- Email: [email protected]
Group gradings on ο(8, ℂ)
Vol. 61 (2008)
REPORTS ON MATHEMATICAL PHYSICS
No.2
GROUP GRADINGS ON 0(8, q CRISTINA DRAPER* Departamento de Matematica Aplicada, Campus El Ejido, Universidad de Malaga, 29013 Malaga (Spain) (e-mail: [email protected])
and ANTONIO VIRUEL t Departamento de Algebra, Geometria y Topologfa, Campus de Teatinos, Universidad de Malaga, 29071 Malaga (Spain) (e-mail: [email protected]) (Received August 17, 2007 -
Revised November 9, 2007)
This is a matricial description of all the fourteen fine group gradings on the exceptional Lie algebra 0(8, iC). Keywords: fine grading, maximal abelian diagonalizable subgroup of automorphisms, exceptional Lie algebra, ()4.
1. Introduction Given a Lie algebra L, a Lie-grading on L is a vector space decomposition L = $iEIL i such that L i f=. 0 and for each pair of indices i, j E I there exists k E I such that LiL j C Lk (see Definition 2.1 below). Besides of its own mathematical interest, gradings on Lie algebras show up in numerous applications of physical nature. As it was already pointed out by Patera et al. in a series of papers on the subject [8-12], gradings are in the background of the choice of bases and the additive quantum numbers, as well as they are a key ingredient when studying contractions that keep "undeformed" certain subalgebra (e.g. see [7]). Gradings on Lie algebras also play a relevant role in quantum mechanics via Jordan algebras [6]: given a pair of indices i, j E I satisfying (LiLj)L i eLi, any element k E L j gives rise to a Jordan algebra structure in L i by defining x
0
y = (xk)y.
*Supported by MCYT grants MTM2004-06580-C02-02 and MTM2004-08115-C04-04, and by JA grants FQM-336 and FQM-1215. tPartially supported by the FEDER-MEC grant MTM2007-6oo16, and by the JA grants FQM-213 and FQM-2863. [265]
266
C.
DRAPER
and
A. VIRUEL
Gradings on the algebras o(n, C) have been computed in [1] and [5] for n i= 8. The case of n = 8 is of special relevance because of its strong symmetry, reflected partly by the triality automorphism. Because of this, 0(8, C) is usually considered as an exceptional Lie algebra. Previous works on gradings on exceptional Lie algebras are [2, 3]. Here we provide a neat description of all the fourteen fine group gradings on 0(8, C) (see Section 3). Our main contribution is the description of the ones produced by groups of automorphisms containing outer order 3 automorphisms. The proof showing that these are all the fine group gradings is lengthy and cannot be included here, so it appears in [4].
2. Basic concepts We introduce the basic concepts we use along this paper. DEFINITION 2.1. A group grading f on a Lie algebra is a decomposition in vector spaces L = tBgEGL g such that g, h E G, and G is generated by Supper) := {g E G I L g i= group grading is of type (h 1, ... , h s ) if it has hi components s is the greatest nonzero dimension of a component.
L over a group G LgL h C L gh for all O}. We say that the of dimension i, and
Note that the group grading f is also a Lie-grading over the set Supper). It is unknown whether or not every Lie-grading on a simple Lie algebra is in fact a group grading, although on nonsimple Lie algebras there are always Lie-gradings which cannot be considered as group gradings. DEFINITION 2.2. Two group gradings f and f' on a Lie algebra L, given by the decompositions L = tBgEGL g and L = tBhEHMh, are equivalent if there is a bijection A: Supper) ---+ Supp(f') and an' automorphism f E Aut(L) such that f(L g)
= MJ...(g).
Here we study gradings up to equivalence. Therefore, the grading group of a group grading is not unique, and it is convenient to choose a preferred one among the grading groups. This will be the best in the sense explained in [2]. REMARK 2. 1. Any group grading f is given by L = tB gEG u L g for certain group G u , called universal group, such that for any equivalent grading L = tBhEHMh there is a group epimorphism p: G u ---+ H and an automorphism f E Aut(L) such that f(L g) = Mp(g). DEFINITION 2.3. A group grading is called fine when it cannot be refined. A refinement of the grading L = tBgEGL g is another group grading L = tBhEHMh such that each component L g is sum of several components Mh. EXAMPLE 2.1. The main example of a fine group grading is the root decomposition in a simple Lie algebra, which turns out to be a grading over 71}, where l is the rank of the algebra. Recall that a group grading on a CC-algebra L is always the simultaneous diagonalization relative to a commuting set of semisimple automorphisms (details
GROUP GRADlNGS ON 0(8, iC)
267
in [2]). Moreover, a grading is fine when it is produced by a maximal subgroup Q of semisimple automorphisms, which is usually called a MAD. In this case, the universal group of the grading is just the group of characters X(Q) = hom(Q, CX). From now on, L will denote the algebra of skewsymmetric matrices 0(8, C) = {x E Matgxg(C) I x+x t = O}. Let eij be the elementary matrix with (i, j)-entry 1, and the remaining entries zero. Define hi) = eji - ei}' Then {hi} Ii, j = 1, ... , 8; i < j} is a basis of L, and the following section describes the fine group gradings on L in terms of this base.
3. The fine group gradings We introduce some useful elements of 0(8, C) to describe the group gradings on L: g(a) = el1 + e22 + e33 + e44 +
+ f(a) =
~ (a - ~) (eS7 + e6g + e7S + eg6),
~ (a + ~) (el1 + e22 + e33 + e44) + (ess + e66 + en + egg) +
h(a) =
~ (a + ~) (ess + e66 + en + egg)
~ (a - ~ )(e13 + e24 + e3l + e42),
~(a + ~)(el1 + e22 + e33 + e44 + ess + e66 + en + egg) +
~ (a - ~) (elS + e26 + e37 + e4g + eSI + e62 + e73 + eg4),
pea) = el1 + e22 + e33 + e44 + eSS + e66 + ~(a + ~)(en + egg)
-
+ ~(a ~)(e7g +eg7), q(a) = el1 + e22 + e33 + e44 +
+
~ (a + ~) (ess + e66)
~ (a - ~) (eS6 + e6S) + en + egg,
rea) = el1 + e22 + ~ (a + ~) (e33 + e44) + ~(a -l)(e34 + e43) + eSS + e66 + en + egg,
268
C. DRAPER and A. VIRUEL
sea) =
1( 1)
2" a + -;;
(ell
i( 1)
+ e22) + 2"
a - -;; (e12
+ e2l) + e33 + eM
+ eSS + e66 + en + e88, II = diag{l, 1, 1, 1, 1, 1, 1, -I},
h = diag{l, 1, 1, 1, 1, 1, -1,
13 = diag{l, 1, 1, 1, 1, -1,1, I},
14 = diag{1, 1, 1, 1, - 1, 1, 1, I},
Is = diag{l, 1, 1, -1, 1, 1, 1, I},
16 = diag{l, 1, -1,1,1,1,1, I},
17 = diag{l, -1,1,1,1,1,1, I},
18 = diag{-1, 1, 1, 1, 1, 1, 1, I},
= ish, g6 = hit, gl
= 161s, g7 = 17151311, g2
= Isf614h, glO = 16lshll, gs
g13
= 14hhit,
+ e2l + e34 + e43 + eS6 + e6S + e78 + e87, g8 = e13 + e3l + e24 + e42 + eS7 + e7S + e68 + e86, gll = e12 + e2I + e34 + e43 - eSS + e66 - en + e88, g12 = ell - e22 + e33 - eM - eS6 + e6S - e78 + e87, g14 = eiS + e26 + e37 + e48 + eSI + e62 + e73 + e84· g3
=
= hlsl4h, g9 = ishl4h, g4
I},
e12
Notice that j;, gi, I(a), g(a), h(a), pea), q(a), rea), sea) E 0(8, C) = {P E Mat8x8(C) I P pI = id} for all a E C*. Thus we can use the adjoint map Ad: 0(8, C) --+ Auto(8, C) to obtain (inner) automorphisms of L, Ad P(x) = p-Ixp. We define Fi = Ad Ii and G i = Adgi . Now, we completely describe the group of automorphisms, denoted by Qi, producing each grading. We also provide the simultaneous diagonalization, which is a straightforward computation by means of any computer algebra software. Grading over Z x Zi: QI = ({Adg(u), G I , G 2, G3, G4 I u E C*}) is an abelian diagonalizab1e subgroup of Aut(L). In fact, it is maximal according to [5] (it is the MAD corresponding to T2(~)' Hence it produces a fine grading, of type (25,0, 1), whose homogeneous comp'onents are: L{!2" 1 -1 , -1 , -I} = (ib3,s - b3,7 - ib4,6 + b4,8}),
+ b7,8), L{!,l,-I,-l,l} = (ib3,6 - b3,8 - ib4,S + b4,7), L{!,-I,I,-I,-I} = (ib1,5 - bI,7 - ib2,6 + b2,8), L{! 1 -11 -I} = (-ib3,s + b3,7 - ib4,6 + b4.8), 2" " L{!,-I,I,-I,I} = (ib l,6 - bl,8 - ib2,S + b2.7), L{!,l,-I,l,l} = (- ib 3,6 + b3,8 - ib4,5 + b4,7), L{!4' 11 "
-1 -I} ,
= (-b5,6 - ibs.8 + ib6,7
269
GROUP GRADINGS ON 0(8, C) L{!,_I,I,I,_I} L{!,_I,I,I,I}
= (-ib1,5
+ b1,7 (-ibI,6 + bl,S -
ib2,6 + b 2,s),
=
ib2,S
+ b S,6), (- ib 3,6 - b 3,s + ib4,s + b4,7), (-ibl,s - b1,7 + ib2,6 + b 2,s), (ib 3,s + b3,7 + ib4,6 + b 4,s), (-ib l ,6 - b1,8 + ib 2,s + b2,7), (ib3,6 + b3,S + ib4,s + b4,7), (ib1,5 + b1,7 + ib2,6 + b 2,s), (-b S,6 + ibs,s - ib 6 ,7 + b 7,s), (ib l ,6 + b1,8 + ib 2,s + b2,7), (bs,s + b6,7), (-ib 3,s - b3,7 + ib4,6 + b 4,s),
L{l,I,I,-I,-I}
= (b l ,2, b3 ,4,
L{2,1,-I,-I,I}
=
L{2,-I,I,-I,-I}
=
L{2,I,-I,I,-I}
=
L{2,-I,I,-I,I}
=
L{2,I,-I,I,I}
=
L{2,-I,I,I,-I}
=
L{4,1,1,-I,-I}
=
L{2,-I,I,I,I}
=
L{l,I,I,I,-I}
=
L{2,1,-I,-I,-1}
=
L{l,-I,-I,-I,-I}
= (b 2 ,3
-
L{l,-I,-I,I,-I}
= (b 2,3
+ bl ,4),
L{l,I,I,-I,I}
=
+ b2,7),
b 7,s
= L{l,-I,-I,-I,I} =
b l ,4),
L{l,I,I,I,I}
(b6,S - b S,7),
L{l,-I,-I,I,1}
(b6,s
+ b S,7),
(b2,4 -
= (b2,4
b1,3),
+ b1,3),
The subindex of each component is not the subindex of the group grading, but the set of eigenvalues of {Adg(2), G I , G2, G3, G4} for which the component is an eigensubspace (e.g. the index 1, -1, -1, -I} is for the element (-2, 0, 1, 1, 1) E ZxZ~). The universal group of the grading is just ZxZ~(= X(QI)). These comments about the universal group and the order and eigenvalues of the automorphisms, are valid for all the gradings in this paper. Grading over Z2 x Z~: Q2 = ({Ad f(u), Adg(v), G3, Gs I u, v E C*}) ~ (C*)2 X Z~ is an abelian diagonalizable subgroup of Aut(L). In fact, it is the MAD corresponding to T4:~ ([5]). The simultaneous diagonalization of L relative to this subgroup of automorphisms is of type (20,4):
{!'
+ b4,7), L{!,~,_I,I} = (b1,5 + ib1,7 - b2,6 - ib2,S + ib3,s - b3,7 - ib4,6 + b 4,s), L{!,~,I,_I} = (-b l,6 - ibl,s - b2,S - ib2,7 - ib3,6 + b 3,s - ib4,S + b4,7), L{!,~,l,l} = (-bl,s - ibl,7 - b2,6 - ib2,s - ib 3,5 + b3,7 - ib4,6 + b4,S), L{!,3,_I,_I} = (-b l,6 + ibl,s + b2,S - ib2,7 - ib3,6 - b3,S + ib4,s + b4,7), L{!,3,-I,1} = (-bl,s + ibl,7 + b2,6 - ib2,s - ib3,s - b3,7 + ib4,6 + b 4,s), L{!,~,_I,_I}
=
(b l,6 + ib1,8 - b2,5
-
ib2,7 + ib3,6 -
b 3,s -
ib4,5
270
C. DRAPER and A. VIRUEL
L{i,3,1,-lj = (b l,6 - ib1,8 L{i,3,1,lj
= (b1,5 - ib1,7
+ bz,s -
+ bZ,6 -
L{Z,j,-I,-1} = (-b l,6 - ib1,8 L{Z,1,-I,lj = (-b1,5 - ib1,7 L{Z,j,I,_lj = (b l,6 L{Z,j,I,lj = (b1,5
ibz,7 ibz,s
+ ib3,6 + b3,s + ib4,s + b4,7),
+ ib3,s + b3,7 + ib4,6 + b4,s),
+ bz,s + ibz,7 + ib3,6 -
+ bZ,6 + ibz,s + ib3,s -
+ ibl,S + bz,s + ibz,7 -
+ ib1,7 + bZ,6 + ibz,s -
ib 3,6 ib3,s
b3,s - ib4,s b 3,7 - ib4,6
+ b3,s -
+ b3,7 -
ib4,s ib4,6
+ b4,7),
+ b4,s),
+ b4,7),
+ b4,s),
+ ib z,7 - ib3,6 - b3,s + ib4,s + b4,7), L{Z,3,-I,lj = (b1,5 - ib 1,7 - bZ,6 + ibz,s - ib 3,s - b3,7 + ib4,6 + b4,s), L{2,3,1,-lj = (-b l ,6 + ib1,8 - bz,s + ibz,7 + ib 3,6 + b 3,s + ib4,s + b4,7), L{2,3,1,lj = (-b1,5 + ib l ,7 - bZ,6 + ibz,s + ib 3,s + b 3,7 + ib4,6 + b4,s),
L{2,3,-I,-lj
= (b l ,6 - ib1,8 - bz,s
+ bZ,4, bS,7 + b6,s), L{I,I,-I,-Ij = (b1,z + b3,4, bS ,6 + b7,s), L{I,I,-I,lj = (-b1,3 + b Z,4, -b S ,7 + b6,S), L{I,l,l,-lj = (b l,4 + b Z,3, bs,s + b6,7), L{!4" I -I , -Ij = (-b1,z - ibl,4 + ibz,3 + b3,4), L{I,9,-I,-lj = (-b S ,6 + ibs,s - ib6,7 + b7,s), L{4,1,-I,-lj = (-b1,2 + ibl,4 - ib z,3 + b3,4) , L{l '9' ! -I -Ij = (-b S ,6 - ibs,s + ib6,7 + b7,s). , L{I,I,I,lj
= (b1,3
Grading over Zi: Q3 = ({Fi I i = 1, ... , 7}) ~ Zi is a MAD of Aut(L), corresponding to TJo~. The simultaneous diagonalization of L relative to it is of type (28): L{I,-l,-l,l,l,l,lj
=
(b6,7),
L{I,-l,l,-l,l,l,lj
= (b S,7),
L{I,-I,l,l,-I,l,lj
= (b4,7),
L{I,-l,l,l,l,-l,lj
= (b3,7),
L{I,-I,I,I,I,I,-Ij
= (bZ,7),
L{I,-l,l,l,l,l,lj
L{l,l,-l,-l,l,I,lj
=
(bS,6),
L{I,l,-l,l,-l,l,lj
= (b 4,6),
L{I,l,-I,l,l,-I,lj
= (b 3,6),
L{1,1,-l,l,l,l,-lj
= (b Z,6),
L{I,l,l,-l,-l,l,lj
= (b 4 ,s),
L{I,l,l,-l,I,I,-lj
=
L{I,l,l,l,-l,-l,1}
= (b 3,4),
L{l,l,_l,l.l,l,lj
= (b1,6),
L{I,l,l,-l,l,-l,lj L{I,l,l,-l,l,l,lj
=
(b3,S),
= (b1,5) ,
L{I, I, I, 1,-1, 1,-1}
= (bz,4),
L{I,l,l,l,-l,l,lj
= (b 1,7),
(bz,s),
= (b1,4),
271
GROUP GRADINGS ON 0(8, C) L{l,l,l,l,l,-l,-lj L{l,l,l,l,l,l,-lj
= (bZ,3),
L{l,l,l,l,l,-l,lj
= (bl,2),
L{-l,-l,l,l,l,l,lj
= (b6,s), L{-l,l,l,l,-l,l,lj = (b 4 ,s),
= (b7,s),
= (bs,s), L{-l,l,l,l,l,-l,l} = (b 3 ,s),
L{-l,l,-l,l,l,l,lj
L{-l,l,l,l,l,l,-lj
= (b1,3),
L{-l,l,l,-l,l,l,lj
= (bz,s),
L{-l,l,l,l,l,l,lj
= (bl,s).
Grading over Z~: Q4 = ({G 6 , G l , G z, G 3 , G 7 )} ~ Z~ is a MAD of Aut(L), corresponding to Td,~' The simultaneous diagonalization of L relative to the quasitorus Q4 is of type (24,0,0,1), with L{l,l,l,-l,-l} = (bl,2, b 3,4, bS ,6, b7,s) and:
= (-b1,8 + bZ,7), L{-I,-I,I,I,-lj = (b1,8 + bZ,7), L{-I,I,-I,-I,-lj = (-b3,s + b4,7), L{-I,I,-I,I,-lj = (b3,s + b4,7), L{-I,I,I,-I,-lj = (-bs,s + b6,7), L{-I,I,I,I,-lj = (bs,s + b6,7), L{l,-l,-l,-l,-l} = (-b l ,4 + bZ,3), L{l,-l,-l,l,-lj = (b l,4 + bZ,3), L{l,-l,l,-l,-l} = (-b l ,6 + bz,s), L{l,-I,I,I,-lj = (bl,6 + bz,s), L{1,1.-I,-I,-lj = (-b 3,6 + b4,s), L{l,l,-l,l,-lj = (b 3,6 + b4,s),
= (-b1,7 + bz,s), L{-I,-I,I,I,lj = (b1,7 + bz,s), L{-I,I,-I,-I,lj = (-b 3,7 + b4,s), L{-I,I,-I,I,lj = (b3,7 + b4,s), L{-I,I,I,-I,lj = (-b S ,7 + b6,s), L{-I,I,I,I,lj = (b S ,7 + b6,s), L{l,-l,-I,-I,lj = (-b1,3 + b Z,4), L{l,-l,-I,l,lj = (b l ,3 + b Z,4), L{l,-I,l,-I,l} = (-b1,5 + b Z,6), L{l,-I,I,I,1} = (b1,5 + bZ,6), L{l,I,-I,-I,lj = (-b 3,s + b4,6), L{l,I,-I,I,lj = (b 3,s + b4,6).
L{-l,-l,l,-l,-lj
L{-I,-I,I,-I,lj
Grading over Z x Zi: Qs = ({Adh(u), G s , G 9 , G 3 , G s I u E C*}} ~ (C*) x Zi is a MAD of Aut( L), corresponding to Tz(~' The simultaneous diagonalization of L relative to the quasitorus Qs is of type '(28): L{l,-l,-l,-I,-Ij L{l,-I,-I,-l,lj
+ b4,s), bS ,7 + b6,s),
= (bl,s - bZ,7 - b3,6
= (-b1,3
+ bZ,4 -
+ bZ,3 + bs,s + b6,7), L{l,-I,-I,I,lj = (b1,3 + bz ,4 + bS ,7 + b6,s), L{l,-I,l,-I,-lj = (-bl,2 + b3,4 - bS ,6 + b7,s), L{l,-I,l,-I,lj = (b1,5 - b Z,6 - b 3,7 + b4,S), L{l,-I,I,I,-Ij = (-b l ,6 - bz,s + b 3,s + b4,7), L{l,-I,I,I,lj = (-bl,s - b Z,6 + b3,7 + b4,s), L{l,-I,-I,I,-lj
= (b l ,4
L{l,l,-l,-I,-Ij
= (-b l ,4
L{I,l,-I,-I,lj
= (-b1,7
+ bZ,3 -
+ bZ,8 -
bS ,8 b 3,s
+ b6,7),
+ b4,6),
272
C. DRAPER and A. VIRUEL
+ b2,7 + b3,6 + b4,s), L{l,I,-I,I,lj = (b l,7 + b 2,s + b3,s + b4,6), L{l,I,I,-I,-lj = (b l,2 + b3,4 + bS,6 + b7,s), L{l,I,I,-I,lj = (-b1,5 + b2,6 - b3,7 + b4,s), L{l,I,I,I,-lj = (b l,6 + b 2,s + b 3,s + b4,7), L{l,I,I,I,lj = (b1,5 + b2,6 + b3,7 + b4,s), L{l,I,-I,I,-lj
= (b1,8
+ ib4,6 - bS,7 + b6,S), L{! -I -I I -Ij = (-b l,4 - ib1,8 - b2,3 - ib2,7 + ib3,6 + ib4,s + bs,s + b6,7), 4' , " L{! -I -I I Ij = (-b1,3 - ib1,7 - b2,4 - ib2,s + ib3,S + ib4,6 + bS,7 + b6,s), 4' , " L{! -I I -I -Ij = (b l,2 + ibl,6 - ib2,s - b3,4 - ib3,s + ib4,7 - bS,6 + b7,s), 4' " , L{!4" I -I , -I , -Ij = (b l,4 + ib1,8 - b2,3 - ib2,7 + ib3,6 - ib4,s - bs,s + b6,7), L{! I I -I -Ij = (-b1,2 - ibl,6 + ib2,S - b3,4 - ib 3,s + ib4,7 + bS,6 + b7,s), 4' " , L{4,-I,-I,-I,lj = (b1,3 - ib l ,7 - b2,4 + ib2,S + ib 3,s - ib4,6 - bS,7 + b6,s), L{4,-I,-I,I,-lj = (-b l,4 + ibl,s - b2,3 + ib 2,7 - ib3,6 - ib4,s + bs,s + b6,7), L{4,-I,-I,I,lj = (-b1,3 + ib1,7 - b2,4 + ib2,s - ib 3,s - ib4,6 + bS,7 + b6,S), L{4,-I,I,-I,-lj = (b l ,2 - ib l ,6 + ib 2,s - b3,4 + ib 3,s - ib4,7 - bS,6 + b7,s), L{4,1,-I,-I,-lj = (b l,4 - ib1,8 - b2,3 + ib2,7 - ib 3,6 + ib4,s - bs,s + b6,7), L{4,1,1,-I,-lj = (-b1,2 + ib l ,6 - ib 2,s - b3,4 + ib 3,s - ib4,7 + bS,6 + b7,s). L{~,_I,_I,_I,lj = (b1,3
+ ibl,7 -
b2,4 - ib2,s - ib3,s
Grading over Z x Z~: Q6 = ({p(u), Fs , F7 , F6 , Fs, F4 I U E C*}) ~ C* x Z~ is again an abelian diagonalizable subgroup of Aut(L), concretely the MAD T2~~' The simultaneous diagonalization of L relative to the quasitorus Q6 is of type (28): L{l,-I,-I,I,I,lj
=
(b l ,2),
L{l,-I,I,-I,I,lj
= (b1,3),
L{l,-I,I,I,-I,lj
= (b l ,4),
L{l,-I,I,I,I,-Ij
= (bI,S),
L{l,I,-I,-I,I,1}
= (b 2,3),
L{l,I,-I,I,I,-lj
= (b 2,s),
L{l,I,I,-I,-I,lj
= (b3,4),
L{l,-I,I,I,I,lj
= (b l ,6),
L{l,I,-I,I,-I,lj L{l,I,-I,I,I,lj
= (b2,4),
= (b 2,6),
L{l, 1.1,-1,1,-1} = (b 3 ,s), L{l,I,I,I,-l,-lj L{l,I,I,I,I,-lj
L{! -I I I I Ij 2'
""
=
(b 4,s),
= (b S,6),
= (ib l ,7 + b1,8),
L{l,I,I,-I,I,lj
= (b 3,6),
= (b4,6), = (b7 .S ),
L{l,I,I,I,-I,1} L{l,I,I,I,I,1}
L{! I -I I I Ij 2"
",
= (ib 2,7
+ b 2,S),
273
GROUP GRADINGS ON 0(8, iC)
+ b3,S), L{1,1,1,I,I,-I} = (ibS,7 + bs,s), L{Z,-I,I,I,I,I} = (-ib l ,7 + bl,s), L{2,I,I,-I,I,I} = (- ib 3,7 + b3,s), L{Z,I,I,I,I,-I} = (-ib s,7 + bs,s),
+ b4,s), L{!,I,I,I,I,I} = (ib 6,7 + b6,s), L{2,I,-I,I,I,1} = (-ibz,7 + bz,s), L{Z,I,I,I,-I,I} = (-ib4,7 + b4,s), L{Z,I,I,I,I,1} = (-ib6,7 + b6,s).
= (ib3,7
L{1,1,1,-I,I,I}
= (ib4,7
L{!,1,1,I,_I,I}
Grading over 'l} x Z~: Q7 = ({p(u), Fs, F 7 , F 6 , q(v) I u, v E C*}} ~ (C*)z x Z~ is a MAD of Aut( L), corresponding to T4(~' producing the following grading, of type (26, 1): ' L{2,I,I,I,3}
L{1,1,1,I,j-} = (-bS,7 L{Z,I,I,I,j-}
=
L{I
=
I I -II}
2' "
,
L{I,-I,-I,I.1}
+ ibs,s + ib6,7 + b6,s),
+ ibs,s - ib6,7 + b6,S), (bS,7 - ibs,s + ib6,7 + b6,s),
= (bS,7
L{!,1,1,I,3} = L{ 2' I -I I I I} ",
+ bl.S), (ib3,7 + b3,s), (ib l,7
= (b1,2) ,
+ b l .6), L{I,-I,I,I,3} = (-ib1,5 + b l ,6), L{II - I I I} = (ibz,s + bZ,6), " "3 L{I,I,-I,I,3} = (-ibz,s + b Z,6), L{I
,
-I I I I} "'3
L{I,I,I,_I,I}
= (ibl,s
=
+ b6,s),
= (-bS,7 - ibs,s - ib6,7
(b 3 ,4),
+ b4 ,6), L{I,I,I,I,3} = (-ib4,s + b4,6), L(Z,I,-I,I,I} = (-ib z,7 + bz,s), L{Z,I,I,I,I} = (- ib4,7 + b4,s). L(I,I,I,I,~} = (ib 4 ,s
L{!
2"
+ bz,s), (i b4,7 + b4,s),
= (ibz,7
I -I I I} "
=
L 1!,1,1,I,I}
L{I,_I,I,_I,I}
= (b1,3) ,
= (b l ,4),
L{I,_I,I,I,I}
= (b Z,3),
L{I,I,_I,_I,I}
= (b Z,4),
Ll,l,-I,I,I}
L{I , I "I
-I I}
'3
L{I,I,I,-I,3} L{I,I,I,I,I}
= =
+ b3,6), (-ib 3,s + b3,6), (ib3,s
= (b S ,6, b7 ,s),
+ b1,8), L{Z,I,I,-I,I} = (-ib3,7 + b3,s), L{Z,-I,I,I,I}
= (-ib l ,7
Grading over Z~ x Z4: Qs = ({Gs, GIO, G l1 , G!2l) ~ Z~ x Z4 is a MAD of Aut(L), corresponding to To(~' which produces a grading of type (24,2) on L:
+ bl,s + ib z,7 = (-ib1,7 + bl,s - ib z,7 = (-ibl,7 - bl,s + ib z,7 -
L{-I,-I,-I,-i}
LH,-I,-I,i} L{-I,-I,I,-i}
= (ib1,7
+ b4,6), bz,s + ib 3,s - b3,6 + ib4,s + b4,6), bz,s + ib 3,s + b3,6 - ib4,s + b4,6),
bz,s - ibo,s - b3,6 - ib4,S
274
C. DRAPER and A. VIRUEL
+ b3,6 + ib4,s + b4,6}, L{-I,I,-I,-ij = (ib1,5 + b l ,6 + ibz,s - bZ,6 - ib 3,7 - b3,s - ib4,7 + b4,S}, L{-I,I,-I,i} = (-ib1,5 + b 1,6 - ibz,s - bZ,6 + ib 3,7 - b3,s + ib4,7 + b4,S}, L{-I,I,I,-i} = (-ib1,5 - b 1,6 + ibz,s - bZ,6 + ib 3,7 + b3,s - ib4,7 + b4,S}, L{-I,I,I,i} = (ib1,5 - b l ,6 - ibz,s - bZ,6 - ib 3,7 + b 3,s + ib4,7 + b4 ,s}, = (ib1,7 - b1,8 - ib z,7
L{-I,-I,!,ij
L{l,-I,-I,-i}
= (-ib l ,7
-
bl,s -
-
bz,s -
ib z,7
ib 3,s
+ bz,s -
ib 3,s - b3,6 - ib4,s
+ b4,6},
+ ib z,7 + bz,s + ib3,s - b3,6 + ib4,s + b4,6}, L{l,-I,I,-i} = (ib1,7 + b1,8 - ibn + bz,s + ib 3,s + b 3,6 - ib4,s + b4,6}, L{l,-I,I,ij = (-ib l ,7 + bl,s + ib z,7 + bz,s - ib 3,s + b 3,6 + ib4,s + b4,6},
L{l,-I,-I,ij
= (ib1,7 -
L{l,I,-I,-i}
= (-ib1,5 - b l ,6 - ibz,s
b1,8
+ bZ,6 -
b3,s - ib4,7 + b4,S}, b3,s + ib4,7 + b4,s},
ib 3,7
-
+ ibz,s + bZ,6 + ib3,7 L{l,I,I,-i} = (ib1,5 + b l ,6 - ibz,s + b Z,6 + ib 3,7 + b 3,s - ib4,7 + b4,s}, L{l,I,I,ij = (-ib1,5 + b l ,6 + ibz,s + bZ,6 - ib 3,7 + b3,s + ib4,7 + b4,s}, L{l,I,-I,i}
= (ib1,5 - b l ,6
= (bs,s + b6,7}, L{-I,I,-I,-l} = (-bl,z + b 3,4}, L{l,-I,-I,I} = (-bs,s + b6,7}, L{l,l,-l,-I} = (bl,z + b3,4},
= (-b1,3 + bZ,4}, L{_I,I,_I,I} = (-b S,6 + b7 ,s}, L{l,I,-I,I} = (b S,6 + b7,s}, L{-I,-I,l,-I} = (b l ,4 + b Z,3, -b S ,7 + b6,s},
L{-l,-I,-I,-l}
L{l,-I,-I,-l}
= (-b l ,4
L{_I,_I,_I,I
+ bZ,3},
L{_I,_I,I,I}
= (b1,3
+ bZ,4,
bS,7
+ b6,s}.
Grading over Z~: Q9 = ({Gs, GIO, G 13 , G 3 , G 7 , G I4 }} ~ Z~ is also a MAD of Aut(L), corresponding to The induced grading is of type (28):
T(?i-
L{_I,_I,_I,_I,_I,_I}
= (bl,s - bZ,7
-
b3,6 + b4 ,s},
+ b4,6} L{_I,_I,_I,I,_I,I} = (-bl,s - bZ,7 + b3,6 + b4,s}, L{_I,_I,_I,I,I,I} = (-b l ,7 - bz,s + b 3,s + b4,6}, L(-I,-I,I,-I,l,-l} = (b1,3 - bZ,4 - bS ,7 + b6,s}, L(-I,-I,I,-I,I,I} = (-b1,3 + b z,4 - bS,7 + b6,s}, L{_I,_I,I,I,_I,_I} = (-b l ,4 - bZ,3 + bs,s + b6,7}, L{-I,-I,-I,-l,l,l}
L{-I,-I,I,l,-I,I}
= (b l ,7
-
bz,s -
= (b l ,4 + bZ,3
b3,s
+ bs,s + b6,7},
= (-b1,3 - bz,4 + bS,7 + b6,s}, = (b1,3 + b2 ,4 + bS,7 + b6,s},
L{-I,-I,I,l,l,-I}
L{-I,-I,I.l,I,lj
+ b4,7}, b3,7 + b4,s},
L{-I,I,-I,-l,-l,l}
= (b l ,6 - bz,s - b3,s
L{-I,I,-I,-I,I,-lj
= (b1,5 - bZ,6 -
GROUP GRADINGS ON 0(8, iC)
275
+ b3,S + b4,7), L{-I,I,-l,l,I,-I} = (-bI,s - b Z,6 + b 3 ,7 + b4 ,s), L{-I,I,I,-I,-l.-I} = (bI,z - b3,4 - b S,6 + b7 ,s), L{-l,l,I,-I,-I,I} = (-bI,z + b3,4 - b S,6 + b7 ,s), L{l,-I,-I,-I,-I,I} = (-b1,8 + b Z,7 - b3,6 + b4,s), L{l,-I,-I,-I,I,-I} = (-b I ,7 + bz,s - b 3 ,s + b4 ,6), L{-l,l,-l,l,-l.-I}
=
L{l,-I,-I,I,-l.-l}
= (b1,8
L{l,-I,-I,I,I,-l}
+ b Z,7 + b3,6 + b4,s), (b I ,7 + bz,s + b 3 ,s + b4 ,6),
=
L{l,-I,I,-I,-I,-I} L{l,-I,I,-I,-I.l}
=
L{l,I,-I,-I,-I,I}
=
L{l,I,-I,-I.l,-l}
=
(-b 1,6 - bz,s
=
(b I ,4 -
+ b6,7), bs,s + b6,7), b3,s + b4,7), b 3,7 + b4,s),
b Z,3 - bs,s
+ b Z,3 (-b 1,6 + bz,s (-bI,5 + b Z,6 -
(-b I ,4
= (b I ,6 + bz,s + b3 ,s + b4 ,7), = (bI,5 + b Z,6 + b 3,7 + b4,s),
L{l,I,-I,I,-I,-I} L{l,I,-l,l,I,-I}
L{l,l,l,-l,-I,-I} L{l,I,I,-I,-l,I}
= (-b1,2 - b3,4
= (b1,2
+ bS,6 + b7,s),
+ b3,4 + bS,6 + b7 ,s).
Grading over Z3 xZz: QIO = ({p(U), Fs, rev), q(w) I u, v, WE C*}) ~ (C*)3 xZz is an abelian diagonalizable subgroup of Aut(L). More precisely, it is the MAD-group T6(~' which produces a grading of type (25,0, 1): L{1,l,Q} = (-bS,7 L{I2' I';' I I}
=
+ ibs,s + ib6,7 + b6,S),
(-b37+ib38+ib47+b48), ' , , ,
+ ib6 ,7 + b6,s), ib3,s + ib4,7 + b4,s),
L{~,I,I,3} = (bS,7 - ibs,s
L ,I,5,I} = (b3,7 1 L{I I I I} = (-b3,s , '5' 3
+ ib3,6 + ib4,s + b4,6),
+ ib4,s + b4,6), L{l,l.!,3} = (b3,s + ib3,6 - ib4,s + b4,6), L{l,I,S,3} = (-b 3,s - ib 3,6 - ib4,s + b4,6), L{Z,q,I} = (b3,7 + ib3,s - ib4,7 + b4,s), L{Z,I,q} = (bS,7 + ibs,s - ib6,7 + b6,s), L{Z,I,I,3} = (-b S,7 - ibs,s - ib 6 ,7 + b6 ,s), L{I,I,S,1}
=
(b3,s -
ib3,6
276
C. DRAPER and A. VIRUEL
L{i,-I,I,I} L{I,_I,!,I}
+ h,s}, = (ib1,3 + bl ,4},
= (ib l ,7
L{ I,-I,I,I} = (b 1,2 } , L{l,-I,S,I}
= (-ib1,3
+ b2,6},
L{I,I,I,~}
=
L{l,I,I,3}
= (-ib 2,s
(ib 2,s
+ b l ,4},
+ b2,6},
L{2,-J,J,I} = (-ibJ,7
+ bJ,s},
+ b2,s}, L(I , - I"3 I ! } = (ib u + b l ,6}, L{l,-I,I,3} = (-ib U + b l ,6}, L{J,J,!, I} = (i b2,3 + b2,4} ,
L( l' I Ill} = (ib 2,7 , ,
L{l,I,I,I}
= (b 3,4, bS ,6, b7,s},
L{l,I,S,I}
= (- ib 2,3
L{2,I,I,I}
=
+ b2,4} , (-ib 2 ,7 + b2,s}.
Grading over Z4: QIJ = ({Adp(u), Ads(v), Adr(w), Adq(z) I u, v, w, Z E C*}} ~ (C*)4 is an abelian diagonalizable subgroup of Aut(L) (corresponding to Ts(,~)' The simultaneous diagonalization of L relative to the quasitorus QII (in fact, a torus) is of type (24,0,0, 1). Of course this is the Cartan grading, in other words, the decomposition in root spaces relative to the Cartan subalgebra L {I,I,I,I} -- (b 1,2, b3,4, bS,6, b}' 7,S· L{IIII}=(-bI7+ibIS+ib27+b2S}, 1'7' , , , , , L{Il' II , '31}=(-bs7+ibss+ib67+b6S}, ' , , ,
+ ib4 ,7 + b4,s}, L{I! ! I} = (-b1,3 + ib l ,4 + ib2,3 + b2,4}, , 7' 5' L{I,~,s,l} = (b1,3 + ib l ,4 - ib2,3 + b2,4}, L{I I ! !} = (-b 3,s + ib3,6 + ib4 ,s + b4 ,6}, , '5' 3 L{l,I,s,~} = (b3,s - ib3,6 + ib4 ,s + b4 ,6}, L{J,7,!,1} = (b1,3 - ib l ,4 + ib2,3 + b2,4} , L{l,7,S,I} = (-b1,3 - ib l ,4 - ib 2,3 + b2,4} , L{2! I I} = (bJ,7 + ib1,8 - ib2,7 + b2 ,s}, '7' , L{2,I,q} = (bs,7 + ibs,s - ib6,7 + b6,s}, L{2,I,S,I} = (-b 3,7 - ib 3,s - ib4 ,7 + b4 ,s},
L{1,I,S,I}
= (b 3 ,7 - ib3,s
+ ib3,s + ib4 ,7 + b4 ,S}, L{i,1,J,3} = (bs,7 - ibs,s + ib6,7 + b6,s}, L{i,7,1,1} = (b l ,7 - ibl,s + ib2,7 + b 2,s}, L(I ! I !} = (-b u + ib l ,6 + ib2,s + b2,6}, '7' , 3 L{l,~,1,3} = (b u + ib 1,6 - ib2,s + b2,6}, L{J,J,!,3} = (b3,s + ib3,6 - ib4 ,s + b4 ,6}, L{l,I,S,3} = (-b 3,s - ib 3 ,6 - ib4 ,s + b4 ,6}, L{I,7,1'1} = (b u - ibJ,6 + ib2,s + b2,6}, L{l,7,1,3} = (-b u - ib 1,6 - ib 2,s + b2,6}, L{2,1,!,1} = (b3,7 + ib3,s - ib 4 ,7 + b4,s}, L{2,1,1,3} = (-b S ,7 - ibs,s - ib 6,7 + b6,s}, L{2,7,1,I} = (-b l ,7 - ib1,8 - ib 2,7 + b2,s}. L{!
I ! I}
2' , 5'
=
(-b3,7
In order to describe the gradings involving a copy of Z3, we introduce some outer automorphisms. First choose a basis of root vectors in L. Such a basis could be the one produced by Q I I, but it is easier to find a basis of root vectors in the isomor-
277
GROUP GRADINGS ON 0(8, iC)
phic Lie algebra i, j
L = {x
E
{I, ... , 4}, we define
E
if i > j, and a3,1, a4,1,
Ci,j
hi
+ Cx t = OJ,
where
C = (0
14
1o4 )
. For
= ei,i-ei+4,i+4, ai,j = ej,i-ei+4.j+4, Ci,j = ej,i+4-ei,j+4
= ei+4,j -ej+4,i if i
< j. Fix B
= {hI, h 2 , h 3 , h4, a2,1, a3,2, a4,3, C3,4,
a4,2, C4,1, C4,2, C2,1, C3,1, C2,3' al,2, a2,3, a3,4, C4,3, al,3, al,4, a2,4, CI,4, C2,4, Cl,2,
cu, C3,2}, and denote by i, j, k = 1, ... ,28. Define HI
Mat8x8(C) I xC
bi
its i-th element. Now let
({Ji,j(b k )
= 8k ,jbi
for
+ ({J3, I + ({J3,2 + ({J5,27 + ({J6,8 + ({J7,9 + ({J8,25 + ({JII, 12 + ({J12,26 + ({J13,18 + ({J14,23 + ({J17,15 + ({J18,20 + ({J19,21 + ({J20,13 + ({J23,24 + ({J24,14 + ({J25,6 + ({J26, II
=
({J2,4
- ({JI, I - ({JI,2 - ({JI,3 -
({JI,4 - ({J4,2 -
- ({J21,10 - ({J22,7 - ({J27,16 -
({J4,4 -
({J9,22 -
({JIO,19 - ({J15,28 -
({J16,5
({J28,17,
+ ({J2,3 + ({J2,4 + ({J4,1 + ({J5,26 + ({J6,16 + ({J7,19 + ({J8,5 + ({J1O,22 + ({JII,13 + ({J12,18 + ({J14,8 + ({J15,23 + ({J16,12 + ({J17,14 + ({J18,28 + ({J19,7 + ({J20,17 + ({J22,IO + ({J23,25 + ({J24,6 + ({J26,20 + ({J27,1l + ({J28,24 - ({JI,I - ({JI,3 - ({JI,4 - ({J3,3 - ({J9,21
H2 =
({J2,2
- ({J13,15 - ({J21,9 tx,y,Z,u
({J25,27 -
2({JI,2,
.{ 2 = dlag 1,1,1,1, x, y, z, u, xy, xyz, yz, uxy, uy, uxy z, uxyz, uyz,
11111 xy' xyz'
1111 x y z u
- , - , -, - ,
Ill}
YZ' uxy' uy' uxy2z' uxyz' uyz
.
Note that HI and H2 are outer order 3 exterior automorphisms. The automorphism H2 fixes a subalgebra of type G2, hence it can be considered as the triality automorphism (we have built it by extending the corresponding automorphism of the Dynkin diagram). And, although there is P E 0 (8, C) such that the composition H2 Ad(P) is conjugated to HI in Aut(L), we have chosen HI because it normalizes the maximal torus of the automorphism group given by {tx,y,Z,u I x, y, z, u E (CX}. Now we can describe the remaining gradings. I
Grading over Z~: If we denote w = -(-1)1, QI2 = ({tl,w2,w2,w2, tw2,I,w2,1' Hd) ~ Z~ is an abelian diagonalizable subgroup of Aut(L) inducing a grading of type (24,2): L{I,I,lU) L{I,I,(2)
=
=
(-wb l -
2 (-w b l -
+ b 18 + b20), L{I,w,(2) = (wb13 + w2bl8 + b 20 ) , 2 L{l,w2,wl = (w b 6 + wb 8 + b 25 ), L{w,I,I) = (-b I 5 - b 17 + b28), L{I,w,1)
=
(b 13
+ b 3 , (1 + W)b2 + b 4 ), 2 2 w b 2 + b3, (1 + ( )b2 + b4),
wb 2
2 (w b 13
+ Wb l8 + b 20 ) , L{I,w2,I} = (b6 + b 8 + b 25 ), 2 L{I,w2,w2} = (wb6 + w b 8 + b 25), 2 L{w,l,w) = (-WbI5 - w b 17 + b 28 ), L{I,w,w}
=
278
C. DRAPER and A. VIRUEL
+ b28), L{UJ,UJ,UJ} = (-w b lO + Wb l9 + b 21 ), L{UJ,UJ2,1} = (b 14 + b23 + b 24 ), 2 L{UJ,UJ2.UJ2} = (Wb 14 + w b 23 + b24) , 2 L(UJ2,I,UJ} = (wb5 - w h6 + b 27 ), L(UJ2.UJ,I} = (b ll + b 12 + b 26 ), 2 L{~,UJ,UJ21 = (wbl l + w b 12 + b26), 2 L{UJ2,UJ2,w} = (-w b 7 - wb 9 + b 22 ), L{UJ,I,UJ2)
=
(-«ib I5 - wb 17 2
+ big + b 21 ), 2 L{w,w,w2} = (-wb lO + w b l 9 + b 21 ), 2 L{w,w2,UJ} = (w b 14 + Wb 23 + b 24 ), L{UJ2,1,1) = (b 5 - b 16 + b 27 ), L{w2,1,(2) = (w2b5 - Wbl 6 + b 27 ), 2 L{w2,UJ,UJ) = (w b ll + Wb 12 + b 26 ). L{UJ2,UJ2,1} = (-b7 - bg + b 22), L(UJ2,UJ2,UJ2) = (-wb 7 - w2b9 + b 22 ). L{w,w.l}
=
(-b lO
Grading over Z~ x 23: Q13 = ({fl,-I,l,l, Ll,-I,-I,-I, H2 }) ~ 2~ x 2 6 is an abelian diagonalizable subgroup of Aut(L) , producing a grading of type (14,7):
+ b 12 - b l6 - b l8 - b 24 + b 28 , -b lO + b 22 ), L(_I,_I,I} = (b 6 + b12 + b l6 + b l 8 + b 24 + b 28, blO + b 22 ), L(_l,l,_1} = (-bll + b13 + bl5 - b23 + b25 + b 27 , bg + b21), L{-I,I,l) = (b ll + b13 - b l5 - b23 - b 25 + b 27, -b9 + b 21 ), L{l,-I,-I} = (-b5 + b 8 - b l4 + b17 - b20 + b 26 , -b7 + bI9), L{l,-I,I} = (b 5 + b 8 + b l4 + b 17 + b20 + b26, b 7 + b I9), L{l,l,-I} = (b 3 , b l + b2 ), 2 2 L{-l,-I,UJ} = (Wb6 + w b 12 + b l6 + Wb l8 + w b 24 + b28), 2 2 L{_I,_I, -UJ) = (wb 6 + w b 12 - b l6 - Wb l 8 - w b 24 + b28), 2 2 L{-l,-I,_UJ2} = (w b 6 + wb 12 - bl6 - w b l8 - Wb 24 + b 28 ), 2 2 L{_l,_I,(2) = (w b 6 + Wb l2 + b 16 + w b 18 + Wb24 + b 28 ), L{_l,l,UJ} = (w2bll + wb 13 - b 15 - w2b23 - Wb25 + b27), 2 2 L{-I,l,-w) = (-w b ll + Wb13 + b l5 - w b 23 + Wb 25 + b27), L{_1,l,-UJ2) = (-wbll + w2b13 + b l5 - Wb23 + w2b25 + b27), 2 2 L{_1,l,UJ2) = (Wbll + w b 13 - b l5 - Wb 23 - w b 25 + b 27 ), 2 L{l,-I,UJ) = {wb 5 + w2b8 + bl4 + wbi7 + w b20 + b26}, 2 L(l,-l,-UJ} = (-wb 5 + w b 8 - b 14 + wb 17 - w2b20 + b 26 ), 2 L{l,_I,_UJ2) = (-w2b5 + wb 8 - b1 4 + w b 17 - wb 20 + b 26 ), 2 2 L{l,-1,UJ2} = (w b 5 + wb8 + b l4 + w b 17 + wb20 + b 26), L{_l,_l,-I}
=
(b 6
279
GROUP GRADINGS ON 0(8, C)
2 -w - W 2w w ) b l - - - b2 - --b3 + b4 , l-w l-w l-w 2 2 -w 2w W ) 2 b2 l 2 + 1 2 b3 + b4 . L{I,I,_w } = ( -l--b + 1 -w - +w - +w
L{I,I,-w} = (
Grading over
Z2 x Z3: QI4 =
({tu,l,u,ft' tl,v,+I' v
Hi. I u, v E C*})
~
(C*)2 X Z3
is an abelian diagonalizable subgroup of Aut(L), which produces a grading of type (26, 1):
L{12' I , I} = (b s + b 17 + b26 ), 2 Lei,l,w2} = (w bs + wb 17 + b26 ), L{13 } = (w2b 13 - wb 23 2' ,w L{I '3' 1 I}
+ b 27 ),
= (b 16 + bls + b24),
2 L(12' I ,w} = (wb s + w b 17
+ b26) ,
= (b 13 - b23 + b27 ), 2 L{13 2} = (wb 13 - w b 23 + b 27 ), 2' ,w 2 L{I , 1 } = (w b 16 + wb ls + b24 ), L{i,3,1}
3'W
2 1 ) + -l-w - b2 + - - b3 + b4 , l-w 2 L{I,I,w2 }=(-1-1_b l + 2b2 + -112b3 + b4 ), -w -w 1 -w 1 +w2 L{I,I,w} = ( - - b l l-w
+ b2S), 2 L{I,3,w2} = (Wb 6 + w b 12 + b2S ), 2 L(2 ''1",w I } = (-w b 11 + wb ls + b 2S ), L{2,1,1} = (b s + b l4 + b 20 ), 2 L{I,j,w2} = (Wb 16 + w b lS + b24 ), L{I,I,I} = (b l + b2, b3),
L{I,3,1}
L{1,j,l}
=
=
(b 6 + b 12
(b 2I ),
L{2,~,I} = (b7),
(w 2b6 + wb 12
+ b2S ), L(2,j,l} = (-b ll + b ls + b2S ), 2 L{2 ''1",w I 2) = (-wb ll + w b lS + b 2S ), L{2,I,w} = (wb s + w 2b l4 + b 20 ), 2 L{2,I,(2) = (w bs + Wh4 + b20 ),
L{I,3,w}
=
Ld,3,1} = (b22 ), L{1,9,1}
=
(b I9 ),
L{2,3,1} = (b9),
L{4,j,1} = (b lO ).
REFERENCES
[I] [2] [3] [4] [5] [6] [7]
Y. A. Bahturin, I. P. Shestakov and M. V. Zaicev: J. Algebra 283 (2005), 849-868. C. Draper and C. Martin: Linear Algebra and its Applications 418 (2006), 85-111. C. Draper and C. Martin: Gradings on the Albert algebra and on f4. Preprint arXiv:math-ra/0703840v1. C. Draper, C. Martin and A. Viruel: Gradings on D4. Preprint. M. Havlfcek, J. Patera and E. Pelantova: Linear Algebra and its Applications 277 (1998), 97-125. P. Jordan: Nachr. Ges. Wiss. Gottingen 209-217 (1933). X.-D. Leng and J. Patera: J. Phys. A. Math. Gen. 27 (1994), 1233-1250.
280 [8] [9] [10] [II] [12]
C. DRAPER and A. VIRUEL
1. Patera and H. Zassenhaus: Linear Algebra and its Applications 112 (1989), 87-159. Patera, E. Pelantova and M. Svobodova: J. Math. Phys. 42 (2001), 3839-3853. Patera, E. Pelantova and M. Svobodova: J. Math. Phys. 43, 6353-6378 (2002). Patera, E. Pelantova and M. Svobodova: J. Math. Phys. 45 (2004), 2188-2198. Patera, E. Pelantova and M. Svobodova: J. Math. Phys. 48 (2007), 093503.
J. J. J. J.
REPORTS ON MATHEMATICAL PHYSICS
No.2
GROUP GRADINGS ON 0(8, q CRISTINA DRAPER* Departamento de Matematica Aplicada, Campus El Ejido, Universidad de Malaga, 29013 Malaga (Spain) (e-mail: [email protected])
and ANTONIO VIRUEL t Departamento de Algebra, Geometria y Topologfa, Campus de Teatinos, Universidad de Malaga, 29071 Malaga (Spain) (e-mail: [email protected]) (Received August 17, 2007 -
Revised November 9, 2007)
This is a matricial description of all the fourteen fine group gradings on the exceptional Lie algebra 0(8, iC). Keywords: fine grading, maximal abelian diagonalizable subgroup of automorphisms, exceptional Lie algebra, ()4.
1. Introduction Given a Lie algebra L, a Lie-grading on L is a vector space decomposition L = $iEIL i such that L i f=. 0 and for each pair of indices i, j E I there exists k E I such that LiL j C Lk (see Definition 2.1 below). Besides of its own mathematical interest, gradings on Lie algebras show up in numerous applications of physical nature. As it was already pointed out by Patera et al. in a series of papers on the subject [8-12], gradings are in the background of the choice of bases and the additive quantum numbers, as well as they are a key ingredient when studying contractions that keep "undeformed" certain subalgebra (e.g. see [7]). Gradings on Lie algebras also play a relevant role in quantum mechanics via Jordan algebras [6]: given a pair of indices i, j E I satisfying (LiLj)L i eLi, any element k E L j gives rise to a Jordan algebra structure in L i by defining x
0
y = (xk)y.
*Supported by MCYT grants MTM2004-06580-C02-02 and MTM2004-08115-C04-04, and by JA grants FQM-336 and FQM-1215. tPartially supported by the FEDER-MEC grant MTM2007-6oo16, and by the JA grants FQM-213 and FQM-2863. [265]
266
C.
DRAPER
and
A. VIRUEL
Gradings on the algebras o(n, C) have been computed in [1] and [5] for n i= 8. The case of n = 8 is of special relevance because of its strong symmetry, reflected partly by the triality automorphism. Because of this, 0(8, C) is usually considered as an exceptional Lie algebra. Previous works on gradings on exceptional Lie algebras are [2, 3]. Here we provide a neat description of all the fourteen fine group gradings on 0(8, C) (see Section 3). Our main contribution is the description of the ones produced by groups of automorphisms containing outer order 3 automorphisms. The proof showing that these are all the fine group gradings is lengthy and cannot be included here, so it appears in [4].
2. Basic concepts We introduce the basic concepts we use along this paper. DEFINITION 2.1. A group grading f on a Lie algebra is a decomposition in vector spaces L = tBgEGL g such that g, h E G, and G is generated by Supper) := {g E G I L g i= group grading is of type (h 1, ... , h s ) if it has hi components s is the greatest nonzero dimension of a component.
L over a group G LgL h C L gh for all O}. We say that the of dimension i, and
Note that the group grading f is also a Lie-grading over the set Supper). It is unknown whether or not every Lie-grading on a simple Lie algebra is in fact a group grading, although on nonsimple Lie algebras there are always Lie-gradings which cannot be considered as group gradings. DEFINITION 2.2. Two group gradings f and f' on a Lie algebra L, given by the decompositions L = tBgEGL g and L = tBhEHMh, are equivalent if there is a bijection A: Supper) ---+ Supp(f') and an' automorphism f E Aut(L) such that f(L g)
= MJ...(g).
Here we study gradings up to equivalence. Therefore, the grading group of a group grading is not unique, and it is convenient to choose a preferred one among the grading groups. This will be the best in the sense explained in [2]. REMARK 2. 1. Any group grading f is given by L = tB gEG u L g for certain group G u , called universal group, such that for any equivalent grading L = tBhEHMh there is a group epimorphism p: G u ---+ H and an automorphism f E Aut(L) such that f(L g) = Mp(g). DEFINITION 2.3. A group grading is called fine when it cannot be refined. A refinement of the grading L = tBgEGL g is another group grading L = tBhEHMh such that each component L g is sum of several components Mh. EXAMPLE 2.1. The main example of a fine group grading is the root decomposition in a simple Lie algebra, which turns out to be a grading over 71}, where l is the rank of the algebra. Recall that a group grading on a CC-algebra L is always the simultaneous diagonalization relative to a commuting set of semisimple automorphisms (details
GROUP GRADlNGS ON 0(8, iC)
267
in [2]). Moreover, a grading is fine when it is produced by a maximal subgroup Q of semisimple automorphisms, which is usually called a MAD. In this case, the universal group of the grading is just the group of characters X(Q) = hom(Q, CX). From now on, L will denote the algebra of skewsymmetric matrices 0(8, C) = {x E Matgxg(C) I x+x t = O}. Let eij be the elementary matrix with (i, j)-entry 1, and the remaining entries zero. Define hi) = eji - ei}' Then {hi} Ii, j = 1, ... , 8; i < j} is a basis of L, and the following section describes the fine group gradings on L in terms of this base.
3. The fine group gradings We introduce some useful elements of 0(8, C) to describe the group gradings on L: g(a) = el1 + e22 + e33 + e44 +
+ f(a) =
~ (a - ~) (eS7 + e6g + e7S + eg6),
~ (a + ~) (el1 + e22 + e33 + e44) + (ess + e66 + en + egg) +
h(a) =
~ (a + ~) (ess + e66 + en + egg)
~ (a - ~ )(e13 + e24 + e3l + e42),
~(a + ~)(el1 + e22 + e33 + e44 + ess + e66 + en + egg) +
~ (a - ~) (elS + e26 + e37 + e4g + eSI + e62 + e73 + eg4),
pea) = el1 + e22 + e33 + e44 + eSS + e66 + ~(a + ~)(en + egg)
-
+ ~(a ~)(e7g +eg7), q(a) = el1 + e22 + e33 + e44 +
+
~ (a + ~) (ess + e66)
~ (a - ~) (eS6 + e6S) + en + egg,
rea) = el1 + e22 + ~ (a + ~) (e33 + e44) + ~(a -l)(e34 + e43) + eSS + e66 + en + egg,
268
C. DRAPER and A. VIRUEL
sea) =
1( 1)
2" a + -;;
(ell
i( 1)
+ e22) + 2"
a - -;; (e12
+ e2l) + e33 + eM
+ eSS + e66 + en + e88, II = diag{l, 1, 1, 1, 1, 1, 1, -I},
h = diag{l, 1, 1, 1, 1, 1, -1,
13 = diag{l, 1, 1, 1, 1, -1,1, I},
14 = diag{1, 1, 1, 1, - 1, 1, 1, I},
Is = diag{l, 1, 1, -1, 1, 1, 1, I},
16 = diag{l, 1, -1,1,1,1,1, I},
17 = diag{l, -1,1,1,1,1,1, I},
18 = diag{-1, 1, 1, 1, 1, 1, 1, I},
= ish, g6 = hit, gl
= 161s, g7 = 17151311, g2
= Isf614h, glO = 16lshll, gs
g13
= 14hhit,
+ e2l + e34 + e43 + eS6 + e6S + e78 + e87, g8 = e13 + e3l + e24 + e42 + eS7 + e7S + e68 + e86, gll = e12 + e2I + e34 + e43 - eSS + e66 - en + e88, g12 = ell - e22 + e33 - eM - eS6 + e6S - e78 + e87, g14 = eiS + e26 + e37 + e48 + eSI + e62 + e73 + e84· g3
=
= hlsl4h, g9 = ishl4h, g4
I},
e12
Notice that j;, gi, I(a), g(a), h(a), pea), q(a), rea), sea) E 0(8, C) = {P E Mat8x8(C) I P pI = id} for all a E C*. Thus we can use the adjoint map Ad: 0(8, C) --+ Auto(8, C) to obtain (inner) automorphisms of L, Ad P(x) = p-Ixp. We define Fi = Ad Ii and G i = Adgi . Now, we completely describe the group of automorphisms, denoted by Qi, producing each grading. We also provide the simultaneous diagonalization, which is a straightforward computation by means of any computer algebra software. Grading over Z x Zi: QI = ({Adg(u), G I , G 2, G3, G4 I u E C*}) is an abelian diagonalizab1e subgroup of Aut(L). In fact, it is maximal according to [5] (it is the MAD corresponding to T2(~)' Hence it produces a fine grading, of type (25,0, 1), whose homogeneous comp'onents are: L{!2" 1 -1 , -1 , -I} = (ib3,s - b3,7 - ib4,6 + b4,8}),
+ b7,8), L{!,l,-I,-l,l} = (ib3,6 - b3,8 - ib4,S + b4,7), L{!,-I,I,-I,-I} = (ib1,5 - bI,7 - ib2,6 + b2,8), L{! 1 -11 -I} = (-ib3,s + b3,7 - ib4,6 + b4.8), 2" " L{!,-I,I,-I,I} = (ib l,6 - bl,8 - ib2,S + b2.7), L{!,l,-I,l,l} = (- ib 3,6 + b3,8 - ib4,5 + b4,7), L{!4' 11 "
-1 -I} ,
= (-b5,6 - ibs.8 + ib6,7
269
GROUP GRADINGS ON 0(8, C) L{!,_I,I,I,_I} L{!,_I,I,I,I}
= (-ib1,5
+ b1,7 (-ibI,6 + bl,S -
ib2,6 + b 2,s),
=
ib2,S
+ b S,6), (- ib 3,6 - b 3,s + ib4,s + b4,7), (-ibl,s - b1,7 + ib2,6 + b 2,s), (ib 3,s + b3,7 + ib4,6 + b 4,s), (-ib l ,6 - b1,8 + ib 2,s + b2,7), (ib3,6 + b3,S + ib4,s + b4,7), (ib1,5 + b1,7 + ib2,6 + b 2,s), (-b S,6 + ibs,s - ib 6 ,7 + b 7,s), (ib l ,6 + b1,8 + ib 2,s + b2,7), (bs,s + b6,7), (-ib 3,s - b3,7 + ib4,6 + b 4,s),
L{l,I,I,-I,-I}
= (b l ,2, b3 ,4,
L{2,1,-I,-I,I}
=
L{2,-I,I,-I,-I}
=
L{2,I,-I,I,-I}
=
L{2,-I,I,-I,I}
=
L{2,I,-I,I,I}
=
L{2,-I,I,I,-I}
=
L{4,1,1,-I,-I}
=
L{2,-I,I,I,I}
=
L{l,I,I,I,-I}
=
L{2,1,-I,-I,-1}
=
L{l,-I,-I,-I,-I}
= (b 2 ,3
-
L{l,-I,-I,I,-I}
= (b 2,3
+ bl ,4),
L{l,I,I,-I,I}
=
+ b2,7),
b 7,s
= L{l,-I,-I,-I,I} =
b l ,4),
L{l,I,I,I,I}
(b6,S - b S,7),
L{l,-I,-I,I,1}
(b6,s
+ b S,7),
(b2,4 -
= (b2,4
b1,3),
+ b1,3),
The subindex of each component is not the subindex of the group grading, but the set of eigenvalues of {Adg(2), G I , G2, G3, G4} for which the component is an eigensubspace (e.g. the index 1, -1, -1, -I} is for the element (-2, 0, 1, 1, 1) E ZxZ~). The universal group of the grading is just ZxZ~(= X(QI)). These comments about the universal group and the order and eigenvalues of the automorphisms, are valid for all the gradings in this paper. Grading over Z2 x Z~: Q2 = ({Ad f(u), Adg(v), G3, Gs I u, v E C*}) ~ (C*)2 X Z~ is an abelian diagonalizable subgroup of Aut(L). In fact, it is the MAD corresponding to T4:~ ([5]). The simultaneous diagonalization of L relative to this subgroup of automorphisms is of type (20,4):
{!'
+ b4,7), L{!,~,_I,I} = (b1,5 + ib1,7 - b2,6 - ib2,S + ib3,s - b3,7 - ib4,6 + b 4,s), L{!,~,I,_I} = (-b l,6 - ibl,s - b2,S - ib2,7 - ib3,6 + b 3,s - ib4,S + b4,7), L{!,~,l,l} = (-bl,s - ibl,7 - b2,6 - ib2,s - ib 3,5 + b3,7 - ib4,6 + b4,S), L{!,3,_I,_I} = (-b l,6 + ibl,s + b2,S - ib2,7 - ib3,6 - b3,S + ib4,s + b4,7), L{!,3,-I,1} = (-bl,s + ibl,7 + b2,6 - ib2,s - ib3,s - b3,7 + ib4,6 + b 4,s), L{!,~,_I,_I}
=
(b l,6 + ib1,8 - b2,5
-
ib2,7 + ib3,6 -
b 3,s -
ib4,5
270
C. DRAPER and A. VIRUEL
L{i,3,1,-lj = (b l,6 - ib1,8 L{i,3,1,lj
= (b1,5 - ib1,7
+ bz,s -
+ bZ,6 -
L{Z,j,-I,-1} = (-b l,6 - ib1,8 L{Z,1,-I,lj = (-b1,5 - ib1,7 L{Z,j,I,_lj = (b l,6 L{Z,j,I,lj = (b1,5
ibz,7 ibz,s
+ ib3,6 + b3,s + ib4,s + b4,7),
+ ib3,s + b3,7 + ib4,6 + b4,s),
+ bz,s + ibz,7 + ib3,6 -
+ bZ,6 + ibz,s + ib3,s -
+ ibl,S + bz,s + ibz,7 -
+ ib1,7 + bZ,6 + ibz,s -
ib 3,6 ib3,s
b3,s - ib4,s b 3,7 - ib4,6
+ b3,s -
+ b3,7 -
ib4,s ib4,6
+ b4,7),
+ b4,s),
+ b4,7),
+ b4,s),
+ ib z,7 - ib3,6 - b3,s + ib4,s + b4,7), L{Z,3,-I,lj = (b1,5 - ib 1,7 - bZ,6 + ibz,s - ib 3,s - b3,7 + ib4,6 + b4,s), L{2,3,1,-lj = (-b l ,6 + ib1,8 - bz,s + ibz,7 + ib 3,6 + b 3,s + ib4,s + b4,7), L{2,3,1,lj = (-b1,5 + ib l ,7 - bZ,6 + ibz,s + ib 3,s + b 3,7 + ib4,6 + b4,s),
L{2,3,-I,-lj
= (b l ,6 - ib1,8 - bz,s
+ bZ,4, bS,7 + b6,s), L{I,I,-I,-Ij = (b1,z + b3,4, bS ,6 + b7,s), L{I,I,-I,lj = (-b1,3 + b Z,4, -b S ,7 + b6,S), L{I,l,l,-lj = (b l,4 + b Z,3, bs,s + b6,7), L{!4" I -I , -Ij = (-b1,z - ibl,4 + ibz,3 + b3,4), L{I,9,-I,-lj = (-b S ,6 + ibs,s - ib6,7 + b7,s), L{4,1,-I,-lj = (-b1,2 + ibl,4 - ib z,3 + b3,4) , L{l '9' ! -I -Ij = (-b S ,6 - ibs,s + ib6,7 + b7,s). , L{I,I,I,lj
= (b1,3
Grading over Zi: Q3 = ({Fi I i = 1, ... , 7}) ~ Zi is a MAD of Aut(L), corresponding to TJo~. The simultaneous diagonalization of L relative to it is of type (28): L{I,-l,-l,l,l,l,lj
=
(b6,7),
L{I,-l,l,-l,l,l,lj
= (b S,7),
L{I,-I,l,l,-I,l,lj
= (b4,7),
L{I,-l,l,l,l,-l,lj
= (b3,7),
L{I,-I,I,I,I,I,-Ij
= (bZ,7),
L{I,-l,l,l,l,l,lj
L{l,l,-l,-l,l,I,lj
=
(bS,6),
L{I,l,-l,l,-l,l,lj
= (b 4,6),
L{I,l,-I,l,l,-I,lj
= (b 3,6),
L{1,1,-l,l,l,l,-lj
= (b Z,6),
L{I,l,l,-l,-l,l,lj
= (b 4 ,s),
L{I,l,l,-l,I,I,-lj
=
L{I,l,l,l,-l,-l,1}
= (b 3,4),
L{l,l,_l,l.l,l,lj
= (b1,6),
L{I,l,l,-l,l,-l,lj L{I,l,l,-l,l,l,lj
=
(b3,S),
= (b1,5) ,
L{I, I, I, 1,-1, 1,-1}
= (bz,4),
L{I,l,l,l,-l,l,lj
= (b 1,7),
(bz,s),
= (b1,4),
271
GROUP GRADINGS ON 0(8, C) L{l,l,l,l,l,-l,-lj L{l,l,l,l,l,l,-lj
= (bZ,3),
L{l,l,l,l,l,-l,lj
= (bl,2),
L{-l,-l,l,l,l,l,lj
= (b6,s), L{-l,l,l,l,-l,l,lj = (b 4 ,s),
= (b7,s),
= (bs,s), L{-l,l,l,l,l,-l,l} = (b 3 ,s),
L{-l,l,-l,l,l,l,lj
L{-l,l,l,l,l,l,-lj
= (b1,3),
L{-l,l,l,-l,l,l,lj
= (bz,s),
L{-l,l,l,l,l,l,lj
= (bl,s).
Grading over Z~: Q4 = ({G 6 , G l , G z, G 3 , G 7 )} ~ Z~ is a MAD of Aut(L), corresponding to Td,~' The simultaneous diagonalization of L relative to the quasitorus Q4 is of type (24,0,0,1), with L{l,l,l,-l,-l} = (bl,2, b 3,4, bS ,6, b7,s) and:
= (-b1,8 + bZ,7), L{-I,-I,I,I,-lj = (b1,8 + bZ,7), L{-I,I,-I,-I,-lj = (-b3,s + b4,7), L{-I,I,-I,I,-lj = (b3,s + b4,7), L{-I,I,I,-I,-lj = (-bs,s + b6,7), L{-I,I,I,I,-lj = (bs,s + b6,7), L{l,-l,-l,-l,-l} = (-b l ,4 + bZ,3), L{l,-l,-l,l,-lj = (b l,4 + bZ,3), L{l,-l,l,-l,-l} = (-b l ,6 + bz,s), L{l,-I,I,I,-lj = (bl,6 + bz,s), L{1,1.-I,-I,-lj = (-b 3,6 + b4,s), L{l,l,-l,l,-lj = (b 3,6 + b4,s),
= (-b1,7 + bz,s), L{-I,-I,I,I,lj = (b1,7 + bz,s), L{-I,I,-I,-I,lj = (-b 3,7 + b4,s), L{-I,I,-I,I,lj = (b3,7 + b4,s), L{-I,I,I,-I,lj = (-b S ,7 + b6,s), L{-I,I,I,I,lj = (b S ,7 + b6,s), L{l,-l,-I,-I,lj = (-b1,3 + b Z,4), L{l,-l,-I,l,lj = (b l ,3 + b Z,4), L{l,-I,l,-I,l} = (-b1,5 + b Z,6), L{l,-I,I,I,1} = (b1,5 + bZ,6), L{l,I,-I,-I,lj = (-b 3,s + b4,6), L{l,I,-I,I,lj = (b 3,s + b4,6).
L{-l,-l,l,-l,-lj
L{-I,-I,I,-I,lj
Grading over Z x Zi: Qs = ({Adh(u), G s , G 9 , G 3 , G s I u E C*}} ~ (C*) x Zi is a MAD of Aut( L), corresponding to Tz(~' The simultaneous diagonalization of L relative to the quasitorus Qs is of type '(28): L{l,-l,-l,-I,-Ij L{l,-I,-I,-l,lj
+ b4,s), bS ,7 + b6,s),
= (bl,s - bZ,7 - b3,6
= (-b1,3
+ bZ,4 -
+ bZ,3 + bs,s + b6,7), L{l,-I,-I,I,lj = (b1,3 + bz ,4 + bS ,7 + b6,s), L{l,-I,l,-I,-lj = (-bl,2 + b3,4 - bS ,6 + b7,s), L{l,-I,l,-I,lj = (b1,5 - b Z,6 - b 3,7 + b4,S), L{l,-I,I,I,-Ij = (-b l ,6 - bz,s + b 3,s + b4,7), L{l,-I,I,I,lj = (-bl,s - b Z,6 + b3,7 + b4,s), L{l,-I,-I,I,-lj
= (b l ,4
L{l,l,-l,-I,-Ij
= (-b l ,4
L{I,l,-I,-I,lj
= (-b1,7
+ bZ,3 -
+ bZ,8 -
bS ,8 b 3,s
+ b6,7),
+ b4,6),
272
C. DRAPER and A. VIRUEL
+ b2,7 + b3,6 + b4,s), L{l,I,-I,I,lj = (b l,7 + b 2,s + b3,s + b4,6), L{l,I,I,-I,-lj = (b l,2 + b3,4 + bS,6 + b7,s), L{l,I,I,-I,lj = (-b1,5 + b2,6 - b3,7 + b4,s), L{l,I,I,I,-lj = (b l,6 + b 2,s + b 3,s + b4,7), L{l,I,I,I,lj = (b1,5 + b2,6 + b3,7 + b4,s), L{l,I,-I,I,-lj
= (b1,8
+ ib4,6 - bS,7 + b6,S), L{! -I -I I -Ij = (-b l,4 - ib1,8 - b2,3 - ib2,7 + ib3,6 + ib4,s + bs,s + b6,7), 4' , " L{! -I -I I Ij = (-b1,3 - ib1,7 - b2,4 - ib2,s + ib3,S + ib4,6 + bS,7 + b6,s), 4' , " L{! -I I -I -Ij = (b l,2 + ibl,6 - ib2,s - b3,4 - ib3,s + ib4,7 - bS,6 + b7,s), 4' " , L{!4" I -I , -I , -Ij = (b l,4 + ib1,8 - b2,3 - ib2,7 + ib3,6 - ib4,s - bs,s + b6,7), L{! I I -I -Ij = (-b1,2 - ibl,6 + ib2,S - b3,4 - ib 3,s + ib4,7 + bS,6 + b7,s), 4' " , L{4,-I,-I,-I,lj = (b1,3 - ib l ,7 - b2,4 + ib2,S + ib 3,s - ib4,6 - bS,7 + b6,s), L{4,-I,-I,I,-lj = (-b l,4 + ibl,s - b2,3 + ib 2,7 - ib3,6 - ib4,s + bs,s + b6,7), L{4,-I,-I,I,lj = (-b1,3 + ib1,7 - b2,4 + ib2,s - ib 3,s - ib4,6 + bS,7 + b6,S), L{4,-I,I,-I,-lj = (b l ,2 - ib l ,6 + ib 2,s - b3,4 + ib 3,s - ib4,7 - bS,6 + b7,s), L{4,1,-I,-I,-lj = (b l,4 - ib1,8 - b2,3 + ib2,7 - ib 3,6 + ib4,s - bs,s + b6,7), L{4,1,1,-I,-lj = (-b1,2 + ib l ,6 - ib 2,s - b3,4 + ib 3,s - ib4,7 + bS,6 + b7,s). L{~,_I,_I,_I,lj = (b1,3
+ ibl,7 -
b2,4 - ib2,s - ib3,s
Grading over Z x Z~: Q6 = ({p(u), Fs , F7 , F6 , Fs, F4 I U E C*}) ~ C* x Z~ is again an abelian diagonalizable subgroup of Aut(L), concretely the MAD T2~~' The simultaneous diagonalization of L relative to the quasitorus Q6 is of type (28): L{l,-I,-I,I,I,lj
=
(b l ,2),
L{l,-I,I,-I,I,lj
= (b1,3),
L{l,-I,I,I,-I,lj
= (b l ,4),
L{l,-I,I,I,I,-Ij
= (bI,S),
L{l,I,-I,-I,I,1}
= (b 2,3),
L{l,I,-I,I,I,-lj
= (b 2,s),
L{l,I,I,-I,-I,lj
= (b3,4),
L{l,-I,I,I,I,lj
= (b l ,6),
L{l,I,-I,I,-I,lj L{l,I,-I,I,I,lj
= (b2,4),
= (b 2,6),
L{l, 1.1,-1,1,-1} = (b 3 ,s), L{l,I,I,I,-l,-lj L{l,I,I,I,I,-lj
L{! -I I I I Ij 2'
""
=
(b 4,s),
= (b S,6),
= (ib l ,7 + b1,8),
L{l,I,I,-I,I,lj
= (b 3,6),
= (b4,6), = (b7 .S ),
L{l,I,I,I,-I,1} L{l,I,I,I,I,1}
L{! I -I I I Ij 2"
",
= (ib 2,7
+ b 2,S),
273
GROUP GRADINGS ON 0(8, iC)
+ b3,S), L{1,1,1,I,I,-I} = (ibS,7 + bs,s), L{Z,-I,I,I,I,I} = (-ib l ,7 + bl,s), L{2,I,I,-I,I,I} = (- ib 3,7 + b3,s), L{Z,I,I,I,I,-I} = (-ib s,7 + bs,s),
+ b4,s), L{!,I,I,I,I,I} = (ib 6,7 + b6,s), L{2,I,-I,I,I,1} = (-ibz,7 + bz,s), L{Z,I,I,I,-I,I} = (-ib4,7 + b4,s), L{Z,I,I,I,I,1} = (-ib6,7 + b6,s).
= (ib3,7
L{1,1,1,-I,I,I}
= (ib4,7
L{!,1,1,I,_I,I}
Grading over 'l} x Z~: Q7 = ({p(u), Fs, F 7 , F 6 , q(v) I u, v E C*}} ~ (C*)z x Z~ is a MAD of Aut( L), corresponding to T4(~' producing the following grading, of type (26, 1): ' L{2,I,I,I,3}
L{1,1,1,I,j-} = (-bS,7 L{Z,I,I,I,j-}
=
L{I
=
I I -II}
2' "
,
L{I,-I,-I,I.1}
+ ibs,s + ib6,7 + b6,s),
+ ibs,s - ib6,7 + b6,S), (bS,7 - ibs,s + ib6,7 + b6,s),
= (bS,7
L{!,1,1,I,3} = L{ 2' I -I I I I} ",
+ bl.S), (ib3,7 + b3,s), (ib l,7
= (b1,2) ,
+ b l .6), L{I,-I,I,I,3} = (-ib1,5 + b l ,6), L{II - I I I} = (ibz,s + bZ,6), " "3 L{I,I,-I,I,3} = (-ibz,s + b Z,6), L{I
,
-I I I I} "'3
L{I,I,I,_I,I}
= (ibl,s
=
+ b6,s),
= (-bS,7 - ibs,s - ib6,7
(b 3 ,4),
+ b4 ,6), L{I,I,I,I,3} = (-ib4,s + b4,6), L(Z,I,-I,I,I} = (-ib z,7 + bz,s), L{Z,I,I,I,I} = (- ib4,7 + b4,s). L(I,I,I,I,~} = (ib 4 ,s
L{!
2"
+ bz,s), (i b4,7 + b4,s),
= (ibz,7
I -I I I} "
=
L 1!,1,1,I,I}
L{I,_I,I,_I,I}
= (b1,3) ,
= (b l ,4),
L{I,_I,I,I,I}
= (b Z,3),
L{I,I,_I,_I,I}
= (b Z,4),
Ll,l,-I,I,I}
L{I , I "I
-I I}
'3
L{I,I,I,-I,3} L{I,I,I,I,I}
= =
+ b3,6), (-ib 3,s + b3,6), (ib3,s
= (b S ,6, b7 ,s),
+ b1,8), L{Z,I,I,-I,I} = (-ib3,7 + b3,s), L{Z,-I,I,I,I}
= (-ib l ,7
Grading over Z~ x Z4: Qs = ({Gs, GIO, G l1 , G!2l) ~ Z~ x Z4 is a MAD of Aut(L), corresponding to To(~' which produces a grading of type (24,2) on L:
+ bl,s + ib z,7 = (-ib1,7 + bl,s - ib z,7 = (-ibl,7 - bl,s + ib z,7 -
L{-I,-I,-I,-i}
LH,-I,-I,i} L{-I,-I,I,-i}
= (ib1,7
+ b4,6), bz,s + ib 3,s - b3,6 + ib4,s + b4,6), bz,s + ib 3,s + b3,6 - ib4,s + b4,6),
bz,s - ibo,s - b3,6 - ib4,S
274
C. DRAPER and A. VIRUEL
+ b3,6 + ib4,s + b4,6}, L{-I,I,-I,-ij = (ib1,5 + b l ,6 + ibz,s - bZ,6 - ib 3,7 - b3,s - ib4,7 + b4,S}, L{-I,I,-I,i} = (-ib1,5 + b 1,6 - ibz,s - bZ,6 + ib 3,7 - b3,s + ib4,7 + b4,S}, L{-I,I,I,-i} = (-ib1,5 - b 1,6 + ibz,s - bZ,6 + ib 3,7 + b3,s - ib4,7 + b4,S}, L{-I,I,I,i} = (ib1,5 - b l ,6 - ibz,s - bZ,6 - ib 3,7 + b 3,s + ib4,7 + b4 ,s}, = (ib1,7 - b1,8 - ib z,7
L{-I,-I,!,ij
L{l,-I,-I,-i}
= (-ib l ,7
-
bl,s -
-
bz,s -
ib z,7
ib 3,s
+ bz,s -
ib 3,s - b3,6 - ib4,s
+ b4,6},
+ ib z,7 + bz,s + ib3,s - b3,6 + ib4,s + b4,6}, L{l,-I,I,-i} = (ib1,7 + b1,8 - ibn + bz,s + ib 3,s + b 3,6 - ib4,s + b4,6}, L{l,-I,I,ij = (-ib l ,7 + bl,s + ib z,7 + bz,s - ib 3,s + b 3,6 + ib4,s + b4,6},
L{l,-I,-I,ij
= (ib1,7 -
L{l,I,-I,-i}
= (-ib1,5 - b l ,6 - ibz,s
b1,8
+ bZ,6 -
b3,s - ib4,7 + b4,S}, b3,s + ib4,7 + b4,s},
ib 3,7
-
+ ibz,s + bZ,6 + ib3,7 L{l,I,I,-i} = (ib1,5 + b l ,6 - ibz,s + b Z,6 + ib 3,7 + b 3,s - ib4,7 + b4,s}, L{l,I,I,ij = (-ib1,5 + b l ,6 + ibz,s + bZ,6 - ib 3,7 + b3,s + ib4,7 + b4,s}, L{l,I,-I,i}
= (ib1,5 - b l ,6
= (bs,s + b6,7}, L{-I,I,-I,-l} = (-bl,z + b 3,4}, L{l,-I,-I,I} = (-bs,s + b6,7}, L{l,l,-l,-I} = (bl,z + b3,4},
= (-b1,3 + bZ,4}, L{_I,I,_I,I} = (-b S,6 + b7 ,s}, L{l,I,-I,I} = (b S,6 + b7,s}, L{-I,-I,l,-I} = (b l ,4 + b Z,3, -b S ,7 + b6,s},
L{-l,-I,-I,-l}
L{l,-I,-I,-l}
= (-b l ,4
L{_I,_I,_I,I
+ bZ,3},
L{_I,_I,I,I}
= (b1,3
+ bZ,4,
bS,7
+ b6,s}.
Grading over Z~: Q9 = ({Gs, GIO, G 13 , G 3 , G 7 , G I4 }} ~ Z~ is also a MAD of Aut(L), corresponding to The induced grading is of type (28):
T(?i-
L{_I,_I,_I,_I,_I,_I}
= (bl,s - bZ,7
-
b3,6 + b4 ,s},
+ b4,6} L{_I,_I,_I,I,_I,I} = (-bl,s - bZ,7 + b3,6 + b4,s}, L{_I,_I,_I,I,I,I} = (-b l ,7 - bz,s + b 3,s + b4,6}, L(-I,-I,I,-I,l,-l} = (b1,3 - bZ,4 - bS ,7 + b6,s}, L(-I,-I,I,-I,I,I} = (-b1,3 + b z,4 - bS,7 + b6,s}, L{_I,_I,I,I,_I,_I} = (-b l ,4 - bZ,3 + bs,s + b6,7}, L{-I,-I,-I,-l,l,l}
L{-I,-I,I,l,-I,I}
= (b l ,7
-
bz,s -
= (b l ,4 + bZ,3
b3,s
+ bs,s + b6,7},
= (-b1,3 - bz,4 + bS,7 + b6,s}, = (b1,3 + b2 ,4 + bS,7 + b6,s},
L{-I,-I,I,l,l,-I}
L{-I,-I,I.l,I,lj
+ b4,7}, b3,7 + b4,s},
L{-I,I,-I,-l,-l,l}
= (b l ,6 - bz,s - b3,s
L{-I,I,-I,-I,I,-lj
= (b1,5 - bZ,6 -
GROUP GRADINGS ON 0(8, iC)
275
+ b3,S + b4,7), L{-I,I,-l,l,I,-I} = (-bI,s - b Z,6 + b 3 ,7 + b4 ,s), L{-I,I,I,-I,-l.-I} = (bI,z - b3,4 - b S,6 + b7 ,s), L{-l,l,I,-I,-I,I} = (-bI,z + b3,4 - b S,6 + b7 ,s), L{l,-I,-I,-I,-I,I} = (-b1,8 + b Z,7 - b3,6 + b4,s), L{l,-I,-I,-I,I,-I} = (-b I ,7 + bz,s - b 3 ,s + b4 ,6), L{-l,l,-l,l,-l.-I}
=
L{l,-I,-I,I,-l.-l}
= (b1,8
L{l,-I,-I,I,I,-l}
+ b Z,7 + b3,6 + b4,s), (b I ,7 + bz,s + b 3 ,s + b4 ,6),
=
L{l,-I,I,-I,-I,-I} L{l,-I,I,-I,-I.l}
=
L{l,I,-I,-I,-I,I}
=
L{l,I,-I,-I.l,-l}
=
(-b 1,6 - bz,s
=
(b I ,4 -
+ b6,7), bs,s + b6,7), b3,s + b4,7), b 3,7 + b4,s),
b Z,3 - bs,s
+ b Z,3 (-b 1,6 + bz,s (-bI,5 + b Z,6 -
(-b I ,4
= (b I ,6 + bz,s + b3 ,s + b4 ,7), = (bI,5 + b Z,6 + b 3,7 + b4,s),
L{l,I,-I,I,-I,-I} L{l,I,-l,l,I,-I}
L{l,l,l,-l,-I,-I} L{l,I,I,-I,-l,I}
= (-b1,2 - b3,4
= (b1,2
+ bS,6 + b7,s),
+ b3,4 + bS,6 + b7 ,s).
Grading over Z3 xZz: QIO = ({p(U), Fs, rev), q(w) I u, v, WE C*}) ~ (C*)3 xZz is an abelian diagonalizable subgroup of Aut(L). More precisely, it is the MAD-group T6(~' which produces a grading of type (25,0, 1): L{1,l,Q} = (-bS,7 L{I2' I';' I I}
=
+ ibs,s + ib6,7 + b6,S),
(-b37+ib38+ib47+b48), ' , , ,
+ ib6 ,7 + b6,s), ib3,s + ib4,7 + b4,s),
L{~,I,I,3} = (bS,7 - ibs,s
L ,I,5,I} = (b3,7 1 L{I I I I} = (-b3,s , '5' 3
+ ib3,6 + ib4,s + b4,6),
+ ib4,s + b4,6), L{l,l.!,3} = (b3,s + ib3,6 - ib4,s + b4,6), L{l,I,S,3} = (-b 3,s - ib 3,6 - ib4,s + b4,6), L{Z,q,I} = (b3,7 + ib3,s - ib4,7 + b4,s), L{Z,I,q} = (bS,7 + ibs,s - ib6,7 + b6,s), L{Z,I,I,3} = (-b S,7 - ibs,s - ib 6 ,7 + b6 ,s), L{I,I,S,1}
=
(b3,s -
ib3,6
276
C. DRAPER and A. VIRUEL
L{i,-I,I,I} L{I,_I,!,I}
+ h,s}, = (ib1,3 + bl ,4},
= (ib l ,7
L{ I,-I,I,I} = (b 1,2 } , L{l,-I,S,I}
= (-ib1,3
+ b2,6},
L{I,I,I,~}
=
L{l,I,I,3}
= (-ib 2,s
(ib 2,s
+ b l ,4},
+ b2,6},
L{2,-J,J,I} = (-ibJ,7
+ bJ,s},
+ b2,s}, L(I , - I"3 I ! } = (ib u + b l ,6}, L{l,-I,I,3} = (-ib U + b l ,6}, L{J,J,!, I} = (i b2,3 + b2,4} ,
L( l' I Ill} = (ib 2,7 , ,
L{l,I,I,I}
= (b 3,4, bS ,6, b7,s},
L{l,I,S,I}
= (- ib 2,3
L{2,I,I,I}
=
+ b2,4} , (-ib 2 ,7 + b2,s}.
Grading over Z4: QIJ = ({Adp(u), Ads(v), Adr(w), Adq(z) I u, v, w, Z E C*}} ~ (C*)4 is an abelian diagonalizable subgroup of Aut(L) (corresponding to Ts(,~)' The simultaneous diagonalization of L relative to the quasitorus QII (in fact, a torus) is of type (24,0,0, 1). Of course this is the Cartan grading, in other words, the decomposition in root spaces relative to the Cartan subalgebra L {I,I,I,I} -- (b 1,2, b3,4, bS,6, b}' 7,S· L{IIII}=(-bI7+ibIS+ib27+b2S}, 1'7' , , , , , L{Il' II , '31}=(-bs7+ibss+ib67+b6S}, ' , , ,
+ ib4 ,7 + b4,s}, L{I! ! I} = (-b1,3 + ib l ,4 + ib2,3 + b2,4}, , 7' 5' L{I,~,s,l} = (b1,3 + ib l ,4 - ib2,3 + b2,4}, L{I I ! !} = (-b 3,s + ib3,6 + ib4 ,s + b4 ,6}, , '5' 3 L{l,I,s,~} = (b3,s - ib3,6 + ib4 ,s + b4 ,6}, L{J,7,!,1} = (b1,3 - ib l ,4 + ib2,3 + b2,4} , L{l,7,S,I} = (-b1,3 - ib l ,4 - ib 2,3 + b2,4} , L{2! I I} = (bJ,7 + ib1,8 - ib2,7 + b2 ,s}, '7' , L{2,I,q} = (bs,7 + ibs,s - ib6,7 + b6,s}, L{2,I,S,I} = (-b 3,7 - ib 3,s - ib4 ,7 + b4 ,s},
L{1,I,S,I}
= (b 3 ,7 - ib3,s
+ ib3,s + ib4 ,7 + b4 ,S}, L{i,1,J,3} = (bs,7 - ibs,s + ib6,7 + b6,s}, L{i,7,1,1} = (b l ,7 - ibl,s + ib2,7 + b 2,s}, L(I ! I !} = (-b u + ib l ,6 + ib2,s + b2,6}, '7' , 3 L{l,~,1,3} = (b u + ib 1,6 - ib2,s + b2,6}, L{J,J,!,3} = (b3,s + ib3,6 - ib4 ,s + b4 ,6}, L{l,I,S,3} = (-b 3,s - ib 3 ,6 - ib4 ,s + b4 ,6}, L{I,7,1'1} = (b u - ibJ,6 + ib2,s + b2,6}, L{l,7,1,3} = (-b u - ib 1,6 - ib 2,s + b2,6}, L{2,1,!,1} = (b3,7 + ib3,s - ib 4 ,7 + b4,s}, L{2,1,1,3} = (-b S ,7 - ibs,s - ib 6,7 + b6,s}, L{2,7,1,I} = (-b l ,7 - ib1,8 - ib 2,7 + b2,s}. L{!
I ! I}
2' , 5'
=
(-b3,7
In order to describe the gradings involving a copy of Z3, we introduce some outer automorphisms. First choose a basis of root vectors in L. Such a basis could be the one produced by Q I I, but it is easier to find a basis of root vectors in the isomor-
277
GROUP GRADINGS ON 0(8, iC)
phic Lie algebra i, j
L = {x
E
{I, ... , 4}, we define
E
if i > j, and a3,1, a4,1,
Ci,j
hi
+ Cx t = OJ,
where
C = (0
14
1o4 )
. For
= ei,i-ei+4,i+4, ai,j = ej,i-ei+4.j+4, Ci,j = ej,i+4-ei,j+4
= ei+4,j -ej+4,i if i
< j. Fix B
= {hI, h 2 , h 3 , h4, a2,1, a3,2, a4,3, C3,4,
a4,2, C4,1, C4,2, C2,1, C3,1, C2,3' al,2, a2,3, a3,4, C4,3, al,3, al,4, a2,4, CI,4, C2,4, Cl,2,
cu, C3,2}, and denote by i, j, k = 1, ... ,28. Define HI
Mat8x8(C) I xC
bi
its i-th element. Now let
({Ji,j(b k )
= 8k ,jbi
for
+ ({J3, I + ({J3,2 + ({J5,27 + ({J6,8 + ({J7,9 + ({J8,25 + ({JII, 12 + ({J12,26 + ({J13,18 + ({J14,23 + ({J17,15 + ({J18,20 + ({J19,21 + ({J20,13 + ({J23,24 + ({J24,14 + ({J25,6 + ({J26, II
=
({J2,4
- ({JI, I - ({JI,2 - ({JI,3 -
({JI,4 - ({J4,2 -
- ({J21,10 - ({J22,7 - ({J27,16 -
({J4,4 -
({J9,22 -
({JIO,19 - ({J15,28 -
({J16,5
({J28,17,
+ ({J2,3 + ({J2,4 + ({J4,1 + ({J5,26 + ({J6,16 + ({J7,19 + ({J8,5 + ({J1O,22 + ({JII,13 + ({J12,18 + ({J14,8 + ({J15,23 + ({J16,12 + ({J17,14 + ({J18,28 + ({J19,7 + ({J20,17 + ({J22,IO + ({J23,25 + ({J24,6 + ({J26,20 + ({J27,1l + ({J28,24 - ({JI,I - ({JI,3 - ({JI,4 - ({J3,3 - ({J9,21
H2 =
({J2,2
- ({J13,15 - ({J21,9 tx,y,Z,u
({J25,27 -
2({JI,2,
.{ 2 = dlag 1,1,1,1, x, y, z, u, xy, xyz, yz, uxy, uy, uxy z, uxyz, uyz,
11111 xy' xyz'
1111 x y z u
- , - , -, - ,
Ill}
YZ' uxy' uy' uxy2z' uxyz' uyz
.
Note that HI and H2 are outer order 3 exterior automorphisms. The automorphism H2 fixes a subalgebra of type G2, hence it can be considered as the triality automorphism (we have built it by extending the corresponding automorphism of the Dynkin diagram). And, although there is P E 0 (8, C) such that the composition H2 Ad(P) is conjugated to HI in Aut(L), we have chosen HI because it normalizes the maximal torus of the automorphism group given by {tx,y,Z,u I x, y, z, u E (CX}. Now we can describe the remaining gradings. I
Grading over Z~: If we denote w = -(-1)1, QI2 = ({tl,w2,w2,w2, tw2,I,w2,1' Hd) ~ Z~ is an abelian diagonalizable subgroup of Aut(L) inducing a grading of type (24,2): L{I,I,lU) L{I,I,(2)
=
=
(-wb l -
2 (-w b l -
+ b 18 + b20), L{I,w,(2) = (wb13 + w2bl8 + b 20 ) , 2 L{l,w2,wl = (w b 6 + wb 8 + b 25 ), L{w,I,I) = (-b I 5 - b 17 + b28), L{I,w,1)
=
(b 13
+ b 3 , (1 + W)b2 + b 4 ), 2 2 w b 2 + b3, (1 + ( )b2 + b4),
wb 2
2 (w b 13
+ Wb l8 + b 20 ) , L{I,w2,I} = (b6 + b 8 + b 25 ), 2 L{I,w2,w2} = (wb6 + w b 8 + b 25), 2 L{w,l,w) = (-WbI5 - w b 17 + b 28 ), L{I,w,w}
=
278
C. DRAPER and A. VIRUEL
+ b28), L{UJ,UJ,UJ} = (-w b lO + Wb l9 + b 21 ), L{UJ,UJ2,1} = (b 14 + b23 + b 24 ), 2 L{UJ,UJ2.UJ2} = (Wb 14 + w b 23 + b24) , 2 L(UJ2,I,UJ} = (wb5 - w h6 + b 27 ), L(UJ2.UJ,I} = (b ll + b 12 + b 26 ), 2 L{~,UJ,UJ21 = (wbl l + w b 12 + b26), 2 L{UJ2,UJ2,w} = (-w b 7 - wb 9 + b 22 ), L{UJ,I,UJ2)
=
(-«ib I5 - wb 17 2
+ big + b 21 ), 2 L{w,w,w2} = (-wb lO + w b l 9 + b 21 ), 2 L{w,w2,UJ} = (w b 14 + Wb 23 + b 24 ), L{UJ2,1,1) = (b 5 - b 16 + b 27 ), L{w2,1,(2) = (w2b5 - Wbl 6 + b 27 ), 2 L{w2,UJ,UJ) = (w b ll + Wb 12 + b 26 ). L{UJ2,UJ2,1} = (-b7 - bg + b 22), L(UJ2,UJ2,UJ2) = (-wb 7 - w2b9 + b 22 ). L{w,w.l}
=
(-b lO
Grading over Z~ x 23: Q13 = ({fl,-I,l,l, Ll,-I,-I,-I, H2 }) ~ 2~ x 2 6 is an abelian diagonalizable subgroup of Aut(L) , producing a grading of type (14,7):
+ b 12 - b l6 - b l8 - b 24 + b 28 , -b lO + b 22 ), L(_I,_I,I} = (b 6 + b12 + b l6 + b l 8 + b 24 + b 28, blO + b 22 ), L(_l,l,_1} = (-bll + b13 + bl5 - b23 + b25 + b 27 , bg + b21), L{-I,I,l) = (b ll + b13 - b l5 - b23 - b 25 + b 27, -b9 + b 21 ), L{l,-I,-I} = (-b5 + b 8 - b l4 + b17 - b20 + b 26 , -b7 + bI9), L{l,-I,I} = (b 5 + b 8 + b l4 + b 17 + b20 + b26, b 7 + b I9), L{l,l,-I} = (b 3 , b l + b2 ), 2 2 L{-l,-I,UJ} = (Wb6 + w b 12 + b l6 + Wb l8 + w b 24 + b28), 2 2 L{_I,_I, -UJ) = (wb 6 + w b 12 - b l6 - Wb l 8 - w b 24 + b28), 2 2 L{-l,-I,_UJ2} = (w b 6 + wb 12 - bl6 - w b l8 - Wb 24 + b 28 ), 2 2 L{_l,_I,(2) = (w b 6 + Wb l2 + b 16 + w b 18 + Wb24 + b 28 ), L{_l,l,UJ} = (w2bll + wb 13 - b 15 - w2b23 - Wb25 + b27), 2 2 L{-I,l,-w) = (-w b ll + Wb13 + b l5 - w b 23 + Wb 25 + b27), L{_1,l,-UJ2) = (-wbll + w2b13 + b l5 - Wb23 + w2b25 + b27), 2 2 L{_1,l,UJ2) = (Wbll + w b 13 - b l5 - Wb 23 - w b 25 + b 27 ), 2 L{l,-I,UJ) = {wb 5 + w2b8 + bl4 + wbi7 + w b20 + b26}, 2 L(l,-l,-UJ} = (-wb 5 + w b 8 - b 14 + wb 17 - w2b20 + b 26 ), 2 L{l,_I,_UJ2) = (-w2b5 + wb 8 - b1 4 + w b 17 - wb 20 + b 26 ), 2 2 L{l,-1,UJ2} = (w b 5 + wb8 + b l4 + w b 17 + wb20 + b 26), L{_l,_l,-I}
=
(b 6
279
GROUP GRADINGS ON 0(8, C)
2 -w - W 2w w ) b l - - - b2 - --b3 + b4 , l-w l-w l-w 2 2 -w 2w W ) 2 b2 l 2 + 1 2 b3 + b4 . L{I,I,_w } = ( -l--b + 1 -w - +w - +w
L{I,I,-w} = (
Grading over
Z2 x Z3: QI4 =
({tu,l,u,ft' tl,v,+I' v
Hi. I u, v E C*})
~
(C*)2 X Z3
is an abelian diagonalizable subgroup of Aut(L), which produces a grading of type (26, 1):
L{12' I , I} = (b s + b 17 + b26 ), 2 Lei,l,w2} = (w bs + wb 17 + b26 ), L{13 } = (w2b 13 - wb 23 2' ,w L{I '3' 1 I}
+ b 27 ),
= (b 16 + bls + b24),
2 L(12' I ,w} = (wb s + w b 17
+ b26) ,
= (b 13 - b23 + b27 ), 2 L{13 2} = (wb 13 - w b 23 + b 27 ), 2' ,w 2 L{I , 1 } = (w b 16 + wb ls + b24 ), L{i,3,1}
3'W
2 1 ) + -l-w - b2 + - - b3 + b4 , l-w 2 L{I,I,w2 }=(-1-1_b l + 2b2 + -112b3 + b4 ), -w -w 1 -w 1 +w2 L{I,I,w} = ( - - b l l-w
+ b2S), 2 L{I,3,w2} = (Wb 6 + w b 12 + b2S ), 2 L(2 ''1",w I } = (-w b 11 + wb ls + b 2S ), L{2,1,1} = (b s + b l4 + b 20 ), 2 L{I,j,w2} = (Wb 16 + w b lS + b24 ), L{I,I,I} = (b l + b2, b3),
L{I,3,1}
L{1,j,l}
=
=
(b 6 + b 12
(b 2I ),
L{2,~,I} = (b7),
(w 2b6 + wb 12
+ b2S ), L(2,j,l} = (-b ll + b ls + b2S ), 2 L{2 ''1",w I 2) = (-wb ll + w b lS + b 2S ), L{2,I,w} = (wb s + w 2b l4 + b 20 ), 2 L{2,I,(2) = (w bs + Wh4 + b20 ),
L{I,3,w}
=
Ld,3,1} = (b22 ), L{1,9,1}
=
(b I9 ),
L{2,3,1} = (b9),
L{4,j,1} = (b lO ).
REFERENCES
[I] [2] [3] [4] [5] [6] [7]
Y. A. Bahturin, I. P. Shestakov and M. V. Zaicev: J. Algebra 283 (2005), 849-868. C. Draper and C. Martin: Linear Algebra and its Applications 418 (2006), 85-111. C. Draper and C. Martin: Gradings on the Albert algebra and on f4. Preprint arXiv:math-ra/0703840v1. C. Draper, C. Martin and A. Viruel: Gradings on D4. Preprint. M. Havlfcek, J. Patera and E. Pelantova: Linear Algebra and its Applications 277 (1998), 97-125. P. Jordan: Nachr. Ges. Wiss. Gottingen 209-217 (1933). X.-D. Leng and J. Patera: J. Phys. A. Math. Gen. 27 (1994), 1233-1250.
280 [8] [9] [10] [II] [12]
C. DRAPER and A. VIRUEL
1. Patera and H. Zassenhaus: Linear Algebra and its Applications 112 (1989), 87-159. Patera, E. Pelantova and M. Svobodova: J. Math. Phys. 42 (2001), 3839-3853. Patera, E. Pelantova and M. Svobodova: J. Math. Phys. 43, 6353-6378 (2002). Patera, E. Pelantova and M. Svobodova: J. Math. Phys. 45 (2004), 2188-2198. Patera, E. Pelantova and M. Svobodova: J. Math. Phys. 48 (2007), 093503.
J. J. J. J.