Graded Lie algebras, representation theory, integrable mappings, and integrable systems

A new class of integrable mappings and chains is introduced. The corresponding $1+2$ integrable systems that are invariant under such integrable mappings are presented in an explicit form. Soliton-type solutions of these systems are constructed in terms of matrix elements of fundamental representations of semisimple $A_n$ algebras for a given group element. The possibility of generalizing this construction to the multidimensional case is discussed.