We have previously described an embedding of the Poincaré Lie algebra into an extension of the Lie field of the group SO0(1, 4), and we used this embedding to construct irreducible representations of the Poincaré group out of representations of SO0(1, 4). Some q generalizations of these results have been obtained by us i.e. we embed classical structures into quantum structures. Here we report o...

In these lectures the relations between symmetries, Lie algebras, Killing vectors and Noether’s theorem are reviewed. A generalisation of the basic ideas to include velocity-dependend co-ordinate transformations naturally leads to the concept of Killing tensors. Via their Poisson brackets these tensors generate an a priori infinitedimensional Lie algebra. The nature of such infinite algebras is...

A Toda equation is specified by a choice of a Lie group and a Z-gradation of its Lie algebra. The Toda equations associated with loop groups of complex classical Lie groups, whose Lie algebras are endowed with integrable Z-gradations with finite dimensional grading subspaces, are described in an explicit form.

A fast diffusion equation is investigated with symmetry point of view. The set of exact non-Lie solutions constructed by M.L. Gandarias [Phys. Lett. A, 2001,V.286, 153-160] are supplemented with the similar ones. The non-classical symmetries of the corresponding potential equation are completely classified with respect to the Lie symmetry group of this equation. As a result, we obtain new wide ...