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The Bianchi variety. (English) Zbl 1201.53084

Summary: The totality Lie(\(V\)) of all Lie algebra structures on a vector space \(V\) over a field \(\mathbb F\) is an algebraic variety over \(\mathbb F\) on which the group GL(\(V\)) acts naturally. We give an explicit description of Lie(\(V\)) for \(\dim V = 3\) which is based on the notion of compatibility of Lie algebra structures.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B99 Lie algebras and Lie superalgebras
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