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On the Dimension of a Composition Algebra

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On the Dimension of a Composition Algebra Doc Math J DMV On the Dimension of a Composition Algebra 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 Mathematics Sub ject Classication Primary A Secondary M I Introduction Let C b e a comp osition algebra over a eld of characteristic 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 quaternion 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 algebras 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 Documenta Mathematica 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 vector space C together with a nondegenerate symmetric bilinear form 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 algebraic structure 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 Documenta Mathematica 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 Documenta Mathematica 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 Documenta Mathematica 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 Documenta Mathematica 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 Documenta Mathematica .
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