14:30 - 15:30 Roman O. Popovych (Faculty of Mathematics, University of Vienna, Austria & Institute of Mathematics, NAS of Ukraine, Kyiv, Ukraine)
Computation of contractions of Lie algebras
Abstract: Limiting processes (contractions) of Lie algebras appear in different areas of physics and mathematics, where Lie algebras arise, e.g., in the study of representations, invariants and special functions. The algebraic counterpart of the notion of contractions of Lie algebras is given by degenerations of Lie algebras.
The main attention in the talk is paid to the practical computation of contractions of Lie algebras. We present a wide list of necessary conditions for the contraction existence. Particular ways for realizing contractions, which are relevant to physics and includes simple and generalized Inönü–Wigner contractions and Saletan (linear) contractions, are discussed and the limitation for using them is clarified. We also plan to present the complete description of contractions of Lie algebras of dimension not greater than four, of five- and six-dimensional nilpotent algebras and of almost Abelian Lie algebras over both the complex and real field.
15:30 - 16:00 break
16:00 - 17:00 Célestin Kurujyibwami (University of Rwanda, Kigali, Rwanda)
Group classification of multidimensional nonlinear Schrödinger equations
Abstract: We describe the equivalence groupoid and the equivalence group of the class N of generalized multidimensional Schrödinger equations with variable mass and show that this class is not normalized. We then partition this class into two disjoint normalized subclasses and derive their corresponding equivalence groups. Restricting to the case of constant mass equal to one, we characterize the point transformations of the subclasses of the class N with respect to specific values of the arbitrary elements, in particular we do this for the class V of multidimensional nonlinear Schrödinger equations with potentials and modular nonlinearities. This class also turns out not to be normalized. We partition it into three normalized subclasses, and this allows us to apply the algebraic method and solve each subclass completely for space dimension two. The group classification in each involves three integers that are invariant with respect to the adjoint action of the equivalence transformations. As a result, a full list of Lie symmetry extensions together with their corresponding families of potentials in the class V is presented.