Realizations of real low-dimensional Lie algebras

Abstract

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject.Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed. On classification of Lie algebrasThe necessary step to classify realizations of Lie algebras is classification of these algebras, i.e. classification of possible commutative relations between basis elements. By the Levi-Maltsev theorem any finite-dimensional Lie algebra over a field of characteristic 0 is a semi-direct sum (the Levi-Maltsev decomposition) of the radical (its maximal solvable ideal) and a semi-simple subalgebra (called the Levi factor) (see, e.g., [25]). This result reduces the task of classifying all Lie algebras to the following problems:1) classification of all semi-simple Lie algebras; 2) classification of all solvable Lie algebras;3) classification of all algebras that are semi-direct sums of semi-simple Lie algebras and solvable Lie algebras.Of the problems listed above, only that of classifying all semi-simple Lie algebras is completely solved in the well-known Cartan theorem: any semi-simple complex or real Lie algebra can be decomposed into a direct sum of ideals which are simple subalgebras being mutually orthogonal with respect to the Cartan-Killing form. Thus, the problem of classifying semi-simple Lie algebras is equivalent to that of classifying all non-isomorphic simple Lie algebras. This classification is known (see, e.g., [14,5]).At the best of our knowledge, the problem of classifying solvable Lie algebras is completely solved only for Lie algebras of dimension up to and including six (see, for example, [48,49,50,51,80,81]). Below we shortly list some results on classifying of low-dimensional Lie algebras.All the possible complex Lie algebras of dimension ≤ 4 were listed by S. Lie himself [37]. In 1918 L. Bianchi investigated three-dimensional real Lie algebras [7]. Considerably later this problem was again considered by H.C. Lee [32] and G. Vranceanu [85], and their classifications are equivalent to Bianchi's one. Using Lie's results on complex structures, G.I. Kruchkovich [27,28,29] classified four-dimensional real Lie algebras which do not contain three-dimensional abelian subalgebras.Complete, correct and easy to use classification of real Lie algebras of dimension ≤ 4 was first carried out by G.M. Mubarakzyanov [49] (see also citation of these results as well as description of subalgebras and invariants of real low-dimensional Lie algebras in [58,59]). At the same year a variant of such classification was obtained by J. Dozias [13] and then adduced in [84]. Analogous results are given in [62]. Namely, after citing classifications of L. Bianchi [7] and G.I. Kruchkovich [27], A.Z. Petrov clas...

Symmetry group classification of ordinary differential equations: Survey of some results

“…Hence, here L 3;4 denotes the fourth Lie algebra of dimension 3 and so on for the other algebras. In contemporary works, the notation A a r ; j is also used (see [46][47][48]). …”

Contractions of low-dimensional Lie algebras

“…A a 3.4 : [e 1 , e 3 ] = e 1 , [e 2 , e 3 ] = ae 2 , 0 < |a| < 1 (indecomposable, solvable); [61,62,65,66].…”

“…The presented approach can be extended to the real case and algebras of greater dimensions. In the above list of algebras, we apply enhanced normalization of series parameters for fourdimensional real Lie algebras, which were proposed in [65,66].…”

“…Low-dimensional Lie algebras are also interesting per se and supply theoretical consideration with substantial examples. In this connection classifications, subalgebras, realizations, invariants, contractions, deformations and other objects concerning low-dimensional Lie algebras were studied [5,22,24,50,55,62,66].…”

On nilpotent and solvable Lie algebras of derivations

“…D is a vector field on C n with polynomial coefficients. Lie algebras of vector fields with polynomial, formal power series, or analytical coefficients were studied intensively by many authors (see, for example, [7], [1], [2], [3], [4], [10], [11]). In general case, when A is an integral domain, subalgebras L of DerA such that L are submodules of the A-module DerA were studied in [6], and sufficient conditions were given for L to be simple.…”