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Norms on structurable algebras. (English) Zbl 0753.17004
An algebra $$A$$ with an involution $$^ -$$ is called a structurable algebra if it satisfies the operator identity $[V_{x,y},V_{z,w}]=V_{\{x,y,z\},w}-V_{z,\{y,x,w\}},$ where $$V_{x,y}z=\{x,y,z\}=(x\overline y)z+(z\overline y)x-(z\overline x)y$$. The class of structurable algebras includes all Jordan algebras (with identical involution), as well as all alternative algebras with involution and a number of interesting exceptional simple algebras. Let $$(A,^ -)$$ be a finite dimensional structurable algebra and $$x\in A$$. The authors call $$x$$ conjugate invertible with the conjugate inverse $$\hat x$$ if $$V_{x,\hat x}=I$$, or equivalently $$V_{\hat x,x}=I$$. The map $$x\mapsto\hat x$$ is a rational map from $$A$$ to $$A$$, and the authors define the conjugate norm $$N$$ of the $$A$$ as the exact denominator (i.e. a denominator of minimal degree) of this map. Then for any $$x\in A$$, $$x$$ is conjugate invertible iff $$N(x)\neq 0$$, and $$N(x)$$ is a semi-invariant for both the structure Lie algebra and the structure group of $$(A,^ - )$$.
The main part of the paper is devoted to the calculation of the norm for each central simple structurable algebra over a field of characteristic $$\neq 2,3$$ or 5, according to the classification of these algebras, given by O. N. Smirnov.

##### MSC:
 17A30 Nonassociative algebras satisfying other identities 17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.) 17B70 Graded Lie (super)algebras
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