Front: [math/0604096] A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra

Title: A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra
Authors: Nikolai Durov, Stjepan Meljanac, Andjelo Samsarov, Zoran Škoda
Categories: math.RT Representation Theory (math.QA Quantum Algebra; math.RA Rings and Algebras; physics.math-ph Mathematical Physics)
Comments: v2: expositional improvements (significant in sections 5,6); v3: minor expositional improvements (including in notation, and in introduction); v4: final version, to appear in Journal of Algebra (4 minor differences from v3 due wrong uploaded file in v3)
MSC: 16G, 17B, 17B40, 14D15, 14L05
Journal reference: J.Algebra 309 (2007) 318-359

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.

Owner: Zoran Skoda
Version 1: Wed, 5 Apr 2006 18:58:24 GMT
Version 2: Mon, 24 Apr 2006 18:04:21 GMT
Version 3: Tue, 29 Aug 2006 16:07:20 GMT
Version 4: Thu, 31 Aug 2006 16:09:15 GMT