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Classification of orbit closures of 4-dimensional complex Lie algebras. (English) Zbl 0932.17005
On an \(n\)-dimensional complex vector space \(\mathcal V\) with a specified basis \(\{e_1,\ldots,e_n\}\), a Lie algebra structure can be uniquely defined by means of the structure constants \(\gamma_{i,j}^k\) with respect to the given basis of \(\mathcal V\), i.e. by \([e_i,e_j]=\sum_{k=1}^n\gamma_{i,j}^k e_k\). In view of the antisymmetry and Jacobi identity for the bracket, the corresponding points \((\gamma_{i,j}^k)\in{\mathbb C}^{n^3}\) constitute an affine algebraic subvariety of \({\mathbb C}^{n^3}\). The group \(GL_n({\mathbb C})\) naturally acts on this variety via change of basis in the vector space \(\mathcal V\). The orbits of this action are exactly the isomorphism classes of the \(n\)-dimensional complex Lie algebras.
The paper under review describes a complete classification of the Zariski closures of the orbits corresponding to this action when \(n\leq 4\). As a consequence, one obtains that belonging to an orbit closure can be always realized (in a certain sense) by means of a one-parameter subgroup when \(n=3\), but this is not the case when \(n=4\).
As another point of interest, we should note that the paper contains the corrected lists of isomorphism classes for the complex Lie algebras of dimensions \(\leq 4\).

17B05 Structure theory for Lie algebras and superalgebras
17B30 Solvable, nilpotent (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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