On the structure of third-power associative absolute valued algebras. (English) Zbl 1369.17004

Cuenca Mira, José Antonio

Bull. Malays. Math. Sci. Soc. (2) 40, No. 3, 1135-1148 (2017).

Summary: This paper deals with pairs of nonzero idempotents \(e\) and \(f\) of a third-power associative absolute valued algebra \(A\) satisfying \((ef)e= e(fe)\) and \((fe)f=f(ef)\) (pairwise flexible idempotents), and the role that they play on the structure of \(A\). We show that if \( g\) is a nonzero idempotent of \(A\) such that the nonzero idempotents commuting with \(g\) are pairwise flexible, then the subalgebra that they generate \(B_g\) is isometrically isomorphic to \(\mathbb {R}\), \(\mathop {\mathbb {C}}\limits^{\star}\), \(\mathop {\mathbb {H}}\limits^{\star}\), or \(\mathop {\mathbb {O}}\limits^{\star}\). Our main theorem proves the equivalence of the following assertions: (i) for every two different nonzero idempotents \(e\) and \(f\), the nonzero idempotents of \(A\) that commute with \((e-f)^2\) are pairwise flexible; (ii) each pair of nonzero idempotents of \( A\) generates a finite-dimensional subalgebra; and (iii) either \(A\) is isometrically isomorphic to \(\mathbb {R}\), \(\mathbb {C}\), \(\mathbb {H}\), \(\mathbb {O}\), \(\mathop {\mathbb {C}}\limits^{\star}\), \(\mathop {\mathbb {H}}\limits^{\star}\) or \(\mathop {\mathbb {O}}\limits^{\star}\), or \(A\) contains a subalgebra \(B\), which contains all idempotents of \(A\) and is isometrically isomorphic to the division absolute valued algebra \(\mathbb {P}\) of the Okubo pseudo-octonions. More consequences on the structure of \(A\) related with the presence of pairwise flexible idempotents are given, among them several generalizations of some well-known theorems.

MSC:

17A80	Valued algebras
17A75	Composition algebras
17A60	Structure theory for nonassociative algebras
17D99	Other nonassociative rings and algebras

Keywords:

absolute valued algebra; idempotent; division algebra; third-power associativity; pairwise flexible idempotents

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