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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.

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