Noncommutative affine spaces and Lie-complete rings

Dosi, Anar
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.


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Citation Formats
A. Dosi, “Noncommutative affine spaces and Lie-complete rings,” COMPTES RENDUS MATHEMATIQUE, pp. 149–153, 2015, Accessed: 00, 2020. [Online]. Available: