Noncommutative affine spaces and Lie-complete rings

Dosi A.

COMPTES RENDUS MATHEMATIQUE, vol.353, no.2, pp.149-153, 2015 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 353 Issue: 2
  • Publication Date: 2015
  • Doi Number: 10.1016/j.crma.2014.10.020
  • Journal Name: COMPTES RENDUS MATHEMATIQUE
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.149-153
  • Middle East Technical University Affiliated: No

Abstract

In this paper, we investigate the structure sheaves of an (infinite-dimensional) affine NC-space A(nc)(x) affine Lie-space A(lich)(x), and their nilpotent perturbations A(nc,q)(x) and A(lich),(x)(q) respectively. We prove that the schemes A(nc)(x) and A(lich)(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 subset of X = Spec(C vertical bar x vertical bar), we obtain the precise descriptions of the algebras O-nc(U), O-nc,(q)(U). O-lich,(q)(U) and O-lich,(q)(U) of noncommutative regular functions on U associated with the schemes A(nc)(x), A(nc,q)(x), A(lich),(x)(q) and 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 Frechet algebras. (C) 2014 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.