) [is] the vector space of F-hermitianmatrices of degree n ...", so that in his notation H3(O) is NOT theJordan algebra J3(O), but is only the vector space on which you canput the Jordan product if you want to make J3(O) from it.

Since the Freudenthal algebra Fr3(O) is closely related to theJordan algebra J3(O), you can say that the 7-grading structure showsthat E8 looks like acombination of Vector, Lie,Jordan, and Cliffordthings.

I very much like that 7-grading because theD4-D5-E6-E7-E8 VoDou Physics model uses:

In other words, unlike West, I am happy with the 7-grading as is,because the D4-D5-E6-E7-E8 VoDou Physicsmodel (unlike M-theory) has a physical interpretation for eachterm in the 7-grading.

 


JohnBaez's Root Vector Geometry of Lie AlgebraGradings:

In a19 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 orderingthe 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     = 64grade(0)     0          14x6 + 8cartan  = 92grade(-1)   -1      8 + 14x4 = 8+56     = 64 grade(-2)   -2          14              = 14       

Unless the gods are playing pranks, this must be thee_8 grading e31 (or a related real form),which is Larsson's

14 + 64 + (so(14) + 1) + 64 + 14

Note that 14 + (so(14) + 1) + 14 = so(16) and64+64 = 128 = dim(so(16)half-spinor).  

 

 

You can find another 5-grading of e_8 using these E8lattice 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                                  1grade(1)    +1/2   8x7                      =  56grade(0)     0    16x7 = 112 + 14 + 8cartan = 134grade(-1)   -1/2   8x7                      =  56grade(-2)   -1                                  1

Unless the gods are playing pranks, this must be the e305-grading of e_8 (or a related realform):

1 + 56 + (e_7 +1) + 56 + 1

 

 

Now, look for a 7-grading of e_8 by using the E8lattice 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       =                 8grade(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   =                28grade(-3)   -3/2        2+2+2   + 2       =                 8

Unless the gods are playing pranks, this must be Larsson's7-grading of e_8

8 + 28 + 56 + 64 + 56 + 28 + 8

 

Can we find a grading of e_8that corresponds to this picture?

 

 

Such a grading would be a 9-grading that looks like
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 |-E8grade(-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 Liealgebra grading, consider these E8 latticecoordinates

	      ±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 basisvectors.

Since the maximum sum value is 2, the highest possible grade is4.

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 doesnot exist.

The reason that 9-level structure of E8 is not a Lie algebragrading is:

Each level of that 9-level structure of E8 is seen from adifferent perspective, while a Lie algebra gradingrequires that, as John Baez says, ".. the vertices line up toform [9] parallel planes ..." from the sameperspective.

The different perspectives of that 9-level structure of E8are:

grade(4)   57 in the 8th dimension grade(3)   27 in the 7th dimensiongrade(2)   16 in the 6th dimensiongrade(1)    8 in the 5th dimensiongrade(0)   24 first 4 dimensions  grade(-1)   8 in the 5th dimensiongrade(-2)  16 in the 6th dimensiongrade(-3)  27 in the 7th dimensiongrade(-4)  57 in the 8th dimension

In order to vary perspective from grade to grade, you would haveto be able to change perspective from grade level to grade level,sort of like shifting the spheres in this

 Chinese Nested Sphere carving.

 


 

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