Structure of canonical variables in the theory of quantum systems with finitely and infinitely many degrees of freedom

It is shown that any representation Of canonical variables, (i.e., a representation of the canonical commutation relations in the Heisenberg form) is a direct integral of irreducible (factor) representations; no assumptions are made concerning the possibility of a transition to the Weyl form of the commutation relations. This theorem is applied to the construction of decompositions into irreducible (factor) representations of any finite-dimensional and some inifinitedimensional Lie algebras by unbounded operators in Hilbert space. The need for such decompositions arises in the harmonic analysis of unitary representations of the corresponding Lie groups.