An Introduction to Nonassociative Algebras by Richard D. Schafer

By Richard D. Schafer

Concise research provides in a quick house a number of the vital principles and leads to the speculation of nonassociative algebras, with specific emphasis on replacement and (commutative) Jordan algebras. Written as an creation for graduate scholars and different mathematicians assembly the topic for the 1st time. "An vital addition to the mathematical literature"—Bulletin of the yank Mathematical Society.

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153)). Then J∼ = H(A), the t2 -dimensional subalgebra of self-adjoint elements in the 2t2 -dimensional algebra A+ . If J is of type AI or AII , and if K is the algebraic closure of F , then JK ∼ = Kt+ where Kt is the algebra of all t × t matrices with elements in K. B, C. Let A be any involutorial central simple associative algebra over F (so the involution is of the first kind). Then J ∼ = H(A), the + subalgebra of self-adjoint elements in A . There are two types (B and C) which may be distinguished by passing to the algebraic closure K of F , so that AK is a total matrix algebra.

Theorem 7. Let A be a finite-dimensional algebra over F (of arbitrary characteristic) satisfying (i) there is a nondegenerate (associative) trace form (x, y) defined on A, and (ii) I2 = 0 for every ideal I = 0 of A. Then A is (uniquely) expressible as a direct sum A = S1 ⊕ · · · ⊕ St of simple ideals Si . Proof: Let S (= 0) be a minimal ideal of A. Since (x, y) is a trace form, S⊥ is an ideal of A. Hence the intersection S ∩ S⊥ is either 0 or S, since S is minimal. We show that S totally isotropic (S ⊆ S⊥ ) leads to a contradiction.

This means that ei C is a totally isotropic subspace (ei C ⊆ (ei C)⊥ ). Hence dim(ei C) ≤ 12 dim C = 4 (Jacobson, p. 170; Artin, p. 122). But x = 1x = e1 x + e2 x for all x in C, so C = e1 C + e2 C. Hence dim(ei C) = 4, and n(x) has maximal Witt index = 4 = 12 dim C. Similarly n (x ) has maximal Witt index = 4. Hence n(x) and n (x ) are equivalent (Artin, ibid). By Theorem 5, C and C are isomorphic. Over any field F there is a Cayley algebra without divisors of zero (take µ = 1 so v 2 = 1). This unique Cayley algebra over F is called the split Cayley algebra over F .

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