A standard way of realizing a Lie algebra is as a family of vector fields closed under commutation. Using the action on the universal enveloping algebra, one finds a realization in a dual form - the double dual. This is an algebraic Fourier transform of a "vector fields realization" of the Lie algebra. On the other hand, in the subject of umbral calculus (canonical boson calculus) the duals-to-vector fields play a primary role. It is shown that the double dual realizations of Lie algebras provide a rich source of examples for the umbral calculus, which, complementarily, provides a canonical construction of polynomial systems associated to the Lie algebra. For any finite-dimensional Lie algebra, take an element in the local Lie group it generates. Then there is an abelian family of operators such that acting on a canonical vacuum state, the abelian group gives the same result as the group element constructed via the given Lie algebra. In other words, they yield the same coherent states. (C) 2000 Academic press. [References: 3]
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