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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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