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

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

2006-04-05

arxiv.org/abs/math/0604096v4

J.Algebra 309 (2007) 318-359

Mathematics

Representation Theory

v2: expositional improvements (significant in sections 5,6); v3: minor expositional improvements (including in notation, and i

Scientific paper

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.

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