Noncommutative affine spaces and Lie-complete rings

Dosi, Anar

doi:10.1016/j.crma.2014.10.020

Functional analysis/Algebraic geometry

Noncommutative affine spaces and Lie-complete rings
[Espaces affines non commutatifs et anneaux de Lie complets]

Présenté par : the Editorial Board

Dosi, Anar ¹

¹ Middle East Technical University, Northern Cyprus Campus, Guzelyurt, KKTC, Mersin 10, Turkey

Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 149-153.

Résumé
Abstract

Dans cette note, nous étudions la structure des faisceaux des NC-espaces $A_{nc}^{x}$ et des Lie espaces $A_{lieh}^{x}$ , affines (de dimension infinie), et de leur perturbations nilpotentes $A_{nc, q}^{x}$ et $A_{lieh, q}^{x}$ , respectivement. Nous montrons que les schémas $A_{nc}^{x}$ et $A_{lieh}^{x}$ sont identiques si et seulement si x est un ensemble fini de variables, c'est-à-dire lorsqu'on traite des espaces affines non commutatifs de dimension finie. Pour chaque ouvert (de Zariski) $U \subset X = Spec (C [x])$ , nous obtenons les descriptions précises des algèbres $O_{nc} (U)$ , $O_{nc, q} (U)$ , $O_{lieh} (U)$ et $O_{lieh, q} (U)$ , de fonctions régulières non commutatives sur U, associées aux schémas $A_{nc}^{x}$ , $A_{nc, q}^{x}$ , $A_{lieh}^{x}$ et $A_{lieh, q}^{x}$ , respectivement. Ces résultats pour $O_{nc} (U)$ généralisent la formule de Kapranov dans le cas où la dimension est finie. De plus, nous montrons que tout anneau Lie complet A est plongé dans $Γ (X, O_{A})$ comme sous-algèbre dense pour la topologie $I_{1}$ -adique associée à l'idéal bilatère $I_{1}$ engendré par tous les commutateurs de A.

In this paper, we investigate the structure sheaves of an (infinite-dimensional) affine NC-space $A_{nc}^{x}$ , affine Lie-space $A_{lieh}^{x}$ , and their nilpotent perturbations $A_{nc, q}^{x}$ and $A_{lieh, q}^{x}$ , respectively. We prove that the schemes $A_{nc}^{x}$ and $A_{lieh}^{x}$ are identical if and only if x is a finite set of variables, that is, when we deal with finite-dimensional noncommutative affine spaces. For each (Zariski) open subset $U \subseteq X = Spec (C [x])$ , we obtain the precise descriptions of the algebras $O_{nc} (U)$ , $O_{nc, q} (U)$ , $O_{lieh, q} (U)$ and $O_{lieh, q} (U)$ of noncommutative regular functions on U associated with the schemes $A_{nc}^{x}$ , $A_{nc, q}^{x}$ , $A_{lieh, q}^{x}$ and $A_{lieh}^{x}$ , respectively. The obtained result for $O_{nc} (U)$ generalizes Kapranov's formula in the finite-dimensional case. Our approach to the matter is based on a noncommutative holomorphic functional calculus in Fréchet algebras.

Reçu le : 2013-04-11
Accepté le : 2014-10-23
Publié le : 2014-12-03

DOI : 10.1016/j.crma.2014.10.020

Affiliations des auteurs :

Dosi, Anar ¹

¹ Middle East Technical University, Northern Cyprus Campus, Guzelyurt, KKTC, Mersin 10, Turkey

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Dosi, Anar. Noncommutative affine spaces and Lie-complete rings. Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 149-153. doi : 10.1016/j.crma.2014.10.020. http://archive.numdam.org/articles/10.1016/j.crma.2014.10.020/

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