Formal power series, operator calculus, and duality on Lie algebras

This paper presents an operator calculus approach to computing with non-commutative variables. First, we recall the product formulation of formal exponential series. Then we show how to formulate canonical boson calculus on formal series. This calculus is used to represent the action of a Lie algebra on its universal enveloping algebra. As applications, Hamilton's equations for a general Hamiltonian, given as a formal series, are found using a double-dual representation, and a formulation of the exponential of the adjoint representation is given. With these techniques one can represent the Volterra product acting on the enveloping algebra. We illustrate with a three-step nilpotent Lie algebra.