Some direct limits of the classical Lie algebras and the projective limits connected with them

The present article considers some concrete classes of direct limits of the classical simple Lie algebras of the types A, B, C, D. With each such direct limit one can connect the projective limit of sets of dominant integral weights, the projective limit of the invariants of joint action in the algebras of the polynomial functions and sometimes the projective limit of the root lattices. The article is devoted to the study of structural mappings in these projective limits and to the proof of non-triviality of the latters.