  The Lie algebra gl(lambda) dependent on the complex parameter lambda is a
continuous version of the Lie algebra gl(inf) of infinite matrices with only
finite number of nonzero entries. The gl(lambda) was first introduced by
B.L.Feigin in [1] in connection with the Lie algebra cohomologies of the
differential operators on the complex line. The paper is devoted to the
representation theory of the gl(lambda). The Shapovalov's form determinant is
calculated; it depends polynomially on lambda and the parameters of the
representation. It is natural to include the Hamiltonian Lie algebra of
functions on the hyperboloid to the range of gl(lambda), relating it to the
value lambda = infinity. Thus, every family of representations of gl(lambda)
depending holomorphically on lambda is associated with a holomorphic vector
bundle on the Riemann sphere. The determinant of the Shapovalov's form thus
determines the corresponding line bundle. The Chern class of this bundle is
calculated in two different ways (for every level on the representation). The
comparison of the two formulas yields new combinatorial identities with power
series.
  The results of Chapter 1 about the Lie algebra gl(inf) may have some
independent interest.
