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. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 251–271, February, 2000.