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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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References:
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