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

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dc.creatorDurov, Nikolai
dc.creatorMeljanac, Stjepan
dc.creatorSamsarov, Andjelo
dc.creatorŠkoda, Zoran
dc.date2006-04-05
dc.date2006-08-31
dc.date.accessioned2026-07-07T07:42:19Z
dc.date.available2026-07-07T07:42:19Z
dc.descriptionGiven 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.
dc.descriptionv2: 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)
dc.identifierhttps://arxiv.org/abs/math/0604096
dc.identifierhttp://arxiv.org/abs/math/0604096
dc.identifierJ.Algebra 309 (2007) 318-359
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/122400
dc.subjectRepresentation Theory
dc.subjectMathematical Physics
dc.subjectQuantum Algebra
dc.subjectRings and Algebras
dc.subject16G, 17B, 17B40, 14D15, 14L05
dc.titleA universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra
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