  A class of simple filtered Lie algebras of polynomial growth with increasing
filtration is distinguished and presentations of these algebras are explicitely
described for the simplest examples.
  Lie (super)algebras of this class appear in relation with
Calogero--Sutherland model, high-spin supergravity, etc.; they are associated
with the associative algebras of twisted differential operators on the big
Schubert cell of the flag varieties.
  The Lie algebra of matrices of complex size introduced by Feigin is a
simplest example of our algebras. Usually, they posess a trace and an invariant
symmetric bilinear form; hence, analogs of dynamical systems such as
Yang-Baxter, KdV, Leznov--Saveliev, etc. are associated with them. In
particular, in the space of pseudodifferential operators there are analogs of
the KdV hierarchies associated with sl(n) for n complex in the same way as the
KdV hierarchy is associated with sl(n) for n integer are those studied by
Gelfand--Dickey and Khesin--Malikov. We briefly describe such dynamical
systems, and generalizations of the classical orthogonal polynomials.
