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 $X_1^0,...,X_n^0$, we give a formula, depending only on the structure constants, representing the infinitesimal generators, $X_i=X_i^{0}t$ in $g{\otimes }_{k}k[[t]]$, 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.

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)

References in corpus (2)

• New realizations of Lie algebra kappa-deformed Euclidean space
• Gauge theories on the kappa-Minkowski spacetime

Cited by in corpus (1)

• Lie algebraic deformations of Minkowski space with Poincare algebra