# zbMATH — the first resource for mathematics

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)
##### Keywords:
classification; variety of Lie algebras; orbit; Zariski closure
Full Text:
##### References:
 [1] Burde, D., Degenerations of nilpotent Lie algebras, J. Lie theory, 9, 193-202, (1999) · Zbl 1063.17009 [2] Fialowski, A.; O’Halloran, J., A comparison of deformations and orbit closure, Comm. algebra, 18, 4121-4140, (1990) · Zbl 0719.17002 [3] Grunewald, F.; O’Halloran, J., Varieties of nilpotent Lie algebras of dimension less than six, J. algebra, 112, 315-325, (1988) · Zbl 0638.17005 [4] Jacobson, N., Lie algebras, (1962), Interscience New York · JFM 61.1044.02 [5] Levy-Nahas, M., Deformation and contraction of Lie algebras, J. math. phys., 8, 1211-1223, (1967) · Zbl 0175.24803 [6] Mubarakzjanov, G.M., On solvable Lie algebras (Russian), Izv. vyssh. uchebn. zaved mat., 1, 114-123, (1963) · Zbl 0166.04104 [7] Patera, J.; Zassenhaus, H., Solvable Lie algebras of dimension ≤4 over perfect fields, Linear algebra appl., 142, 1-17, (1990) · Zbl 0718.17010 [8] Rhomdani, M., Classification of real and complex nilpotent Lie algebras of dimension 7, Linear and multilinear algebra, 24, 167-189, (1989) [9] Seeley, C., Degenerations of 6-dimensional nilpotent Lie algebras over $$C$$, Comm. algebra, 18, 3493-3505, (1990) · Zbl 0709.17006 [10] Steinhoff, C., Klassifikation und degeneration von Lie algebren, (1997), Diplomarbeit Düsseldorf [11] Turkowski, P., Solvable Lie algebras of dimension six, J. math. phys., 31, 1344-1350, (1990) · Zbl 0722.17012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.