Noncommutative affine spaces and Lie-complete rings

Dosi, Anar

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

Dans cette note, nous étudions la structure des faisceaux des NC-espaces ${\mathbb{A}}_{\mathfrak{nc}}^{\mathbf{x}}$ et des Lie espaces ${\mathbb{A}}_{\mathfrak{lieh}}^{\mathbf{x}}$, affines (de dimension infinie), et de leur perturbations nilpotentes ${\mathbb{A}}_{\mathfrak{nc},q}^{\mathbf{x}}$ et ${\mathbb{A}}_{\mathfrak{lieh},q}^{\mathbf{x}}$, respectivement. Nous montrons que les schémas ${\mathbb{A}}_{\mathfrak{nc}}^{\mathbf{x}}$ et ${\mathbb{A}}_{\mathfrak{lieh}}^{\mathbf{x}}$ 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) $U\subset X=\mathrm{Spec}(\mathbb{C}[\mathbf{x}])$, nous obtenons les descriptions précises des algèbres ${\mathcal{O}}_{\mathfrak{nc}}(U)$, ${\mathcal{O}}_{\mathfrak{nc},q}(U)$, ${\mathcal{O}}_{\mathfrak{lieh}}(U)$ et ${\mathcal{O}}_{\mathfrak{lieh},q}(U)$, de fonctions régulières non commutatives sur U, associées aux schémas ${\mathbb{A}}_{\mathfrak{nc}}^{\mathbf{x}}$, ${\mathbb{A}}_{\mathfrak{nc},q}^{\mathbf{x}}$, ${\mathbb{A}}_{\mathfrak{lieh}}^{\mathbf{x}}$ et ${\mathbb{A}}_{\mathfrak{lieh},q}^{\mathbf{x}}$, respectivement. Ces résultats pour ${\mathcal{O}}_{\mathfrak{nc}}(U)$ 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 $\Gamma (X,{\mathcal{O}}_A)$ 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 ${\mathbb{A}}_{\mathfrak{nc}}^{\mathbf{x}}$, affine Lie-space ${\mathbb{A}}_{\mathfrak{lieh}}^{\mathbf{x}}$, and their nilpotent perturbations ${\mathbb{A}}_{\mathfrak{nc},q}^{\mathbf{x}}$ and ${\mathbb{A}}_{\mathfrak{lieh},q}^{\mathbf{x}}$, respectively. We prove that the schemes ${\mathbb{A}}_{\mathfrak{nc}}^{\mathbf{x}}$ and ${\mathbb{A}}_{\mathfrak{lieh}}^{\mathbf{x}}$ 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 $U\subseteq X=\mathrm{Spec}(\mathbb{C}[\mathbf{x}])$, we obtain the precise descriptions of the algebras ${\mathcal{O}}_{\mathfrak{nc}}(U)$, ${\mathcal{O}}_{\mathfrak{nc},q}(U)$, ${\mathcal{O}}_{\mathfrak{lieh},q}(U)$ and ${\mathcal{O}}_{\mathfrak{lieh},q}(U)$ of noncommutative regular functions on U associated with the schemes ${\mathbb{A}}_{\mathfrak{nc}}^{\mathbf{x}}$, ${\mathbb{A}}_{\mathfrak{nc},q}^{\mathbf{x}}$, ${\mathbb{A}}_{\mathfrak{lieh},q}^{\mathbf{x}}$ and ${\mathbb{A}}_{\mathfrak{lieh}}^{\mathbf{x}}$, respectively. The obtained result for ${\mathcal{O}}_{\mathfrak{nc}}(U)$ 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.

[1] Dosi, A.A. Noncommutative holomorphic functions in elements of a Lie algebra and noncommutative localizations, Izv. Math. RAN, Volume 73 (2009) no. 06, pp. 77-100

[2] Dosi, A.A. Formally-radical functions in elements of a nilpotent Lie algebra and noncommutative localizations, Algebra Colloq., Volume 17 (2010) no. Spec. 1, pp. 749-788

[3] Dosi, A.A. Taylor functional calculus for supernilpotent Lie algebra of operators, J. Oper. Theory, Volume 63 (2010) no. 1, pp. 101-126

[4] Dosi, A.A. Taylor spectrum and transversality for a Heisenberg algebra of operators, Mat. Sb., Volume 201 (2010) no. 3, pp. 39-62

[5] Dosiev, A.A. Algebras of power series of elements of a Lie algebra and the Slodkowski spectra, J. Math. Sci., Volume 124 (2004) no. 2, pp. 4886-4908 (translated from Zapiski Nauchnykh Seminarov POMI)

[6] Dosiev, A.A. Cartan–Slodkowski spectra, splitting elements and noncommutative spectral mapping theorems, J. Funct. Anal., Volume 230 (2006) no. 2, pp. 446-493

[7] Dosiev, A.A. Local left invertibility for operator tuples and noncommutative localizations, J. K-Theory, Volume 4 (2009) no. 1, pp. 163-191

[8] Etingof, P.; Kim, J.; Ma, X. On universal Lie nilpotent associative algebras, J. Algebra, Volume 321 (2009) no. 2, pp. 697-703

[9] Feigin, B.; Shoiket, B. On $[A,A]/[A,[A,A]]$ and a $W_n$-action on the consecutive commutators of free associative algebra, Math. Res. Lett., Volume 14 (2007) no. 5–6, pp. 781-795

[10] Grothendieck, A. Éléments de géométrie algébrique, I. Le langage des schémas, Publ. Math. Inst. Hautes Études Sci., Volume 4 (1960), pp. 5-228

[11] Hartshorne, R. (Graduate Texts in Mathematics), Volume vol. 52, Springer Verlag (1977), p. 496

[12] Kapranov, M. Noncommutative geometry based on commutator expansions, J. Reine Angew. Math., Volume 505 (1998), pp. 73-118

[13] Rosenberg, A.L. Noncommutative Algebraic Geometry and Representation of Quantum Groups, Kluwer Academic Publishers, 1995

[14] Smoktunowicz, A. A simple nil ring exists, Commun. Algebra, Volume 30 (2002) no. 1, pp. 27-59

[15] Van Oystaeyen, F.; Verschoren, A.H. Non-Commutative Algebraic Geometry, Lecture Notes in Mathematics, vol. 887, Springer-Verlag, Berlin, 1981