Vyacheslav Boyko

Realizations of real low-dimensional Lie algebras

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.

Computation of invariants of Lie algebras by means of moving frames

A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. The algorithm is applied, in particular, to computation of invariants of real low-dimensional Lie algebras. A number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature. Bases of invariants of the real six-dimensional solvable Lie algebras with four-dimensional nilradicals are newly calculated and listed in a table.

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients

Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.

Invariants of Solvable Lie Algebras with Triangular Nilradicals and Diagonal Nilindependent Elements

The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartan’s method of moving frames.

Equivalence groupoids of classes of linear ordinary differential equations and their group classification

Admissible point transformations of classes of

Invariants of triangular Lie algebras with one nil-independent diagonal element

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Tetrazolecalix[4]arenes as new ligands for palladium(II)

th order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational form, the Laguerre-Forsyth form, the first and second Arnold forms) are exhaustively described. Using these results, the group classification of such equations is revisited within the algebraic approach in three different ways.

Nonlinear Representations for Poincaré and Galilei algebras and nonlinear equations for electromagnetic fields

The invariants of solvable triangular Lie algebras with one nil-independent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartans method of moving frames and the special technique developed for triangular and closed algebras in Boyko et al (J. Phys. A: Math. Theor. 2007 40 7557). The conjecture of Tremblay and Winternitz (J. Phys. A: Math. Gen. 2001 34 9085) on the number and form of elements in the bases is completed and proved.