An Introduction to Nonassociative Algebras by Richard D. Schafer

By Richard D. Schafer

An creation to Nonassociative Algebras Richard D. Schafer

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For y = b3 + vb4 implies that (x, y) = 12 [n(x + y) − n(x) − n(y)] = 12 [n(b1 + b3 ) − µn(b2 + b4 ) − n(b1 ) + µn(b2 ) − n(b3 ) + µn(b4 )] = (b1 , b3 ) − µ(b2 , b4 ). Hence (x, y) = 0 for all y = b3 + vb4 implies (b1 , b3 ) = µ(b2 , b4 ) for all b3 , b4 in B. Then b4 = 0 implies (b1 , b3 ) = 0 for all b3 in B, or b1 = 0 since n(b) is nondegenerate on B; similarly b3 = 0 implies (b2 , b4 ) = 0 (since µ = 0) for all b4 in B, or b2 = 0. That is, x = 0; n(x) is nondegenerate on A. When is A alternative?

Then J0 is a nilalgebra, so that (x, y) = trace Rx1 y1 by (20 ) since trace Rz0 = 0 by (8). Hence x in J1/2 + J0 implies x1 = 0, (x, y) = 0 for all y in J, so x is in J⊥ . That is, J1/2 + J0 ⊆ J⊥ = N = 0, or J = J1 , e = 1. If J contains 1 and e1 = 1, then e2 = 1 − e1 , is an idempotent, and the Peirce decompositions relative to e1 and e2 coincide (with differing subscripts). We introduce a new notation: J11 = J1,e1 (= J0,e2 ), J12 = J1/2,e1 (= J1/2,e2 ), J22 = J0,e1 (= J1,e2 ). More generally, if 1 = e1 + e2 + · · · + et for pairwise orthogonal idempotents ei , we have the refined Peirce decomposition (21) J= Jij i≤j of J as the vector space direct sum of subspaces Jii = J1,ei (1 ≤ i ≤ t), Jij = J1/2,ei ∩ J1/2,ej (1 ≤ i < j ≤ t); that is, (22) Jii = {x | x ∈ J, xei = x}, Jij = Jji = {x | x ∈ J, xei = 21 x = xej }, i = j.

D+ ∼ = Ct + , t ≥ 2. Then Ct + contains an idempotent e11 = 1, a contradiction. AII . D+ is the set H(Zt ) of self-adjoint elements in Zt , Z a quadratic extension of C, where the involution may be taken to be a → g −1 a g with g a diagonal matrix. Hence H(Zt ) contains e11 = 1, a contradiction. B. D+ ∼ = H(Ct ), the involution being a → g −1 a g with g diagonal; hence H(Ct ) contains e11 = 1, a contradiction. C. D+ ∼ = H(C2t ), the involution being a → g −1 a g, g = 0 1t ; H(C2t ) contains the idempotent e11 + et+1,t+1 = 1, a −1t 0 contradiction.

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