By Richard D. Schafer
An creation to Nonassociative Algebras Richard D. Schafer
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Additional resources for An Introduction to Nonassociative Algebras
T , we have t e, xej = C el xkl e, = x i j . k,1= 0 For i = 1, ... , r, we have c t t el x = k,l= 0 e, xkl = C xil . l=O Then - e, xe = x i o , where e = el for j = 1, ... ,t. j=1 t =x -C t t t xoo = x e,xej i,j= 1 + + et . Similarly, xej - exej = xoj j= 1 t t I=1 j= 1 - 1( e r x - q x e ) - C (xe, - exe,) - exe - (ex - exe) - (xe - exe) = x - ex - xe + exe. But these x i j are in aij(i, j = 0,1, ... 15). The expression + ( x - ex - xe + exe) is a refinement of the decomposition of x relative to a single idempotent e.
Hence 'illis associative. 16. I be a finite-dimensional simple alternative algebra over (an arbitrary field) F satisfying: (i) 1 = el + e2 for (orthogonal) primitive idempotents e,; (ii) %,l(=e,91e,)=Fe, (i=l,2); (iii) 2l is not associative. 38) with p = 1. Proof. Let 23 be any semisimple subalgebra of which contains el and e, . Xi,. Also (ii) implies 23 = Fel b12 2321 Fe2. ) It follows from the flexible law that Yzlxlz = (x1z Y yz1)ez for all x1z in b12 , YZl in b21. For this certainly holds ify,, = 0.
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.
An Introduction to Nonassociative Algebras by Richard D. Schafer