Non - Associative Division Algebras

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( ii ) k^ß is commutative if and only if on = ß, andk^ ß is associative precisely ifoL = ß and char (A:) = 3.

( iii ) Let char(Â:) = 3. Then xx^ e к and x^xe к for each x e k^ß-, in particular, k^ß satisfies the identities {xx^)y = y{xx^) and {x^x)y = y{x^x). Furthermore, an element xek^ ß satisfies x^x^ ekx if and only if

xek^kbKjk^ - kb^Kjk^k { b^ - b^ ) .

Proof (i) k^ß is a division algebra if and only if all non-zero right multiplications q^ = (jc 1-^ xa) on A are invertible, and det(^J = c{x,y,z) for a = x-^by -^b^^z, x,y,zE k. Assertions (ii) and (iii) are easily verified; for example, if x^x^ e kx and хфк + кЬ, then x = X-i- fib-\- b^ up to a scalar factor, and then x^jc^ g kx implies ju^(l — )u)(a + j5) = 0, hence /x = 0 or /i = 1, as a + jS ф 0 by (i). a

To give some examples, let к = A:o(0 be the field of rational functions over an arbitrary field ко, and let a, ß be non-zero Ä:o-multiples of t with a -h jî + 0. Then k^ß is a division algebra by (2.2) (i). See also Lex [20] p. 33.

( 2 . 3 ) Proposition. Let к be afield of characteristic 3 and let oc,ßEk such that k^ß is a division algebra. Then the group PsMij^{k^ß) of all k-linear automorphisms ofk^ß is trivial.

Proof Let gEA\xii^{k^ß). By (2.2)(iii) we have b^ = À-^fлe with X^fiEk and ^G {è,è^6 + è2}.Now'jß = b4 = фЧу = (6^)2è^implies/8 = Я^ + /i^^^e, which gives

0 = P-\-ß{fi-iy for e = b,

0 = p-ß-a{(x^ß)^^ for e = 6^ and

0 = X^ + ßifi-lУ-ф-hß)fi^ for e = b-\-b^. These equations contradict (2.2) (i) unless e = è, A = 0, /г = 1, and then g — id. n

Propositions (2.1) and (2.3) show that there exist division algebras admitting many derivations, but no automorphisms. Of course, purely inseparable field extensions A\k have the same property.

( 2 . 4 ) Gradings. Let Л be a 3-dimensional division algebra over к with a nontrivial grading A = kbi® kb2 Ф kb^ (cp. Patera-Zassenhaus [22] p. 88/89). We may assume that kb^ = k, as b^ ^2 ^ ^^з ^^^ ^i ^з ^ ^^2 imply b^ b^ g kb^, and it turns out that A is isomorphic to some algebra k^ß as in (2.2). In fact, (2.2) (iii) allows to determine all (non-trivial) gradings of division algebras k^ß of characteristic 3; the following is a complete list (with repetitions):

k^^ß = k®k{b^-y)® k{b^ + 2yb + y^)

= кф k{b^ + y) Ф kiyb"" + (a + ß)b + y^)

= Ä:фÄ:(62^-è + 7)фÄ:((l-7)è^-(a + i8-h7)è + }'2 + a + J8),