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...

Application of calixarene derivatives in bio-medical researches is reviewed in this article. Antiviral, bactericidal, antithrombothic, antituberculosis, anticancer activity as well as specific protein complexation, membranotropic properties and toxicity of modified calixarenes are discussed.

A purely algebraic algorithm for computation of invariants (generalized Casimir operators) of Lie algebras by means of moving frames is discussed. Results on the application of the method to computation of invariants of low-dimensional Lie algebras and series of solvable Lie algebras restricted only by a required structure of the nilradical are reviewed.

An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], which deals with low-dimensional Lie algebras, here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable Lie algebras of general dimension n < ∞ restricted only by a required structure of the nilradical.Specifically, invariants are calculated here for families of real/complex solvable Lie algebras. These families contain, with only a few exceptions, all the solvable Lie algebras of specific dimensions, for whom the invariants are found in the literature.

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.