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

Autori
Durov, Nikolai ; Samsarov, Andjelo ; Meljanac, Stjepan ; Škoda, Zoran

Izvornik
Journal of algebra (0021-8693) 309 (2007), 1; 318-359

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
deformations of algebras; Lie algebras; Weyl algebra; Bernoulli numbers; representations; formal schemes

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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