By Richard D. Schafer
An advent to Nonassociative Algebras Richard D. Schafer
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Extra resources for An Introduction to Nonassociative Algebras
K X i j = BkiXij, xijek = 6 j k X i j for k = 1, ... ,1 ) III. , t ) . The 21ijare clearly subspaces of %. We prove first the uniqueness of an expression (i, j = 0,1, For i, j = 1, ... , 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.
Flo~ol is associative. 17) Proof. %lo9101(resp. 2fol%lo) is an ideal of all(resp. 22). 19). If B = 0, then '2l = all@ ao0 with all# 0, Uoo# 0, a contradiction. Hence 8 = a,and e a e = all= 'illlo sol. 23) implies that 9110aO1 is associative. 15. Let 'illbe a jinite-dimensional simple alternative algebra, and let 1 = el + e, for pairwise orthogonal idempotents e, (i = 1, ... , t). I f t 2 3, then 2I is associatiue. + Proof. , e,. , t, we wish to show first that a:i= a;l = 0 (i = 2, ... , t). Let e = el + e, # 1 since t 2 3.
35) implies C1 = e z , Zz = el. Hence b=Be,- Caiui- CBjwj+aez. 45) that + C aiui g12 + 1Bj w j h 1 + B h 1 = B h z l - 1atg1, ui - C P j h 1 wj + agz1 bu = agiz = ub. 40) holds for all b in 23. 38) with p = 1, and b + ub is a homomorphic image in 2l of the (2n)-dimensional algebra constructed by the Cayley-Dickson process from the algebra 23 of dimension n = 2 + 2k. Beginning with the 2-dimensional algebra Fe, @ Fez , the process does not terminate until we have 23 = a. By (iii) a # Fe, @ Fez. The case k = 0 gives a homomorphic image in 2l of the (simple) algebra Fz of all 2 x 2 matrices over F.