BERNARDONI FCACCIATORI, SERGIO LUIGICERCHIAI B. LSCOTTI A.mostra contributor esterni nascondi contributor esterni 

Fabio Bernardoni, Sergio L. Cacciatori, Bianca L. Cerchiai, and Antonio Scotti

Citation: Journal of Mathematical Physics 49, 012107 (2008); doi: 10.1063/1.2830522 View online: http://dx.doi.org/10.1063/1.2830522

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/49/1?ver=pdfcov Published by the AIP Publishing

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Mapping the geometry of the E

group

Fabio Bernardoni,^1,a兲Sergio L. Cacciatori,^2,5,b兲Bianca L. Cerchiai,^3,c兲and Antonio Scotti^4,d兲

1Departament de Física Teórica, IFIC, Universitat de València-CSIC Apt. Correus 22085, E-46071 València, Spain

2Dipartimento di Scienze Fisiche e Matematiche, Università dell’Insubria, Via Valleggio 11, I-22100 Como, Italy

3Theory Group, Lawrence Berkeley National Laboratory, Building 50A5104, 1 Cyclotron Road, Berkeley, California 94720, USA

4Dipartimento di Matematica, Università di Milano, Via Saldini 50, I-20133 Milano, Italy

5INFN, Sezione di Milano Via Celoria 16, I-20133 Milano, Italy

共Received 2 October 2007; accepted 10 December 2007; published online 30 January 2008兲

In this paper, we present a construction for the compact form of the exceptional Lie group E₆ by exponentiating the corresponding Lie algebra e₆, which we realize as the sum of f₄, the derivations of the exceptional Jordan algebra J₃of dimension 3 with octonionic entries, and the right multiplication by the elements of J₃ with vanishing trace. Our parametrization is a generalization of the Euler angles for SU共2兲 and it is based on the fibration of E6via an F₄subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for F₄. An interesting first application of these results lies in the fact that we are able to determine an explicit expression for the Haar invariant measure on the E₆ group manifold. © 2008 American Institute of Physics. 关DOI:10.1063/1.2830522兴

I. INTRODUCTION

The standard model 共SM兲 provides a very good description of elementary particle physics.

However, despite its success, there are reasons to go beyond it, for example, the recent discovery of neutrino oscillations, the fine tuning of the mixing matrices, the hierarchy problem, the diffi- culty in including gravity, and so on.

A starting point could be the fact that the renormalization flow of the coupling constants suggests the unification of gauge interactions at energies of the order of 10¹⁵ GeV, which can be improved共fine tuned兲 by supersymmetry. It is natural to expect the gauge group G of the grand unification theory共GUT兲 theory to be a simple group. Obviously, at low energies, the GUT model must reproduce the SM physics, so that not only G should contain SU共3兲c⫻SU共2兲L⫻U共1兲Y共the SM gauge group兲 but it should also predict the correct spectra after spontaneous symmetry break- ing. The increasing accuracy in the analysis of particle spectra imposes even more restrictions in the possible choices for G. For example, it is already known that the current estimate for the lower bound of the proton lifetime rules out some of the GUT model candidates. Recently, the particular structure of the neutrino mixing matrix seems to suggest that a good candidate for a GUT could be based on the semidirect product between the exceptional group E₆⁴and the discrete group S₄.^3,6

In general, while the local properties of group G, which for a Lie group are all encoded in the corresponding Lie algebra, are enough to perform a perturbative analysis, in order to obtain nonperturbative results, such as instantonic calculations or lattice simulations, a knowledge of the

a兲Electronic mail: fabio.bernardoni@ific.uv.es

b兲Electronic mail: sergio.cacciatori@uninsubria.it

c兲Electronic mail: blcerchiai@lbl.gov

d兲Electronic mail: antonio.scotti@gmail.com

49, 012107-1

0022-2488/2008/49共1兲/012107/19/$23.00 © 2008 American Institute of Physics

entire group structure is required, in particular, of the invariant measure on the group in a suitable global parametrization. There are many ways to give such an explicit expression for the Haar measure on a Lie group. However, for large dimensions, it becomes quite hard to find a realization, which is able at the same time to provide both a reasonably simple form for the measure and an explicit determination for the range of the angles. In this paper, we solve this problem for the exceptional Lie group E₆, by constructing a generalization of the Euler parametrization, with a technique we have introduced in Ref.2and fully developed in Ref.1. In Sec. II, we explain how the e₆algebra can be represented using a theorem due to Chevalley and Schafer. The construction of the group and the determination of the corresponding Haar measure is made in Sec. III.

II. THE CONSTRUCTION OF THE e₆ALGEBRA

Our starting point for the construction of the exceptional algebra e₆ is a theorem due to Chevalley and Schafer^5,7which we rewrite here for convenience.

Theorem 2.1: The exceptional simple Lie algebra f₄of dimension 52 and rank 4 over K is the derivation algebra D of the exceptional Jordan algebra J of dimension 27 over K. The excep- tional simple Lie algebra e₆of dimension 78 and rank 6 over K is the Lie algebra

D+兵RY其, Tr Y = 0, 共2.1兲

spanned by the derivations of J and the right multiplications of elements Y of trace 0.

We will refer to Ref.8 for the notations. Then, the exceptional Jordan algebra is the algebra J₃. For our purposes, K will beR or C. The right multiplication is RY共X兲=Y ⴰX and the trace is the sum of diagonal elements. The exceptional Jordan algebra is 27 dimensional, just like the principal fundamental representation of e₆.

In our previous paper,¹we determined the algebra of derivations D and used it to obtain an irreducible representation of the exceptional algebra f₄. To obtain a representation of e₆, we only need to determine the matrix representation of RY. This is a very simple task and we solved it in R²⁷ with the product inherited from J₃by means of the linear isomorphism

⌽:J3→ R²⁷, A哫 ⌽共A兲,

⌽共A兲 ª

冢

冣

where A is the Jordan matrix,

A =

冢

冣

and␳is the linear isomorphism between the octonionsO and R⁸given by¹

␳^:O→ R, o = o⁰+兺

i=1 7

oⁱii哫␳共o兲,

1See Ref.1for more details.

␳共o兲 ª

冢

冣

After choosing a 26 dimensional base of traceless Jordan matrices, we used the MATHEMATICA

program in Appendix A to find the matrices which complete the base for f₄to a base for the whole e₆algebra. In particular, if we work in the real case, we obtain the split form of e₆, with signature 共52, 26兲. However, multiplying the 26 added generators by i, we find that the algebra remains real, but the Killing form becomes the compact one.

III. THE CONSTRUCTION OF THE GROUP E₆

A. The generalized Euler parametrization for E₆

To give an Euler parametrization for E₆, we start by choosing its maximal subgroup. It is H = F₄, the group generated by the first 52 matrices. LetP be the linear complement of f4in e₆. We then search for a minimal linear subset V ofP, which, under the action of Ad共F4兲, generates the whole P. Looking at the structure constants, we see that V can be chosen as the linear space generated by c₅₃, c₇₀. Note that they commute.

If we then write the general element g of E₆in the form

g = exp共h˜兲exp共v兲exp共h兲, h,h˜ 苸 f4, v苸 V, 共3.1兲 we have a redundancy of dimension 28. As argued in Ref.1, we expect to find a 28 dimensional subgroup of F₄, which, acting by adjunction, defines an automorphism of V. This can be done by noticing that V commutes with the first 28 matrices ci, i = 1 , . . . , 28, which generate an SO共8兲 subgroup of F₄.

We found convenient to introduce the change of base

˜c₅₃ª¹2c₅₃+^冑₂³c₇₀, 共3.2兲

˜c₇₀ª −^冑2³c₅₃+¹₂c₇₀. 共3.3兲 Thus, we have

g关x1, . . . ,x₇₈兴 = B关x1, . . . ,x₂₄兴e^x25c˜53e^x26˜c70F₄关x27, . . . ,x₇₈兴,

where B = F₄/SO共8兲. We chose for F4the Euler parametrization given in Ref.1so that we found for B,

B关x1, . . . ,x₂₄兴 = BF₄关x1, . . . ,x₁₆兴B9关x17, . . . ,x₂₃兴e^x24c45, 共3.4兲 where

BF₄关x1, . . . ,x₁₅兴 = B9关x1, . . . ,x₇兴e^x8c45B₈关x9, . . . ,x₁₄兴e^x15˜c30e^x16c22, 共3.5兲

B₉关x1, . . . ,x₇兴 = e^x1c˜3e^x2c˜16e^x3˜c15e^x4˜c35e^x5c˜5e^x6c˜1e^x7c˜30, 共3.6兲

B₈关x1, . . . ,x₆兴 = e^x1˜c3e^x2˜c16e^x3˜c15e^x4˜c35e^x5˜c5e^x6˜c1, 共3.7兲 and the tilded matrices are the ones introduced in Ref.1.

B. Determination of the range for the parameters

To determine the range of the parameters, we will use the topological method we have developed in Ref.2. Let us first determine the volume of E₆by means of the Macdonald formula.

1. The volume of E₆

The rational homology of the exceptional Lie group E₆is that of a product of odd dimensional spheres,¹⁰H_*共E6兲=H*共兿_i=1⁶ S^di兲, with⁴

d₁= 3, d₂= 9, d₃= 11, d₄= 15, d₅= 17, d₆= 23. 共3.8兲 The simple roots of E₆are

r₁= L₁+ L₂, 共3.9兲

r₂= L₂− L₁, 共3.10兲

r₃= L₃− L₂, 共3.11兲

r₄= L₄− L₃, 共3.12兲

r₅= L₅− L₄, 共3.13兲

r₆=L₁− L₂− L₃− L₄− L₅+冑3L₆

2 , 共3.14兲

where Li, i = 1 , . . . , 6, is an orthonormal base for the Cartan algebra. The volume of the fundamen- tal region is then

Vol共fE₆兲 =2

L. 共3.15兲

Indeed, we computed the 36 positive roots, all having length冑2. They have exactly the structure given in Ref.8, with Li= ei, the canonical base of R⁶. The Macdonald formula^9,11 gives for the volume of the compact form of E₆,

Vol共E6兲 = 冑3 · 2¹⁷·␲⁴²

3¹⁰· 5⁵· 7³· 11. 共3.16兲

2. The invariant measure on E₆

With the chosen generalized Euler parametrization, the invariant measure on E₆decomposes into the product of the measure of F₄and the one on M = E₆/F4. The invariant measure on F₄was computed in Ref.1so that we need to compute here only the induced measure on M.

Using the notation of Ref.1, let us define

JMª␲_P共e^−x26˜c70e^−x25˜c53B关x1, . . . ,x₂₄兴⁻¹d共B关x1, . . . ,x₂₄兴e^x25˜c53e^x26c˜70兲兲, 共3.17兲 where␲_Pis the projection on the subspace generated by c_j, j = 53, . . . , 78. The metric induced on M by the bi-invariant metric on E₆is then

ds_M² = −1

6Tr共JM丢JM兲, 共3.18兲

and the invariant measure on E₆ is then

d␮E₆=兩det共JMi

j 兲兩d␮F₄兿

l=1 26

dxl, 共3.19兲

where J_Mi^j is the 26⫻26 matrix defined by

JM= 兺

i,j=1 26

J_Mi^j ci_jdxⁱ, 共3.20兲

with ci_j a base兵c˜53, c˜₇₀, c₅₄, . . . , c₆₉, c₇₁, . . . , c₇₈其 of P.

In order to compute 兩det共JMi

j 兲兩, it is convenient to introduce the notations

␻关x,y,z兴 = e^xc45e^yc˜53e^zc˜70, 共3.21兲

J_␻ª␻^−1d␻^, 共3.22兲

J₉ª B9−1dB₉, 共3.23兲

JF₄ª BF₄

−1dBF₄, 共3.24兲

so that

JM关x1, . . . ,x₂₆兴 =␻关x24,x₂₅,x₂₆兴⁻¹B₉关x17, . . . ,x₂₃兴⁻¹JF₄关x1, . . . ,x₁₆兴B9关x17, . . . ,x₂₃兴␻关x24,x₂₅,x₂₆兴 +␻关x24,x₂₅,x₂₆兴⁻¹J₉关x17, . . . ,x₂₃兴␻关x24,x₂₅,x₂₆兴 + J_␻关x24,x₂₅,x₂₆兴. 共3.25兲 Some remarks are in order now.

共1兲 The following relations are true:

e^−␣c˜53cLe^␣c˜53= cos␣^cL+ sin␣^cL+26, 共3.26兲 if L = 45, . . . , 52. Moreover, c˜₇₀ commutes with cI, I = 45, . . . , 52, 71, . . . , 78, and with

˜c₅₃.

共2兲 c˜53 and c˜₇₀ commute with the so共8兲 algebra generated by the matrices cI, I

= 1 , . . . , 21, 30, . . . , 36.

共3兲 The adjoint action of e^x24c45on the above so共8兲 algebra generates in addition the matrices cJ, J = 46, . . . , 52.

共4兲 From JF₄苸f4 and B₉傺F4, it follows that

˜J

F₄ª B9−1J_F

4B₉苸 f4.

共5兲 The adjoint action of ␻ ^{on the f}4 matrices generates all the remaining matrices of e₆. In particular, the projection of␻^−1cL␻, L = 1 , . . . , 52, on cJ, J = 54, . . . , 69, is different from zero only if L = 22, . . . , 29, 37, . . . , 44.

共6兲 The adjoint action of the SO共8兲 group, corresponding to the above so共8兲 algebra, gives a rotation both on the indices I =兵22, ... ,29其 and J=兵37, ... ,44其. More precisely,

SO共8兲⁻¹cISO共8兲 = RI

LcL, 共3.27兲

SO共8兲⁻¹cJSO共8兲 = R˜J

KcK, 共3.28兲

where L runs from 22 to 29 and K runs from 37 to 44, and R , R˜ are both orthogonal matrices.

To verify these, it suffices to note that

e^−xcAcIe^xcA= cosx

2cI⫾ sinx

2cI_A, 共3.29兲

e^−xcAc_Je^xcA= cosx

2c_J⫾ sin x 2c_J

A, 共3.30兲

where A = 1 , . . . , 21, 30, . . . , 36, I , IA苸兵22, ... ,29其, J,JA苸兵37, ... ,44其. In particular, det共R丢R˜ 兲

= 1.

From the first three points, we see that the matrix JMtakes the form

M =

冢

冣

where on the rows we indicate the coefficients of dxI with I = 1 , . . . , 26 starting from the bottom and on the columns the projections on c˜_A, cL, following the order A = 53, 70, L

= 71, . . . , 78, 54, . . . , 69. The asterisks indicate the elements which do not contribute to the deter- minant,

det JM= det A det C det D.

In particular, A is a 3⫻3 block obtained by projecting J_␻on c˜₅₃, c˜₇₀, c₇₁. Point 1 implies

A =

冢

冣

so that

det A = sin x₂₅. 共3.33兲

The C block is a 7⫻7 matrix obtained by projecting␻^−1J9␻^{on c}K, K = 72, . . . , 78. From point 1, it follows that it can be obtained by multiplying the projection of e^−x24c45J₉e^x24c45 on cL, L

= 46, . . . , 52, by sin x₂₅. If we call C˜ the matrix corresponding to such a projection, we find det C = sin⁷x₂₅det C˜ . The determinant of C˜ can be computed directly and gives

det C = sin x₂₀cos x₂₁cos x₂₂sin²x₂₂sin²x₂₃cos⁴x₂₃sin⁷x₂₄sin⁷x₂₅. 共3.34兲 The D block requires some further discussion. It is a 16⫻16 matrix obtained by projecting

␻^−1˜JB_F

4␻^{on c}I, I = 54, . . . , 69. First, from points共4兲 and 共5兲 of our remarks, we see that only the cL

with L = 22, . . . , 29, 37, . . . , 44 in J˜

F₄contribute to the determinant. Let us define the 16⫻16 matrix U with

U_A^Bª −¹6Tr共␻^−1cA␻^cB兲, 共3.35兲

A = 22, . . . ,29,37, . . . ,44, 共3.36兲

B = 54, . . . ,69. 共3.37兲 Moreover, let D˜ be the matrix obtained by projecting J_F

4on cLwith L = 22, . . . , 29, 37, . . . , 44, and

Qª

冉

冊

where R and R˜ are the rotation matrices defined at point 共6兲. Then, we can deduce from points 共4兲–共6兲 that

det D = det共U · Q · D˜ 兲 = det U det D˜. 共3.39兲 The matrix U and its determinant are easily computed. Indeed, we have

␻^−1c22␻^{= cos}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c23␻^{= − cos}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c24␻^{= − cos}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c25␻^{= cos}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c26␻^{= − cos}冉^x2²⁴冊^sin

冉

冊

冉

冊

␻^−1c27␻^{= cos}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c28␻^{= cos}冉^x2²⁴冊^sin

冉

冊

冉

冊

␻^−1c29␻^{= − cos}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c37␻^{= − sin}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c38␻^{= − sin}冉^x2²⁴冊^sin

冉

冊

冉

冊

␻^−1c39␻^{= − sin}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c40␻^{= − sin}冉^x2²⁴冊^sin

冉

冊

冉

冊

␻^−1c41␻^{= − sin}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c42␻^{= − sin}冉^x224冊^sin

冉

冊

冉

冊

␻^−1c43␻^{= − sin}冉^x2²⁴冊^sin

冉

冊

冉

冊

␻^−1c44␻^{= − sin}冉^x224冊^sin

冉

冊

冉

冊

where ellipses stay for terms which vanish when projected on c_B, B = 54, . . . , 61, 70, . . . , 78. From these, it follows

det U = sin⁸

冉

冊

冉

冊

At this point, we need to compute det D˜ . This can be done by noticing that J_F

4coincides exactly with the current JM of F₄ in Ref.1. We then find

det D˜ = 2⁷sin¹⁵ x₁₆

2 cos⁷x₁₆

2 sin x₄cos x₅cos x₆sin²x₆cos⁴x₇sin²x₇sin⁷x₈

⫻ sin x12cos x₁₃cos x₁₄sin²x₁₄cos²x₁₅sin⁴x₁₅. 共3.41兲 We can finally write down the invariant measure on the base space,

d␮M= 2⁷sin x₄cos x₅cos x₆sin²x₆cos⁴x₇sin²x₇sin⁷x₈

⫻ sin x12cos x₁₃cos x₁₄sin²x₁₄cos²x₁₅sin⁴x₁₅sin¹⁵ x₁₆

2 cos⁷x₁₆ 2

⫻ sin x20cos x₂₁cos x₂₂sin²x₂₂sin²x₂₃cos⁴x₂₃sin⁷x₂₄

⫻sin⁸x₂₅sin⁸

冉

冊

冉

冊

Note that the periods of the variables are 4␲ so that one should take the range x_i=关0,4␲兴 for i

= 1 , 2 , 3 , , 9 , 10, 11, 17, 18, 19. However, as in Ref.1, it is easy to show directly from the param- etrization that they can all be restricted to关0,2␲兴. The range of xiis then

x₁苸 关0,2␲兴, x2苸 关0,2␲兴, x3苸 关0,2␲兴, x4苸 关0,␲兴,

x₅苸

冋

册

冋

册

冋

册

x₉苸 关0,2␲兴, x10苸 关0,2␲兴, x11苸 关0,2␲兴, x12苸 关0,␲兴,

x₁₃苸

冋

册

冋

册

冋

册

x₁₇苸 关0,2␲兴, x18苸 关0,2␲兴, x19苸 关0,2␲兴, x20苸 关0,␲兴,

x₂₁苸

冋

册

冋

册

冋

册

x₂₅苸

冋

册

The remaining parameters xj, j = 27, . . . , 78, will run over the range for F₄, as given in Ref.1. The volume of the whole closed cycle V so obtained is then

Vol共V兲 = Vol共F4兲冕R

d␮M= 冑3 · 2¹⁷·␲⁴²

3¹⁰· 5⁵· 7³· 11, 共3.44兲 where R is the range of parameters xi, i = 1 , . . . , 26. This is the volume of E₆, so that we cover the group exactly once.²

IV. CONCLUSIONS

In this paper, we have performed an explicit construction for the compact form of the simple Lie group E₆. This is particularly interesting because recently it has been argued that this group could be the most promising for unification in GUT theories.^3,6 To parameterize the group, we have used the generalized Euler angle method, a technique we introduced in Refs.2and1to give the most simple expression for the invariant measure on the group, while at the same time still being able to provide an explicit expression for the range of the parameters. Both these require- ments are necessary in order to minimize the computation power needed for computer simulations, for example, of lattice models.

Our results can be easily extended to the GUT group E₆⁴’S4. Also, a modified Euler param- etrization could be applied to evidentiate the subgroups related to the correct symmetry breaking, see Ref.3. Finally, our parametrization could be used for a straightforward geometrical analysis of the exceptional Lie group E₆ and its quotients.

ACKNOWLEDGMENTS

S.L.C. is grateful to Giuseppe Berrino and Stefano Pigola for helpful comments. B.L.C. would like to thank O. Ganor for useful discussions. This work has been supported in part by the Spanish Ministry of Education and Science共Grant No. AP2005-5201兲 and in part by the Director, Office of Science, Office of High Energy and Nuclear Physics of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

APPENDIX A: THE e₆MATRICES

The matrices we found usingMATHEMATICA, and orthonormalized with respect to the scalar product具a,b典ª−¹6Tr共ab兲, was computed by means of the followings programs.

TheMATHEMATICAprogram

% % % % the octonionic products

2Obviously, there is a subset of vanishing measure which is multiply covered.

QQ关1, 1兴 = e关1兴; QQ关1, 2兴 = e关2兴; QQ关1, 3兴 = e关3兴;

QQ关1, 4兴 = e关4兴; QQ关1, 5兴 = e关5兴; QQ关1, 6兴 = e关6兴;

QQ关1, 7兴 = e关7兴; QQ关1, 8兴 = e关8兴; QQ关2, 1兴 = e关2兴;

QQ关2, 2兴 = − e关1兴; QQ关2, 3兴 = e关5兴; QQ关2, 4兴 = e关8兴;

QQ关2, 5兴 = − e关3兴; QQ关2, 6兴 = e关7兴; QQ关2, 7兴 = − e关6兴;

QQ关2, 8兴 = − e关4兴; QQ关3, 1兴 = e关3兴; QQ关3, 2兴 = − e关5兴;

QQ关3, 3兴 = − e关1兴; QQ关3, 4兴 = e关6兴; QQ关3, 5兴 = e关2兴;

QQ关3, 6兴 = − e关4兴; QQ关3, 7兴 = e关8兴; QQ关3, 8兴 = − e关7兴;

QQ关4, 1兴 = e关4兴; QQ关4, 2兴 = − e关8兴; QQ关4, 3兴 = − e关6兴;

QQ关4, 4兴 = − e关1兴; QQ关4, 5兴 = e关7兴; QQ关4, 6兴 = e关3兴;

QQ关4, 7兴 = − e关5兴; QQ关4, 8兴 = e关2兴; QQ关5, 1兴 = e关5兴;

QQ关5, 2兴 = e关3兴; QQ关5, 3兴 = − e关2兴; QQ关5, 4兴 = − e关7兴;

QQ关5, 5兴 = − e关1兴; QQ关5, 6兴 = e关8兴; QQ关5, 7兴 = e关4兴;

QQ关5, 8兴 = − e关6兴; QQ关6, 1兴 = e关6兴; QQ关6, 2兴 = − e关7兴;

QQ关6, 3兴 = e关4兴; QQ关6, 4兴 = − e关3兴; QQ关6, 5兴 = − e关8兴;

QQ关6, 6兴 = − e关1兴; QQ关6, 7兴 = e关2兴; QQ关6, 8兴 = e关5兴;

QQ关7, 1兴 = e关7兴; QQ关7, 2兴 = e关6兴; QQ关7, 3兴 = − e关8兴;

QQ关7, 4兴 = e关5兴; QQ关7, 5兴 = − e关4兴; QQ关7, 6兴 = − e关2兴;

QQ关7, 7兴 = − e关1兴; QQ关7, 8兴 = e关3兴; QQ关8, 1兴 = e关8兴;

QQ关8, 2兴 = e关4兴; QQ关8, 3兴 = e关7兴; QQ关8, 4兴 = − e关2兴;

QQ关8, 5兴 = e关6兴; QQ关8, 6兴 = − e关5兴; QQ关8, 7兴 = − e关3兴;

QQ关8, 8兴 = − e关1兴;

% % % % the Jordan algebra product

Qm关x_គ, y₋兴 ª Sum关Sum关x关关i兴兴y关关j兴兴QQ关i, j兴, 兵i, 8其兴, 兵j, 8其兴

QP关x−, y₋兴 ª 兵Coefficient关Qm关x, y兴, e关1兴兴, Coefficient关Qm关x, y兴, e关2兴兴, Coefficient关Qm关x, y兴, e关3兴兴, Coefficient关Qm关x, y兴, e关4兴兴,

Coefficient关Qm关x, y兴, e关5兴兴, 关Qm关x, y兴, e关6兴兴, Coefficient关Qm关x, y兴, e关7兴兴, Coefficient关Qm关x, y兴, e关8兴兴其

OctP关a−, b₋兴 ª 兵兵Sum关QP关Part关Part关a, 1兴, i兴, Part关Part关b, i兴, 1兴兴, 兵i, 3其兴, Sum关QP关Part关Part关a, 1兴, i兴, Part关Part关b, i兴, 2兴兴, 兵i, 3其兴, Sum关QP关Part关Part关a, 1兴, i兴, Part关Part关b, i兴, 3兴兴, 兵i, 3其兴其, 兵Sum关QP关Part关Part关a, 2兴, i兴, Part关Part关b, i兴, 1兴兴, 兵i, 3其兴, Sum关QP关Part关Part关a, 2兴, i兴, Part关Part关b, i兴, 2兴兴, 兵i, 3其兴, Sum关QP关Part关Part关a, 2兴, i兴, Part关Part关b, i兴, 3兴兴, 兵i, 3其兴其, 兵Sum关QP关Part关Part关a, 3兴, i兴, Part关Part关b, i兴, 1兴兴, 兵i, 3其兴, Sum关QP关Part关Part关a, 3兴, i兴, Part关Part关b, i兴, 2兴兴, 兵i, 3其兴, Sum关QP关Part关Part关a, 3兴, i兴, Part关Part关b, i兴, 3兴兴, 兵i, 3其兴其其

OctPS关a−, b₋兴 ª 1/2共OctP关a, b兴 + OctP关b, a兴兲

% % % % correspondence between the Jordan algebra andR²⁷

FF关AA−兴 ª 兵Part关兵Part关Part关Part关AA, 1兴, 1兴, 1兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 2兴, 1兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 2兴, 2兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 2兴, 3兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 2兴, 4兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 2兴, 5兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 2兴, 6兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 2兴, 7兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 2兴, 8兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 3兴, 1兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 3兴, 2兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 3兴, 3兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 3兴, 4兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 3兴, 5兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 3兴, 6兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 3兴, 7兴其, 1兴, Part关兵Part关Part关Part关AA, 1兴, 3兴, 8兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 2兴, 1兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 3兴, 1兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 3兴, 2兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 3兴, 3兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 3兴, 4兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 3兴, 5兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 3兴, 6兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 3兴, 7兴其, 1兴, Part关兵Part关Part关Part关AA, 2兴, 3兴, 8兴其, 1兴, Part关兵Part关Part关Part关AA, 3兴, 3兴, 1兴其, 1兴其

FFi关vv−兴 ª 兵兵兵Part关vv, 1兴, 0, 0, 0, 0, 0, 0, 0其, 兵Part关vv, 2兴, Part关vv, 3兴, Part关vv, 4兴, Part关vv, 5兴, Part关vv, 6兴, Part关vv, 7兴, Part关vv, 8兴, Part关vv, 9兴其,

兵Part关vv, 10兴, Part关vv, 11兴, Part关vv, 12兴, Part关vv, 13兴, Part关vv, 14兴, Part关vv, 15兴, Part关vv, 16兴, Part关vv, 17兴其其, 兵兵Part关vv, 2兴, − Part关vv, 3兴, − Part关vv, 4兴,

− Part关vv, 5兴, − Part关vv, 6兴, − Part关vv, 7兴, − Part关vv, 8兴,

− Part关vv, 9兴其, 兵Part关vv, 18兴, 0, 0, 0, 0, 0, 0, 0其, 兵Part关vv, 19兴, Part关vv, 20兴, Part关vv, 21兴, Part关vv, 22兴, Part关vv, 23兴, Part关vv, 24兴, Part关vv, 25兴, Part关vv, 26兴其其, 兵兵Part关vv, 10兴, − Part关vv, 11兴, − Part关vv, 12兴, − Part关vv, 13兴, − Part关vv, 14兴,

− Part关vv, 15兴, − Part关vv, 16兴, − Part关vv, 17兴其, 兵Part关vv, 19兴, − Part关vv, 20兴,

− Part关vv, 21兴, − Part关vv, 22兴, − Part关vv, 23兴, − Part关vv, 24兴, − Part关vv, 25兴,

− Part关vv, 26兴其, 兵Part关vv, 27兴, 0, 0, 0, 0, 0, 0, 0其其其

% % % % construction of the matrices

MT =兵兵兵mt关1兴, 0, 0, 0, 0, 0, 0, 0其, 兵mt关2兴, mt关3兴, mt关4兴, mt关5兴, mt关6兴, mt关7兴, mt关8兴, mt关9兴其, 兵mt关10兴, mt关11兴, mt关12兴, mt关13兴, mt关14兴, mt关15兴, mt关16兴, mt关17兴其其, 兵兵mt关2兴, − mt关3兴,

− mt关4兴, − mt关5兴, − mt关6兴, − mt关7兴, − mt关8兴, − mt关9兴其,

兵mt关18兴, 0, 0, 0, 0, 0, 0, 0其, 兵mt关19兴, mt关20兴, mt关21兴, mt关22兴, mt关23兴, mt关24兴, mt关25兴, mt关26兴其其, 兵兵mt关10兴, − mt关11兴, − mt关12兴, − mt关13兴, − mt关14兴, − mt关15兴,

− mt关16兴, − mt关17兴其, 兵mt关19兴, − mt关20兴, − mt关21兴, − mt关22兴, − mt关23兴, − mt关24兴,

− mt关25兴, − mt关26兴其, 兵− mt关1兴 − mt关18兴, 0, 0, 0, 0, 0, 0, 0其其其;

h = Array关hh, 27兴;

M = Array关mm, 兵27, 27其兴;

imm = FF关OctPS关MT, FFi关h兴兴兴;

Do关Do关mm关i, j兴 = Coefficient关Part关imm, i兴, Part关h, j兴兴, 兵i, 27其兴, 兵j, 27其兴

Do关econi关j兴 = D关M, mt关j兴兴, 兵j, 26其兴;

Mno关1兴 = econi关1兴;

Do关Do关AA关j, i兴 = 0, 兵i, 26其兴, 兵j, 26其兴

Do关Do关AA关j, i兴 = − Tr关econi关j兴 . Mno关i兴兴/Tr关Mno关i兴 . Mno关i兴兴, 兵i, j − 1其兴;Mno关j兴 = econi关j兴 + Sum关AA关j, i兴Mno关i兴, 兵i, j − 1其兴, 兵j, 2, 26其兴

Do关Mnno关i兴 = −冑6Mno关i兴/Sqrt关Tr关Mno关i兴 . Mno关i兴兴兴, 兵i, 26其兴;

% % % % rotation to give the irreducible 26 representations XXX =兵兵1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

− 1, 0, 0, 0, 0, 0, 0, 0, 0, 0其/Sqrt关2兴,

兵0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0其,

兵1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, − 2其/Sqrt关6兴, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0其,

兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0其, 兵0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0其,

兵1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1其/Sqrt关3兴其;

Do关cc关i + 52兴 = XXX . Mnno关i兴 . Transpose关XXX兴, 兵i, 26其兴;

The matrices cc关i兴, i=53, ... ,78, so obtained, must be added to the 52 matrices cc关i兴 determined in Ref.1to give a base of the e₆algebra.

APPENDIX B: STRUCTURE CONSTANTS

The structure constants s_IJ^K are defined by 关CI, C_J兴=兺_K=1⁷⁸ s_IJ^KC_K. We checked that the coeffi- cients s_IJKªsIJ

K are completely antisymmetric in the indices. We write only the nonvanishing terms, up to symmetries, which are not yet written in Ref.1,

s_1,54,55=¹₂, s_1,56,58= −¹₂, s_1,57,61= −¹₂, s_1,59,60= −¹₂,

s_1,62,63= −¹₂, s_1,64,66=¹₂, s_1,65,69=¹₂, s_1,67,68=¹₂,

s_1,71,72= − 1, s_2,54,56=¹₂, s_2,55,58=¹₂, s_2,57,59= −¹₂,

s_2,60,61= −¹₂, s_2,62,64= −¹₂, s_2,63,66= −¹₂, s_2,65,67=¹₂,

s_2,68,69=¹₂, s_2,71,73= − 1, s_3,54,58=¹₂, s_3,55,56= −¹₂,

s_3,57,60= −¹₂, s_3,59,61=¹₂, s_3,62,66=¹₂, s_3,63,64= −¹₂,

s_3,65,68= −¹₂, s_3,67,69=¹₂, s_3,72,73= − 1, s_4,54,57=¹₂,

s_4,55,61=¹₂, s_4,56,59=¹₂, s_4,58,60= −¹₂, s_4,62,65= −¹₂,

s_4,63,69= −¹₂, s_4,64,67= −¹₂, s_4,66,68=¹₂, s_4,71,74= − 1,

s_5,54,61=¹₂, s_5,55,57= −¹₂, s_5,56,60=¹₂, s_5,58,59=¹₂,

s_5,62,69=¹₂, s_5,63,65= −¹₂, s_5,64,68=¹₂, s_5,66,67=¹₂,

s_5,72,74= − 1, s_6,54,59=¹₂, s_6,55,60= −¹₂, s_6,56,57= −¹₂,

s_6,58,61= −¹₂, s_6,62,67=¹₂, s_6,63,68= −¹₂, s_6,64,65= −¹₂,

s_6,66,69= −¹₂, s_6,73,74= − 1, s_7,54,58=¹₂, s_7,55,56= −¹₂,

s_7,57,60=¹₂, s_7,59,61= −¹₂, s_7,62,66= −¹₂, s_7,63,64=¹₂,

s_7,65,68= −¹₂, s_7,67,69=¹₂, s_7,71,75= − 1, s_8,54,56= −¹₂,

s_8,55,58= −¹₂, s_8,57,59= −¹₂, s_8,60,61= −¹₂, s_8,62,64= −¹₂,

s_8,63,66= −¹₂, s_8,65,67= −¹₂, s_8,68,69= −¹₂, s_8,72,75= − 1,

s_9,54,55=¹₂, s_9,56,58= −¹₂, s_9,57,61=¹₂, s_9,59,60=¹₂,

s_9,62,63=¹₂, s_9,64,66= −¹₂, s_9,65,69=¹₂, s_9,67,68=¹₂,

s_9,73,75= − 1, s_10,54,60=¹₂, s_10,55,59=¹₂, s_10,56,61= −¹₂,

s_10,57,58= −¹₂, s_10,62,68=¹₂, s_10,63,67=¹₂, s_10,64,69= −¹₂,