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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 \(\leq 4\).

MSC:
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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