A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra

Abstract

Given an n-dimensional Lie algebra g over a field k ⊃ Q , together with its vector space basis X 1 0 , … , X n 0 , we give a formula, depending only on the structure constants, representing the infinitesimal generators, X i = X i 0 t in g ⊗ k k [ [ t ] ] , where t is a formal variable, as a formal power series in t with coefficients in the Weyl algebra A n . Actually, the theorem is proved for Lie algebras over arbitrary rings k ⊃ Q . We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of coth ( x / 2 ) . The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.