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

摘要

Given an n-dimensional Lie algebra g over a field k superset of 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 circle times(k) k[t], where I 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 superset of 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. (c) 2006 Elsevier Inc. All rights reserved.