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...

Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semiinvariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras.An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for both the complex and real Lie algebras of dimensions not greater than four are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases.Levels and colevels of low-dimensional Lie algebras are discussed in detail. Properties of multi-parametric and repeated contractions are also investigated.

Complete sets of bases of differential invariants, operators of invariant differentiation, and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of finite-dimensional Lie algebras on the real plane are revisited.

The orbits of Weyl groups W (An) of simple An type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of An. Matrices transforming points of the orbits of W (An) into points of subalgebra orbits are listed for all cases n ≤ 8 and for the infinite series of algebra-subalgebra pairs An ⊃ A n−k−1 × A k × U 1 , A 2n ⊃ Bn, A 2n−1 ⊃ Cn, A 2n−1 ⊃ Dn. Numerous special cases and examples are shown.