Representations of General Commutative Relations. Conservation Laws for Quantum Hamiltonians

In this paper we consider general commutative relations and construct its irreducible representations. Also we prove classified theorems. These commutative relations appear, in particle, in connection with nonstandard Poisson brackets. As applications we consider quantum Hamiltonians, which are written for creation and annihilation operators with arbitrary commutative relations. For this Hamiltonians we prove a criterion of presence of conservation laws depending on particle number operators.