the octonions and g_2

8/2/2019 The Octonions and G_2

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Intr. alla Teoria dei Gruppi

Tesina

The Octonions and G2:an "Exceptional" Connexion

Student:

Nicols Cuello

Professor:

Dr. Lorenzo Magnea

March 2, 2012


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Contents

1 Introduction 3

1.1 The Octonion Genesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Different Octonions Constructions 6

2.1 An Eight Dimensional Vector Space over the Reals . . . . . . . . . . . . 62.2 The Cayley-Dickson Construction . . . . . . . . . . . . . . . . . . . . . . 8

3 The Exceptional Lie Algebra G2 113.1 The Algebra of Derivations on O . . . . . . . . . . . . . . . . . . . . . . 123.2 Dimension ofD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Proof of the Simplicity ofD: Main Lines . . . . . . . . . . . . . . . . . . 16

3.3.1 D is Semisimple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 D is Simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Conclusions 18

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Chapter 1

Introduction

1.1 The Octonion Genesis

The Hamilton Quaternions

The starting point for the Octonions birth was the construction of the Quaternionsby Hamilton in 1843. Inspired by the relation between C and 2-dimensional geometry,he tried for many years to add and multiply triplets searching for a bigger algebra whichcould have the same relation with the 3-dimensional geometry. Adding triplets was nota real problem, in fact it is equivalent to the sum of two elements in a 3-dimensionalvector space. The crucial point was to find out how to multiply them, it became a realobsession for Hamilton who related this fact in a letter to his son: Every morning in theearly part of the above-cited month, on my coming down to breakfast, your (then) littlebrother William Edwin, and yourself, used to ask me: Well, Papa, can you multiplytriplets? Whereto I was always obliged to reply, with a sad shake of the head: No, Ican only add and subtract them.

Today it easy to understand why this research couldnt met with success, in fact itis impossible to build a 3-dimensional normed division algebra. The story tells that hekept searching a way to do this multiplication until the 16th October 1843, when, whilehe was walking with his wife along the Canal River, he had a sort of revelation... Asstruck by the lightning, this equation came to his mind:

i2 = j2 = k2 = ijk = 1

Hamilton understood that the basis he was looking for from the beginning for hisalgebra was in fact {1,i ,j ,k}, to his surprise this algebra was not three dimensional butfour. So he left behind the triplets and started exploring this just-born 4-dimensionalnormed division algebra. His discovery was received with big enthusiasm because of theirapplications to geometry, even a school of quaternionsts was funded... But what reallyinterest us is that this also gave some new ideas about the possibility of constructingbigger algebras.

Graves versus Cayley

Paternity controversies about some concepts or ideas seem to be very frequent in sci-ence... It does really matter who was the first that discovered an amazing property or

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solved an extremely hard problem. Among the most famous controversies, we may high-light the infinitesimal calculus, which opposed Newton to Leibniz, and the Klein-Gordonformula which actually had been written a few years before by Erwin Schrodinger. Theoctonions had had their own controversy: the story tells that John T. Graves wrote a

letter on October 1843 to his friend Hamilton complimenting him for his discovery andasking the following question: If with your alchemy you can make three pounds of gold,why should you stop there? He was clearing talking about the possibility of construct-ing bigger algebras, in fact a few months later he wrote again to Hamilton describing anew 8-dimensional algebra, which he called the octaves, and exposing some ideas abouta hypothetic extension to 2n-dimensional algebras. Hamilton proposed him to publicizehis interesting results on January 1844, the problem was that he personally assumed thereviewing task of the paper, and being too much busy with work on the quaternions, hekept putting it off. It is curious to see that the non associativity of octonions was onlypointed out six months later by Hamilton...

In the meantime Cayley, fresh out of Cambridge, published in March 1845 a paperwhere, apparently as an afterthought, he tacked on a brief description of the octonions...

Just for the record, this paper is remembered as one of his worst works excluding the briefdiscussion about 8-dimensional algebra. The Graves recovery was almost immediate,but even showing his correspondence with Hamilton there was no way of reversing thesituation, it was too late: octonions were already known as Cayley numbers by thecommunity of mathematicians. Lacking a clear application to geometry or physics, theoctonions remained a curiosity until 1925 when Elie Cartan described the triality (thesymmetry between vectors and spinors in 8-dimensional Euclidean space). But all theattempts done to apply octonionic quantum mechanics to nuclear and particle physicsthe years later met with little success.

Forgotten and then rediscovered

In the 1980s the octonions became again a trendy subject of study after the astonishingdiscovery that the octonions explain some curious features of string theory, neverthelessthis would take us too far away from the main scope of this article (and thereby from myunderstanding) so we will not develop this subject. However, even if their importance inphysics remains unproved, the octonions still a very powerful tool to tie together somealgebraic structures that otherwise appear as isolated and inexplicable exceptions.

1.2 Preliminaries

We present here some definitions and we recall some important results for quaternionswhich will turn very useful throughout this article.

Definition 1. Algebra: A is an algebra over R if A is a real vector space having adistributive multiplication map with the properties thatR is in the center ofA and 1 isthe multiplicative identity ofA.

Note that we do not require the associativity, as a matter of fact octonions form analgebra and they are not associative.

Definition 2. Division Algebra: A finite dimensional algebraA is a division algebra ifgiven a, b inA such that ab = 0, then either a = 0 or b = 0.

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Definition 3. Normed Division Algebra: A finite dimensional algebraA is a normeddivision algebra if it is a normed vector space with norm N such thatN(ab) = N(a)N(b)holds for all a, b inA.

An important theorem written by Hurwitz asserts that there exist only four normed

division algebras: the reals R, the complexes C, the quaternions H and the octonionsO, cf. [1].

Conjugation Function and Norm

Each complex number (a, b) can be uniquely written as a linear combination a + ibwhere a and b are real numbers. The multiplication between two complexes is given bythe following rule: (a, b)(c, d) = (ac db)(ad + cb). The conjugation works as follows:(a, b) = (a, b), note that we have also (a, b)(c, d) = (a, b) (c, d) for all a,b,c,d R. Ofparticular interest are two maps from C to R: the norm and trace. Here we define themfor complexes, however they can be easily extended for quaternions and octonions. Thenorm on C is given by N(a, b) = (a, b)(a, b) = (a + ib)(a ib) = a2 + b2, and the trace

by tr(a, b) = (a, b) + (a, b) = (a + ib) + (a ib) = 2a.

Quaternions

Since the octonions were built as an extension of quaternions a quick review couldbe very useful at this stage. The quaternions, H, can be written as the elements ofa 4-dimensional vector space with basis {1, i, j, k}: a + bi + cj + dk. The followingHamiltons rules show that the quaternions form an algebra: ij = k = ji, jk = i =kj, ki = j = ik, i2 = j2 = k2 = 1. This operations are synthesized in the Fig.1.1, note that the arrow sense is important because it defines the sign of the product,as a consequence we see quaternions do not commute anymore, unlike complexes. Theconjugation can be extended in the following way a + bi + cj + dk = a ib jc kd

and, as for complexes, we have xy = x y with x, y H.

Figure 1.1: The Quaternion Multiplication

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Chapter 2

Different Octonions Constructions

There exist three different ways of constructing the Octonions: one given by the mul-tiplication table of the algebra but not really practical, another one which ensues fromthe consideration that complex numbers can be considered like a couple of real numbers,and finally the last based on a very geometrical view implying Clifford Algebras. Wewill expose in detail the two firsts and and omit the third one, we follow here the samescheme than J. Baez in his excellent article about Octonions [1].

2.1 An Eight Dimensional Vector Space over the Reals

The Multiplication Table: The simplest way to think of octonions is as an eightdimensional vector space where the basis is given by {1, e1, e2, e3, e4, e5, e6, e7}. Wecan construct the multiplication table even if it is hardly lightening...

1 e1 e2 e3 e4 e5 e6 e71 1 e1 e2 e3 e4 e5 e6 e7

e1 e1 1 e3 e2 e5 e4 e7 e6e2 e2 e3 1 e1 e6 e7 e4 e5e3 e3 e2 e1 1 e7 e6 e5 e4e4 e4 e5 e6 e7 1 e1 e2 e3e5 e5 e4 e7 e6 e1 1 e3 e2e6 e6 e7 e4 e5 e2 e3 1 e1e7 e7 e6 e5 e4 e3 e2 e1 1

Table 2.1: Multiplication table

We must however highlight some interesting properties:

Every element in O is a linear combination of these basis elements where thescalars are in the field R. By the distributive laws, this completely defines octonionmultiplication;

e1, . . . , e7 are square roots of 1;

ei and ej anticommute when i = j: eiej = ejei

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This means that {1, e7, e2, e3} is a subalgebra. As this could be done for sevendifferent combinations of four basis elements, we can affirm there exist 7 subalgebrasinto the octonions:

The external lines: {1, e6, e1, e7}, {1, e7, e2, e5}, {1, e5, e3, e6},

The inner lines: {1, e1, e4, e5}, {1, e2, e4, e6}, {1, e3, e4, e7}

The inner cercle: {1, e1, e2, e3}

As a final comment we may say that each one of this subalgebras is isomorphic tothe quaternions, to see this we can define a map from any octonion set of four differentbasis elements to the quaternions as it is done in [2]. This will prove useful in the nextsection where we are going to construct octonions from quaternions, in fact octonionscan be considered as a vector space over any of these quaternions.

Figure 2.1: The Fano Plane

2.2 The Cayley-Dickson Construction

This construction will allow us to understand a little bit better the relation between R,C, H and O. We have already mentioned that any complex number can be consideredas a couple of reals, as an extension we can consider quaternions as couples of complexnumbers and octonions as pairs of quaternions as well. This procedure is known asthe Cayley-Dickson Construction and fortunately its much more lightening than themultiplication table.

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we say as well that x H is a ordered pair of complexes and x C is a ordered pair ofreals. If we keep applying this procedure we can obtain algebras of dimension 16, 32, 64and so on. The matter is that, with each step, the sons get worse than the fathers! Forinstance:

from R to C we loose the property than every element is its own conjugate,

from C to H we loose commutivity,

from H to O we loose associativity,

finaly, at sedonians, we loose the division algebra property.

Now that we have studied two different constructions for the octonions, it is time toswitch to the Exceptional Lie Algebra G2. At the end of the next chapter we will findout the exceptional relation that links octonions and G2.

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Chapter 3

The Exceptional Lie Algebra G2

For this incursion into the Exceptional Lie Algebras, we will first need to recall somealgebraic definitions, then, once established these concepts, we proceed to the calculationof the Algebra of Derivations on O which will lead us to the dimension of D. Finally,we will describe the more significative steps in the demonstration of the simplicity of D,this result will show up the relation between G2 and the octonions. We follow here thesame major lines than K.E. McLewin in her thesis of Master Degree [2].

Preliminary definitions

Definition 5. Lie Algebra: L is a Lie Algebra over a fieldF if L is a vector space overF and it has an operation called bracket [ , ] satisfying:

bilinearity: [x + x, y] = [x, y] + [x, y], [x, y + y] = [x, y] + [x, y] and [x,y] =[x,y] = [x, y] for all x, x, y , y L and F

antisymmetry: [x, y

] = [y, x

] the Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0

Definition 6. Commute: Given a Lie algebra L, two elements x, y L are said tocommute if [x, y] = 0.

Definition 7. Lie Ideal: Given a Lie algebra L, I L is a Lie ideal if given anyx L, y I, [x, y] I.

Definition 8. Simple Lie Algebra: a Lie Algebra L is a simple Lie algebra if [L, L] = 0and if L has no ideals other than (0) and itself.

Definition 9. Semisimple Lie Algebra: A Lie Algebra L is semisimple if it is the directsum of simple Lie Algebras. Specifically, r

i=1Li, where implies [Li, Lj] = 0 for all

i = j and each Li is simple.

As we have studied in Intr. alla Teoria dei Gruppi, there exist two types of familiesof Lie Algebras which have been classified by Lie itself. The first category is composed bythe complex simple Lie Algebras which are infinite: Al, Bl, Cl and Dl. All these algebrasare assumed to be over the complexes. The 5 exceptional Lie Algebras E6, E7, E8, F4and G2 form the second category. G2 is the algebra we are particularly interested in, isa Simple Lie Algebra and it has dimension 14, this result will turn to be very importantto prove the link with the algebra of derivations over O.

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3.1 The Algebra of Derivations on O

The main scope of this section will be to show that the set of derivations over O is a LieAlgebra, in order to do this we first need to define the derivation over an algebra:

Definition 10. Derivation over A: given an algebra A over a field F, we define aderivation overA as a function D : A A such that:

D(x + y) = D(x) + D(y), (3.1)

D(x) = D(x), (3.2)

D(xy) = xD(y) + D(x)y, (3.3)

for all F, and for all x, y A.

We denote D(A) to be the set of derivations over A. To gain an insight into deriva-tions, we recall some propositions concerning D which will be useful for the incomingdemonstrations. All the detailed proofs can be found in [2], we skip them to concentrateon crucial results of the reasoning.

Proposition 1. LetA be a division algebra overR. If D D(A), then D() = 0 forall R.

Proposition 2. If D D(O), then tr(D(x)) = 0 for all x O.

Proposition 3. D(x) = D(x) for all x O.

To get to D(O) it is instructive to take a D D(H) and to extend it in such a way thatthe extended version belongs to D(O). In the first chapter we said that quaternions werea 4 dimensional vector space with basis {1, e1, e2, e3}. As D is linear, it is fully defined

by its behavior on the basis elements. The trick here is to extend this derivation usingthe fact that an octonion can be considered as an ordered pair of quaternions: a + berwith a, b H. Multiplying a generic quaternion (expressed as a linear combinationof these basis elements) by one of the remaining basis elements from octonions yieldsanother quaternion (expressed as a linear combination of the 4 remaining different basiselements). To do this, we use the multiplication table for octonions given in chapter 2.We have liberty on this choice: e4, e5, e6 or e7. Lets choose e4, specifically we define:D(e4) = ce4 where c = c0 + c1e1 + c2e2 + c3e3 H. Then:

ce4 = (c0 + c1e1 + c2e2 + c3e3)e4

= c0e4 + c1e5 c2e6 + c3e7 O

Thus our extension is well defined since D(a + be4) = D(a) + bD(e4) + D(b)e4. Thisextension permits to found the following result according to what was mentioned before:

Proposition 4. Let D D(H). If we define D(e4) = ce4 from some c H, thenD D(O).

The proof of this proposition is relegated to appendix. This "extended division"stills having the properties of derivation. To prove that D(O) is indeed a Lie Algebra ithas to fulfill the 3 conditions given in definition 5. We have to define the multiplicative

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law for elements in D(O), we choose the bracket on D as follows: if E, D D(O), then[D, E] = DE ED. Using this definition it is straightforward to show that it is bilinear,antisymmetric and, since our bracket was defined using the commutator, it satisfies theJacobi identity. This leads to the following theorem:

Theorem 1. D(O) is a Lie Algebra.

From now on we will use the notation D for D(O).

3.2 Dimension ofD

We would like to show now that, as G2, D has dimension 14. To do this we start with ageneric D D, the splitting of this algebra into the basis elements would have the form7

i=1 iei, 0 = 0 because of the proposition 2. Again, as D is a linear map, it is fullydetermined by its effect on the basis elements. Since we can construct e3, e5, e6 and e7from e1, e2 and e4 using the multiplication relations given in chapter 2, if we define:

D(e1) =7

i=1

eii, (3.4)

D(e2) =7

i=1

eii, (3.5)

D(e4) =

7i=1

eii, (3.6)

then we have the derivation rules that follow:

D(e3) = D(e1)e2 + e1D(e2)

D(e5) = D(e1)e4 + e1D(e4)

D(e6) = D(e4)e2 + e4D(e2)

D(e7) = D(e3)e4 + e3D(e4)

= D(e1e2)e4 + e3D(e4)

= [D(e1)e2 + e1D(e2)]e4 + e3D(e4)

The explicit form of each D(ei) for i = 1, 2, ..., 7 can be computed using these rela-tions (see Appendix B), note that they depend on 7 3 = 21 variables. The results are

given in the table 3.1.The number of variables can be decreased searching for the relations between these

different coefficients. As this result will bridge the gap between octonions and the G2 wewill recall in detail the proof of the following proposition:

Proposition 5. Suppose D D and we let

D(e1) =7

i=1

eii, D(e2) =7

i=1

eii, D(e4) =7

i=1

eii

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Substituting:

(7

i=1

iei)e2 + e1(7

i=1

iei) + (7

i=1

iei)e1 + e2(7

i=1

iei) = 0

and since e2i = 1 for all i,

2 +7

i=1,i=2

ieie2 1 +7

i=1,i=1

ie1ei 1 +7

i=1,i=1

ieie1 2 +7

i=1,i=2

ie2ei = 0

Using eiej = ejei, we remark that the two first sums cancel with the other two:

2 + 1 = 0

For the other two relations the same calculation can be done, the starting pointis D(tr(e1e4)) = 0 and D(tr(e2e4)) = 0, this leads to 1 + 4 = 0 and 2 + 4 = 0

respectively.There remains the last relation: 6 + 5 3 = 0. We consider now: D(tr(e3e4)) = 0and using the same properties as before we found:

D(e3e4 + e3e4) = 0

D(e3e4) + D(e4e3) = 0

D(e3e4) + D(e4e3) = 0

It can be expanded in this way:

D(e1e2)e4 + e3D(e4) + D(e4)e3 + e4D(e1e2) = 0

{D(e1)e2 + e1D(e2)}e4 + e3D(e4) + D(e4)e3 + e4{D(e1)e2 + e1D(e2)} = 0

Splitting the D(ei) into sum we found, by an analogous computation as above, therelation sought:

6 + 5 3 = 0.

Plugging this relations in the table 3.1 we obtain the table 3.2. The fact that thenumber of independent variables is equal to 14 starts to show up the relation with G2.

D(e1) D(e2) D(e3) D(e4) D(e5) D(e6) D(e7)1 0 0 0 0 0 0 0

e1 0 2 3 4 5 6 7

e2 2 0 3 4 5 6 7e3 3 3 0 6 + 5 7 4 4 + 7 5 6e4 4 4 6 5 0 5 6 7e5 5 5 7 + 4 5 0 7 + 2 3 6e6 6 6 4 7 6 2 7 0 3 + 5e7 7 7 5 + 6 7 3 + 6 3 5 0

Table 3.2: Linearly Independent Form of Derivations on the Octonions

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Explicitly these variables are: 2, . . . , 7 ; 3, . . . , 7 and 5, . . . , 7. However, for themoment we have just shown that D has at maximum 14 dimensional so we need toprove that it is exactly 14. The following two theorems, cf. [2] once again for the proofs,permit to establish it:

Theorem 2. Every octonion map of the form in table 3.2 is a derivation.

Theorem 3. A linear mapping on the octonions D has the form in table 3.2 if and onlyif D is a derivation over the octonions.

As the mentioned table has exactly 14 independent variables we can finally assertthat D, the algebra of derivations over the octonions, has dimension 14.

3.3 Proof of the Simplicity of D: Main Lines

Note that semisimplicity is a necessary condition for simplicity. The next step of the

proof is to show that D has no nonzero abelian ideals. From Intr. alla Teoria dei Gruppione could think to compute the Cartan-Killing metric g and then apply the Cartanscriterium computing the determinant of g. However, as octonions arent associativewe cannot construct a standard matrix representation. We mentioned in chapter 2 thata modified version of representations can be found, nevertheless to my knowledge theredoesnt exist a way to extend this criterium to these representations.

As the detailed proof of the simplicity of D would lead us too far of the main scopeof this article, we will just show how we can get to the final result by enumerating themost important results and quickly explaining some steps.

3.3.1 D is Semisimple

We recall the two definitions of the properties we want to prove.

Definition 11. Semisimple Algebra: an algebra is Semisimple if and only if it has nononzero abelian ideals.

Definition 12. Simple Algebra: an algebra is Simple if and only if it has no nonzeroideals.

It can be proved by contradiction that D has no nonzero abelian ideals: we supposeit exists a nonzero abelian ideal and we show that this implies that the ideal is {0},which is a contradiction.

The explicit proof needs to define the basis of D. Using the table 3.2, the 14 basis

elements of D can be easily computed. For instance, let D1 be the derivation definedby table 3.2 in which 2 = 1 and all other i, i, i are equal to zero. The other basiselements could be defined in a similar way, we get then the basis {D1, . . . , D14} incorrespondence with {2, . . . , 7, 2, . . . , 7, 4, . . . , 7}.

Defining a restriction D within these basis elements it can be shown, cf. [2], that Dhas no nonzero abelian ideals, thus:

Proposition 6. D is semisimple.

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The next step is the complexification ofD, DC, some parts of the proof will need someconcepts concerning Lie algebras over the complexes. Making some basic considerationswe get: dimC(DC) = dimR(D) = 14. Using Lie Algebra properties, it follows that Dsemisimple implies:

Proposition 7. DC is semisimple.

3.3.2 D is Simple

To show that D is simple it is sufficient to show DC is simple. As DC is semisimple itcan be written as the direct sum of simple Lie Algebras: DC =

ri=1Di for some r N

and each Di simple. Thus, if D DC, then D =

Di with Di Di. In order todemonstrate the simplicity of DC, we have to show that r = 1. To do this, a detailedstudy of DC is required, which is omitted here.

We obtain at the end of the trail that DC is a direct sum of either one or two simpleLie Algebras. As dimC(DC) = 14, either DC is a simple Lie algebra of dimension 14 or

DC is a direct sum of two simple Lie Algebras whose dimensions add up to 14. So wehave to search, among the Lie Algebras we introduced at the beginning of this chapter,two which verify this condition or directly a Lie Algebra of dimension 14. Note that G2has dimension 14 as mentioned in the preliminaries, so it is the perfect candidate. Usingthe results about the dimensions of Simple Lie Algebras studied in Intr. alla Teoria deiGruppi, we get that the Simple Lie Algebras of dimension less than or equal to 14 are:

A1: dimension 3,

A2: dimension 8,

B2: dimension 10,

G2: dimension 14.

Among these algebras, it is impossible to obtain 14 by the sum of two elements. . . Thecloser we can get is 3 + 10 = 13. At this level, we can assert that r = 2, thus r = 1 andthis enables us to conclude:

Theorem 4. DC is a simple Lie algebra. Furthermore, DC is isomorphic to the excep-tional Lie algebra G2.

And finally,

Theorem 5. D is a real Lie algebra of type G2

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Appendix A

Isomorphism between C C and H

Here u1, u2, v1, v2 C and a1, a2, b1, b2, c1, c2, d1, d2 R

(u1, u2)(v1, v2) = (u1v1 v2u2, v2u1 + u2v1)

= ((a1, a2)(c1, c2) (d1, d2)(b1, b2), (d1, d2)(a1, a2) + (b1, b2)(c1, c2))

= ((a1c1 a2c2, a1c2 + a2c1) (d1b1 + b2d2, d1b2 d2b1),

(d1a1 d2a2, d1a2 + d2a1) + (b1c1 + c2b2, b2c1 + b1c2))

= ((a1c1 a2c2 d1b1 b2d2, a1c2 + a2c1 d1b2 + b1d2),

(d1a1 + b1c2 d2a2 + c2b2, d1a2 + d2a1 + b2c1 b1c2))

The multiplication of two ordered pairs of complexes is a new pair of ordered com-plexes. Thus, applying we obtain a quaternion expressed in the basis elements{1,i ,j ,k}:

((u1, u2)(v1, v2)) = a1c1 a2c2 b1d1 b2d2

i (a1c2 + a2c1 d1b2 + d2b1)

j (d1a1 + b1c1 d2a2 + c2b2)

k (d1a2 + d2a1 + b2c1 b1c2)

On the other side,

(u1, u2)(v1, v2) = (a1 + a2 i + b1 j + b2 k)(c1 + c2 i + d1 j + d2 k)

= a1c1 a2c2 b1d1 b2d2i (a1c2 + a2c1 + b1d2 b2d1)

j (a1d1 + b1c1 + b2c2 a2d2)

k (a1d2 + b2c1 + a2d1 b1c2)

Hence, we have shown that

((u1, u2)(v1, v2)) = (u1, u2)(v1, v2)

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Isomorphism between HH and O (main steps)

We want to show that: ((h1, h2), (h3, h4)) = (h1, h2)(h3, h4) where hi H.To do this we start computing explicitly (h1, h2). Using this result, (h3, h4) can

be easily found substituting 1 by 3 and 2 by 4. As we are applying to an ordered pairof quaternions, we obtain the expression of an octonion in terms of the basis elements{1, e1, e2, . . . , e7}. When we multiply the two octonions obtained, we get an octonionagain by the multiplication rules established in chapter 2. A tedious, but straightforward,calculation leads to the generic expression.

On the other side, we have to compute (h1, h2)(h3, h4). Using the relations definingthe multiplication and the conjugation, this can be expressed as an ordered pair of twonew quaternions: (h, h). When we apply , we obtain the expression of an octonionin terms of the basis elements {1, e1, e2, . . . , e7} again. Comparing it with the precedentresult, we found that both are equal, which completes the demonstration.

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Appendix B

Proof of proposition 4, from [2]

Proposition. Let D D(H). If we define D(e4) = ce4 from some c H, then D D(O).

Proof: D is linear on O from our construction. We have to check that the derivation

condition is fulfilled for all x, y O:

D(xy) = x D(y) + D(x) y

However, since this condition already holds for H and since D is linear on O, we needonly prove D(eia) = eiD(a) + D(ei)a holds for i = 4, . . . , 7 and for arbitrary a H. Wewill only show the calculation for i = 4 since the proof is nearly identical for the others.Let a = a0 + a1e1 + a2e2 + a3e3 + a4e4 + a5e5 + a6e6 + a7e7.

D(e4a) = a0D(e4) + a1D(e4e1) + a2D(e4e2) + a3D(e4e3) +

+a4D(e4e4) + a5D(e4e5) + a6D(e4e6) + a7D(e4e7)

D(e4a) = a0D(e4) + a1e4D(e1) + a1D(e4)e1 + a2e4D(e2) + a2D(e4)e2 +

+a3e4D(e3) + a3D(e4)e3 + a4e4D(e4) + a4D(e4)e4 + a5e4D(e5) +

+a5D(e4)e5 + a6e4D(e6) + a6D(e4)e6 + a7e4D(e7) + a7D(e4)e7

D(e4a) =7

i=1

e4(aiD(ei)) + D(e4)a0 +7

i=1

D(e4)(aiei)

D(e4a) = e47

i=1

aiD(ei) + D(e4)(a0 +7

i=1

aiei)

D(e4a) = e4D(a) + D(e4)a.

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Explicit form of D(ei) for i = 1, 2, . . . , 7

D(e1) = 1e1 + 2e2 + 3e3 + 4e4 + 5e5 + 6e6 + 7e7

D(e2) = 1e1 + 2e2 + 3e3 + 4e4 + 5e5 + 6e6 + 7e7

D(e3) = D(e1)e2 + e1D(e2)

= 1e1e2 + 2e2e2 + 3e3e2 + 4e4e2 + 5e5e2 + 6e6e2 + 7e7e2 +

+1e1e1 + 2e1e2 + 3e1e3 + 4e1e4 + 5e1e5 + 6e1e6 + 7e1e7

= 1e3 2 3e1 + 4e6 5e7 6e4 + 7e5

1 + 2e3 3e2 + 4e5 5e4 + 6e7 7e6

= (2 1) + (3)e1 + (3)e2 + (1 + 2)e3 + (6 5)e4 +

+(7 + 4)e5 + (4 7)e6 + (5 + 6)e7

D(e4) = 1e1 + 2e2 + 3e3 + 4e4 + 5e5 + 6e6 + 7e7

D(e5) = D(e1)e4 + e1D(e4)

= 1e1e4 + 2e2e4 + 3e3e4 + 4e4e4 + 5e5e4 + 6e6e4 + 7e7e4 +

+1e1e1 + 2e1e2 + 3e1e3 + 4e1e4 + 5e1e5 + 6e1e6 + 7e1e7= 1e5 2e6 + 3e1 4 5e1 + 6e2 7e3

1 + 2e3 3e2 + 4e5 5e4 + 6e7 7e6

= (4 1) + (5)e1 + (6 3)e2 + (7 + 2)e3 + (5)e4 +

+(1 + 4)e5 + (2 7)e6 + (3 + 6)e7

D(e6) = D(e4)e2 + e4D(e2)

= 1e1e2 + 2e2e2 + 3e3e2 + 4e4e2 + 5e5e2 + 6e6e2 + 7e7e2 +

+1

e4

e1

+ 2

e4

e2

+ 3

e4

e3

+ 4

e4

e4

+ 5

e4

e5

+ 6

e4

e6

+ 7

e4

e7

= 1e3 2 3e1 + 4e6 5e7 6e4 + 7e5

1e5 + 2e6 3e7 4 + 5e1 6e2 7e3

= (2 4) + (3 + 5)e1 + (6)e2 + (1 7)e3 + (6)e4 +

+(7 1)e5 + (4 + 2)e6 + (5 3)e7

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D(e7) = D(e3)e4 + e3D(e4)

= (2 1)e4 3e1e4 3e2e4 + (1 + )e3e4 +

+(6 3)e4e4 + (7 + 4)e5e4 + (4 7)e6e4 + (5 + 6)e7e41e3e1 + 2e3e2 + 3e3e3 + 4e3e4 + 5e3e5 + 6e3e6 + 7e3e7

= (6 + 5 3) + (7 4 2)e1 + (4 4 2)e2 + (5 6)e3

+(2 1 7)e4 + (3 6)e5 + (3 + 5)e6 + (1 + 2 + 4)e7

D(e1) D(e2) D(e3) D(e4) D(e5) D(e6) D(e7)

1 0 0 2 1 0 4 1 4 2 6 + 5 3e1 1 1 3 1 5 5 3 7 4 2e2 2 2 3 2 6 3 6 4 7 + 1e3 3 3 1 + 2 3 7 + 2 7 + 1 5 6e4 4 4 6 5 4 5 6 2 1 7e5 5 5 7 + 4 5 1 + 4 1 + 7 3 6e6 6 6 4 7 6 2 7 2 + 4 3 + 5e7 7 7 5 + 6 7 3 + 6 3 5 1 + 2 + 4

Table 1: General Form of Derivations on the Octonions

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Bibliography

1. John C. Baez, The Octonions, Bull. Amer. Math. Soc. 39, 145-205, 2002.

2. K.E. McLewin, Octonions and the Exceptional Lie Algebra G2, 2004.

3. R.D. Schafer, An Introduction to Nonassociative Algebras, Massachusetts Instituteof Technology, 1961.

4. Wikipedia, Split-octonion and Octonion articles.

5. J. Daboul, R, Delbourgo, Matrix Representation of Octonions and Generalizations,arxiv: hep-th/9906065v1.

the octonions and g_2

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