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

Given an n-dimensional Lie algebra over a field , together with its vector space basis , we give a formula, depending only on the structure constants, representing the infinitesimal generators, in , where t is a formal variable, as a formal power series in t with coefficients in the Weyl algebra An. Actually, the theorem is proved for Lie algebras over arbitrary rings . 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.