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On classification of metabelian Lie algebras. (English) Zbl 0923.17015

This paper completes the classification of complex metabelian Lie algebras of dimension up to 9. Let \(L\) be such an algebra and set \(V=L^2\) and \(U=L/V\). \(L\) has signature \((\dim U, \dim V)\). The focus of this paper is on algebras of signatures \((5,5)\) and \((6,3)\). The classification problem reduces to classifying the orbits in \(\wedge^2(U^*\otimes V)\) under the action of \(\mathrm{GL}(U)\times\mathrm{GL}(V)\). It is found that there are three families for \((5,5)\) and seven for \((6,3)\) and these are described. For algebras of smaller signature, the families are found from the main results and match that which is already known. Tables listing the algebras are provided.

MSC:

17B30 Solvable, nilpotent (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
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