By Richard D. Schafer
An creation to Nonassociative Algebras Richard D. Schafer
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Extra info for An Introduction to Nonassociative Algebras
7 the radical of 21ii which are properly nilpotent in 21ii. 29) these elements are properly nilpotent in 2l, implying 'illi c 'ill n 21ii. Conversely, if x i i E 91iiis in %, then xii is a fortiori in ai. Hence 'illi = 'ill n aii. 9. 14) be the Peirce decomposition of 2l relative to e. Then 2110 Sol+ 9100is contained in the radical 8 of 3. + Proof. 7, we have seen that Each of these subspaces is contained in $ = I%. 10. Every jinite-dimensional semisimple alternative algebra 2l# 0 has an identity element 1.
That is, b is nilpotent. 3. W(91) may be employed in studying an arbitrary algebra %. W(%). Let 5 be a subspace of (f = 6 ( V ) , the associative algebra of all linear operators on some vector space V over F. By the Lie enveloping algebra of 5 we mean the (Lie) subalgebra of 6 - generated by 5; that is, the intersection of all subalgebras of (f- containing $3. Defining fii inwe see that the Lie enveloping ductively by bl = $, Bi+, = [B1, algebra of $ is ail, m II. Arbitrary Nonassociative Algebras 20 as may be seen from anticommutativity and the Jacobi identity.
2. Any alternative nilalgebra PI of jinite dimension over F is nilpotent. Proof. Let b be a subalgebra of % ’ which is maximal with respect to the property that %* is nilpotent. Such a maximal b exists, since the subalgebra ( 0 ) has this property and 2l is finite-dimensional. 1. Nilpotent Algebras 31 Assume that b is a proper subalgebra of %. Then there exists an element x not in b such that xb*E b. 1 1) For b*'= 0 implies that %b*' = 0 E 23, and there exists a smallest integer m 2 1 such that %b*"'E 9.