A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra
Abstract
Given a $n$dimensional Lie algebra $g$ over a field $k \supset \mathbb Q$, together with its vector space basis $X^0_1,..., X^0_n$, we give a formula, depending only on the structure constants, representing the infinitesimal generators, $X_i = X^0_i t$ in $g\otimes_k k [[t]]$, 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 $k\supset Q$. 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 rightinvariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 April 2006
 DOI:
 10.48550/arXiv.math/0604096
 arXiv:
 arXiv:math/0604096
 Bibcode:
 2006math......4096D
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Mathematical Physics;
 Mathematics  Quantum Algebra;
 Mathematics  Rings and Algebras;
 Mathematical Physics;
 16G;
 17B;
 17B40;
 14D15;
 14L05
 EPrint:
 v2: expositional improvements (significant in sections 5,6)