On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras

Tange, Rudolf

doi:10.5802/aif.3271

On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras
[Sur la première cohomologie restreinte d’une algèbre de Lie réductif et de ses sous-algèbres de Borel]

Tange, Rudolf ¹

¹ University of Leeds School of Mathematics LS2 9JT, Leeds (UK)

Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1295-1308.

Résumé
Abstract

Soit $k$ un corps algébriquement clos de charactéristique $p > 0$ and soit $G$ un groupe réductif connexe sur $k$ . Soit $B$ un sous-groupe de Borel de $G$ et soit $𝔤$ et $𝔟$ les algèbres de Lie de $G$ et $B$ . Notons les premiers noyaux de Frobenius de $G$ et $B$ par $G_{1}$ et $B_{1}$ . De plus, notons les algèbres des fonctions régulières sur $G$ et $𝔤$ par $k [G]$ et $k [𝔤]$ , et de même pour $B$ et $𝔟$ . Le groupe $G$ agit sur $k [G]$ par conjugaison et sur $k [𝔤]$ par l’action adjointe. De même, $B$ agit sur $k [B]$ par l’action de conjugaison et sur $k [𝔟]$ par l’action adjointe. Nous montrons que, sous certaines hypothèses, les groupes de cohomologie $H^{1} (G_{1}, k [𝔤])$ , $H^{1} (B_{1}, k [𝔟])$ , $H^{1} (G_{1}, k [G])$ et $H^{1} (B_{1}, k [B])$ sont nuls. Nous étendons aussi nos résultats à la cohomologie pour les noyaux de Frobenius supérieurs.

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive group over $k$ . Let $B$ be a Borel subgroup of $G$ and let $𝔤$ and $𝔟$ be the Lie algebras of $G$ and $B$ . Denote the first Frobenius kernels of $G$ and $B$ by $G_{1}$ and $B_{1}$ . Furthermore, denote the algebras of regular functions on $G$ and $𝔤$ by $k [G]$ and $k [𝔤]$ , and similarly for $B$ and $𝔟$ . The group $G$ acts on $k [G]$ via the conjugation action and on $k [𝔤]$ via the adjoint action. Similarly, $B$ acts on $k [B]$ via the conjugation action and on $k [𝔟]$ via the adjoint action. We show that, under certain mild assumptions, the cohomology groups $H^{1} (G_{1}, k [𝔤])$ , $H^{1} (B_{1}, k [𝔟])$ , $H^{1} (G_{1}, k [G])$ and $H^{1} (B_{1}, k [B])$ are zero. We also extend all our results to the cohomology for the higher Frobenius kernels.

Reçu le : 2018-02-19
Révisé le : 2018-04-28
Accepté le : 2018-06-12
Publié le : 2019-06-03

Zbl

DOI : 10.5802/aif.3271

Classification : 20G05, 20G10
Keywords: Cohomology, Frobenius kernel, reductive group
Mot clés : Cohomologie, noyau de Frobenius, groupe réductif

Affiliations des auteurs :

Tange, Rudolf ¹

¹ University of Leeds School of Mathematics LS2 9JT, Leeds (UK)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the first restricted cohomology of a reductive {Lie} algebra and its {Borel} subalgebras},
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     pages = {1295--1308},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Tange, Rudolf. On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1295-1308. doi : 10.5802/aif.3271. https://aif.centre-mersenne.org/articles/10.5802/aif.3271/

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