Vector Fields and Their Duals

Abstract

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.