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On the Dimension of a Composition

Markus Rost

Received December

Communicated by Ulf Rehmann

Abstract The p ossible dimensions of a comp osition algebra are

or We give a tensor categorical argument

Sub ject Classication Primary A Secondary

M

I Introduction

Let C b e a comp osition algebra over a eld of dierent from let V

b e its pure subspace consisting of the vectors orthogonal to and let d dim V

We show that the following relation holds in the groundeld

dd d d

This is not very surprising since the only p ossibilities for C are either the ground eld

a separable quadratic extension a algebra or an o ctonion algebra The

pro of of the relation given in this note seems to b e dierent from former approaches

cf It works on a tensor categorical level In characteristic one recovers

the determination of the p ossible dimensions of a comp osition algebra

Our starting problem was to understand comp osition from a tensor cat

egorical p oint of view Instead of comp osition algebras we lo oked at the equivalent

notion of vector pro duct algebras These algebras can b e obtained b e rewriting the

axioms of a comp osition algebra in terms of the pure vectors Vector pro duct alge

bras allow to use diagrammatic tensor calculus in a handy way Using a graphical

technique we foundjust by playing arounda pro of of the relation on dim V These

notes contain alone the algebraic calculations which were extracted from the graph

considerations After these notes had b een written we noticed an identity in vector

pro duct algebras which p erhaps makes the result less mysterious So there is more to

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Markus Rost

say ab out the topic than explained in this text We hop e to come back to this at an

other place Anyway the text is completely selfcontained and contains an argument

on the p ossible dimensions

Throughout the pap er we assume char

Acknowledgements I am indebted to B Eckmann and T A Springer for useful

comments T A Springer suggested to use the relation which reduced the

amount of the calculations considerably Moreover I thank the FIM at ETH Z urich

for its hospitality

I I Composition Algebras and Vector Products

We rst recall a denition

Composition algebras

A comp osition algebra consists of a C together with

a nondegenerate symmetric h i on C

a linear map C C C x y x y

an element e C

such that with N x hx xi

e x x e x

Nx y N x N y

For our purp ose we have to consider the following

Vector product algebras

A vector pro duct algebra consists of a vector space V together with

a nondegenerate symmetric bilinear form h i on V

a linear map V V V x y x y

such that

hx y z i is alternating in x y z

x y x hx xiy hx y ix

The vector pro duct is anticommutative since implies x x Therefore

x y x x y x Hence the choice of the arrangement of the brackets in the

lefthand side of is not essential

B Eckmann has considered continous vector pro ducts in B Eckmann Stetige

Losungen linearer Gleichungssysteme Comment Math Helv 15

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On the Dimension of a Composition Algebra

see also B Eckmann Continous solutions of linear equations An old problem its

history and its solution Exp o Math 9 He used the axioms

hx xi hx y i

hx y xi hx y y i N x y det

hy xi hy y i

They are p erhaps more close to the intuitive idea of a vector pro duct Under presence

of they are easily seen to b e equivalent to

Vector pro duct algebras and comp osition algebras are equivalent notions

?

Namely given a comp osition algebra C let V hei and put

x y y x i x y

Conversely given a vector pro duct algebra V put C heiV and dene the pro duct

on C by

ii ae x be y ab hx y i e ay bx x y

The rewriting formulas i and ii identify comp osition algebras and vector pro duct

algebras on a tensor categorical level This means that the comp osition rule

gives after p olarization and decomp osition with resp ect to C heiV the same tensor

equations as and the p olarization of

This equivalence b etween comp osition algebras and vector pro duct algebras seems to

provide a convenient way to comprise some wellknown rules in comp osition algebras

For the asso ciator in C one nds

x y z x y z x y z hx z iy hy z ix

for x y z V

I I I A Relation for the Contraction of h i

Let V b e a nitedimensional vector pro duct algebra and let e b e an orthonormal

i i

basis of V over some algebraic closure Put

X

he e i d

i i

i

Proposition One has the relation

dd d d

In the following we will tacitly apply in the formulation

a hx y z i hx y z i

b y x x y

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Markus Rost

The relation will b e used also in the following forms which are obtained by

p olarizing and from

a x y z x y z hx z iy hx y iz hz y ix

hx y z ti hy z t xi

b

hx z ihy ti hx y ihz ti hy z iht xi

Other relations to b e used are

X X X

he v ie dv v d v he e iv e v e

i i i i i i

i i i

and

X X X

he e i dd e e e e d he e e e i

i i i j i j i j i j

i ij ij

To warm up we rst consider vector pro duct algebras which corresp ond to asso ciative

comp osition algebras

Proposition Supp ose that the following sharp ening of holds

x y z hx z iy hy z ix

Then

dd d

Pro of Consider

X

A e e e e e e

i k i j k j

ijk

By we have

X

2 2

A d he e i dd

k k

k

On the other hand using and one nds

X

A e e e e e e

i k i j k j

ijk

X

he e ie e he e e ie e e

i j k i k i j i k j

ijk

X X

he e e e i he e e ihe e e i

k i k i k i j i k j

ik ijk

X

he e e e i dd

k i k i

ik

So

A A dd d 

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On the Dimension of a Composition Algebra

Let us start with the pro of of Prop osition

Put

X

u e e v hu v

i i

i

The following formula has b een introduced by T A Springer

hu v d u v

To check it one uses a with x u y e and z e v and nds

i i

X X

hu e v ie u e e v hu v

i i i i

i i

X X

he v e iu hu e ie v

i i i i

i i

X

hv u e ie d u v

i i

i

X

hv e e iu u v

i i

i

d u v u v u v d u v

Formulas and make it easy to compute the sum

X

B he e he e

i k k i

ik

X

2 2

d he e e e i dd d

i k k i

ik

We next compute B in a dierent way One has

X

B e e e e e e e e

i j j k k l l i

ijk l

Applying b shows

0 0

B B C D D

where

X

0

B e e e e e e e e

j k k l l i i j

ijk l

X

C he e e e ihe e e e i

i j k l j k l i

ijk l

X

D he e e e ihe e e e i

i j j k k l l i

ijk l

X

0

D he e e e ihe e e e i

j k k l l i i j

ijk l

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Markus Rost

0 0

By reindexing one nds B B and D D Therefore

B C D

We compute C and D

X

C e e e e e e e e

i j k l j k l i

ijk l

X

e e e e e e

j k l j k l

jk l

X

e e e e e e

j k l j k l

jk l

X X

d he e e e e e e e

k l k l k l k l

k l k l

X

d d he e i dd d

l l

l

X

D e e e e e e e e

i j j k k l l i

ijk l

X

e e e e e e

j j k k l l

jk l

X

2

d d he e i dd

k k

k

Hence

2

B dd d dd dd d

Finally

2

B B dd d dd d

2

dd d d dd d d 

References

Hurwitz A Ub er die Komp osition der quadratischen Formen Math Ann

Jacobson N Basic Algebra I W H Freeman and Co San Francisco

Markus Rost

Fakultat f ur Mathematik

Universitat Regensburg

Universitatsstrae

Germany

markusrostmathematikuniregensburgde

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