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

Research paper by **Nikolai Durov, Stjepan Meljanac, Andjelo Samsarov, Zoran Škoda**

Indexed on: **31 Aug '06**Published on: **31 Aug '06**Published in: **Mathematics - Representation Theory**

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#### 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 right-invariant vector fields
in specific coordinates, in a (new) variant of formal group scheme theory. The
third proof uses coderivations and Hopf algebras.