Representations of alternative algebras
Representations of alternative algebras
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DOI：
10.1090/s0002-9947-1952-0045101-x
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发表时间：
1952
影响因子：
1.3
通讯作者：
R. D. Schafer
中科院分区：
文献类型：
--
作者：
R. D. Schafer
In this paper we apply to alternative algebras a definition of representation given by S. Eilenberg for nonassociative algebras satisfying multilinear identities. The corresponding alternative module generalizes the notion of a two-sided Sl-module as used in the study of associative algebras. Our chief object is to use the representation theory to obtain the generalization to alternative algebras of the theorem of A. Malcev on the strict conjugacy of semisimple components in Wedderburn decompositions. Since every alternative algebra gives rise to a (special) Jordan algebra, and every representation yields (Jordan) representations of this algebra, we can use recent results of N. Jacobson on representations of Jordan algebras. Doing this restricts our principal theorems to algebras of characteristic 0. Following certain preliminaries concerning derivations and associators, we prove the complete reducibility of representations of semisimple alternative algebras. We next prove the first Whitehead lemma for alternative algebras, generalizing G. Hochschild's result for associative algebras. This is sufficient to prove the Malcev theorem in case the square of the radical is {0 {. For all other types of algebras for which this theorem is known (Lie, associative, and Jordan), an inductive argument then suffices to complete the proof for an arbitrary radical. In the case of alternative algebras, however, without a stronger form of the Whitehead lemma a certain associativity condition would invalidate the inductive argument. Using the complete reducibility, we prove that this stronger form holds, and employ it in the proof of the Malcev theorem. In the concluding section we prove a generalization of a theorem due to Hochschild which, although independent of the representation theory, is related to our other results: an alternative algebra (of characteristic 0) is semisimple if and only if its derivation algebra is semisimple or {0}. We are indebted to Professor Jacobson for allowing us to see his paper, General representation theory of Jordan algebras, in manuscript form, and also for giving us valuable advice in connection with the proof of Theorem 2. 1. Representations and semidirect sums. A (nonassociative) algebra SI over a field F is called alternative in case
