Deformations of four-dimensional Lie algebras. (English) Zbl 1127.14014
The authors study the moduli space of four-dimensional complex Lie algebras. It is a continuation of an earlier paper on the moduli space of three-dimensional complex Lie algebras [the authors, Commun. Contemp. Math. 7, No. 2, 145–165 (2005; Zbl 1075.14007)].
The classification of complex Lie algebras of low dimension is well-known. Up to dimension four, the classification was given by G. M. Mubarakzyanov [Izv. Vyssh. Uchebn. Zaved., Mat. 1963, No. 1(32), 114–123 (1963; Zbl 0166.04104)].
It is known that the set of isomorphism classes of complex Lie algebras of dimension \(n\) can be identified with the set of orbits of the affine complex \(N\)-spaces under the action of \(\text{GL}(n)\), where \(N = (n-1)n^2/2\). But this orbit space is not a geometric quotient in general and has a complicated algebraic structure [see H. Bjar and O. A. Laudal, Compos. Math. 75, No. 1, 69–111 (1990; Zbl 0708.14005)].
The main result in this paper is a computation of the versal deformations of any complex Lie algebra of dimension four. To compute the versal deformations, the authors use Lie algebra cohomology and an obstruction theory based on the point of view that Lie algebras are quadratic codifferentials defined on a vector space.
The authors also identify the moduli space of complex Lie algebras of dimension four with a collection of orbifolds and points. This gives a geometric picture of the various deformations, but it is not an algebraic moduli space.
The classification of complex Lie algebras of low dimension is well-known. Up to dimension four, the classification was given by G. M. Mubarakzyanov [Izv. Vyssh. Uchebn. Zaved., Mat. 1963, No. 1(32), 114–123 (1963; Zbl 0166.04104)].
It is known that the set of isomorphism classes of complex Lie algebras of dimension \(n\) can be identified with the set of orbits of the affine complex \(N\)-spaces under the action of \(\text{GL}(n)\), where \(N = (n-1)n^2/2\). But this orbit space is not a geometric quotient in general and has a complicated algebraic structure [see H. Bjar and O. A. Laudal, Compos. Math. 75, No. 1, 69–111 (1990; Zbl 0708.14005)].
The main result in this paper is a computation of the versal deformations of any complex Lie algebra of dimension four. To compute the versal deformations, the authors use Lie algebra cohomology and an obstruction theory based on the point of view that Lie algebras are quadratic codifferentials defined on a vector space.
The authors also identify the moduli space of complex Lie algebras of dimension four with a collection of orbifolds and points. This gives a geometric picture of the various deformations, but it is not an algebraic moduli space.
Reviewer: Eivind Eriksen (Oslo)
MSC:
| 14D15 | Formal methods and deformations in algebraic geometry |
| 17B55 | Homological methods in Lie (super)algebras |
| 17B56 | Cohomology of Lie (super)algebras |
| 13D10 | Deformations and infinitesimal methods in commutative ring theory |
| 14B12 | Local deformation theory, Artin approximation, etc. |
References:
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