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A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra. (English) Zbl 1173.17014
Summary: Given an \(n\)-dimensional Lie algebra \(\mathfrak g\) over a field \(k\supset\mathbb Q\), together with its vector space basis \(X_1^0, X_2^0,\dots, X_n^0\), we give a formula, depending only on the structure constants, representing the infinitesimal generators, \(X_i=X_i^0t\) in \(\mathfrak g\otimes_kk[[t]]\), 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 \(k\supset\mathbb Q\).
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.

MSC:
17B35 Universal enveloping (super)algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
14L05 Formal groups, \(p\)-divisible groups
14L15 Group schemes
16S80 Deformations of associative rings
11B68 Bernoulli and Euler numbers and polynomials
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