Boson subalgebras and classification of boson state vectors

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Abstract
In the construction and classification of the possible state vectors of a limited number of boson modes, the use of subalgebras of invariant operators can simplify the procedure. The subalgebra of all the invariant operators (invariant subalgebra) and the subalgebra generated by the invariant-pair operators (invariant-pair subalgebra) are both considered. The invariant-pair subalgebra has the decisive advantage of allowing the easy evaluation of matrix elements. The construction problem is reduced to the problem of constructing the invariant-pair-free states, and a general procedure for determining these states is presented.