The Wedderburn principal theorem for alternative algebras

Abstract

Except for a generalization of the so-called Wedderburn principal theorem, the structure theory of alternative algebras over an arbitrary field is as complete as that for associative algebras. It is our purpose here to fill this one gap in the alternative theory. The principal theorem.A non-associative algebra 31 of order n over an arbitrary field % is called alternative in casefor all a, x in 21. It is clear that associative algebras are alternative.The most famous examples of alternative algebras which are not associative are the so-called Cayley-Dickson algebras of order 8 over $. Let S be an algebra of order 2 over % which is either a separable quadratic field over 5 or the direct sum 5 ©3-There is one automorphism z->z of S (over %) which is not the identity automorphism. The associative algebra O = 3~\~US with elements for /JT^O in § is called a quaternion algebra. For q in the form (1), the correspondence for 77^0 in %, where q-»g is the involution (3) of Q. Most of our knowledge of alternative algebras is due to M. Zorn.

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“…Since the proof there makes no assumption on the characteristic, formula (20) and Lemmas 2 and 3 are valid for arbitrary F. Also our formula (21) is implied by [2, Lemma 3.2 (i)]. (9) The center of a nonassociative algebra is the set of all elements c in the nucleus for which [x, c]=0 for every x. The center is a characteristic subalgebra, and in the case of a simple alternative algebra is a field.…”

“…)* is nilpotent may be verified by only a slight modification of the proof of [9, Lemma l]. Formulas for 5 and T analogous to (9) and (10) For proof, we observe that 31+ is a semisimple Jordan algebra and that 5+ defined by (7) In the proof of the Malcev theorem in the next section, we shall require a stronger form of the first Whitehead lemma for alternative algebras. 5.…”

“…The Malcev theorem. We have shown in [9] that the Wedderburn principal theorem for associative algebras generalizes to alternative algebras over an arbitrary field F. That is, if 31 is an alternative algebra with radical 9?, and if 3Í/9? is separable, then 31 may be expressed as the direct sum (37) 21 = © + 9?, where © is a separable subalgebra of 21 isomorphic to 2Í/9?.…”

The Wedderburn principal theorem for generalized alternative algebras. I

“…But then it foUows from the proof of Theorem 2 in [8] and the proof of Theorem 3 in [3] that A is an alternative algebra. Since in this case the theorem is known from [6] to be vaUd for A, our induction is complete.…”

On the Wedderburn principal theorem

“…It is known that associative [1], alternative [15] and Jordan [3], [13] algebras have Wedderburn decompositions. In addition, if A is commutative and A -N separable with no nodal subalgebras such that either the simple summands of A -N have degree =3 or A is stable, then A has a Wedderburn decomposition [5].…”