Commuting Polynomials in Quantum Canonical Operators and Realizations of Lie Algebras

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Abstract
It is pointed out that the problem of realizing Lie algebras through polynomials in quantum canonical operators is not equivalent to its classical counterpart because the polynomial Lie algebras taken with respect to the classical and quantum Lie brackets are not isomorphic. Yet there are still many results which are common to both. To show this, the properties of commuting polynomials in quantum canonical operators are analyzed. This makes possible an extension from the classical to the quantum domain of a number of theorems on realizations of semisimple Lie algebras. At the same time it is stressed that differences can arise in the classical and quantum solutions, and some of these are described.