Non-degenerate bilinear alternating maps \(f:V\times V \to V\), \(\dim(V)=3\), over an algebraically closed field. (English) Zbl 1057.15029
Let \(V\) be a three-dimensional vector space over an algebraically closed field \(\mathbb{F}\) of characteristic different from \(2\). The goal of the paper is the classification of the non-degenerate alternating maps \(f:V{\times}V \to V\) over \(\mathbb{F}\) under the tensorial action of the full linear group \(GL(V)\) in \(\bigwedge^{2}V^{*}{\otimes}V\). The authors obtain this classification in terms of a scalar invariant \(\sigma_{2}(f)\) attached to each non-degenerate \(f \in \bigwedge^{2}V^{*}{\otimes}V\). The main result of the paper states that if \(f,f' :V{\times}V \to V\) are two bilinear alternating maps, then they are equivalent if and only if, either \(\sigma_{2}(f)=\sigma_{2}(f') \neq 0\), or \(\sigma_{2}(f)=\sigma_{2}(f') = 0\) and \(n_{f}=n_{f'}\), where \(n_{f}\) and \(n_{f'}\) are some discrete invariants attached to \(f\) and \(f'\).
Reviewer: Nikolai I. Osetinski (Moskva)
MSC:
15A72 | Vector and tensor algebra, theory of invariants |
15A63 | Quadratic and bilinear forms, inner products |
15A69 | Multilinear algebra, tensor calculus |
Keywords:
algebraically closed field; bilinear alternating map; invariant under a tensorial action; general linear groupReferences:
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