scholarly journals The Fundamental Group of a Noncommutative Space

2021 ◽  
Author(s):  
Walter D. van Suijlekom ◽  
Jeroen Winkel
Keyword(s):  
Fundamental Group ◽  
Classical Case ◽  
Group Scheme ◽  
Noncommutative Space ◽  
General Notion ◽  
Noncommutative Torus ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Tannakian Category

AbstractWe introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded algebras and show that the category of flat connections on such modules forms a Tannakian category. As such this category can be realised as the category of representations of an affine group scheme G, which in the classical case is (the pro-algebraic completion of) the usual fundamental group. This motivates us to define G to be the fundamental group of the noncommutative space under consideration. The needed assumptions on the differential graded algebra are rather mild and completely natural in the context of noncommutative differential geometry. We establish the appropriate functorial properties, homotopy and Morita invariance of this fundamental group. As an example we find that the fundamental group of the noncommutative torus can be described as the algebraic hull of the topological group $(\mathbb Z+\theta \mathbb Z)^{2}$ ( ℤ + 𝜃 ℤ ) 2 .

scholarly journals A strictly commutative model for the cochain algebra of a space

2020 ◽  
Vol 156 (8) ◽  
pp. 1718-1743
Author(s):  
Birgit Richter ◽  
Steffen Sagave
Keyword(s):  
Homotopy Theory ◽  
Rational Homotopy ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Simplicial Sets ◽  
Differential Graded Algebras ◽  
Chain Complexes ◽  
Simplicial Set ◽  
New Construction ◽  
Integral Version

AbstractThe commutative differential graded algebra $A_{\mathrm {PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal {I}}(X)$ of $A_{\mathrm {PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal {I}$ to model $E_{\infty }$ differential graded algebras (dga) by strictly commutative objects, called commutative $\mathcal {I}$-dgas. We define a functor $A^{\mathcal {I}}$ from simplicial sets to commutative $\mathcal {I}$-dgas and show that it is a commutative lift of the usual cochain algebra functor. In particular, it gives rise to a new construction of the $E_{\infty }$ dga of cochains. The functor $A^{\mathcal {I}}$ shares many properties of $A_{\mathrm {PL}}$, and can be viewed as a generalization of $A_{\mathrm {PL}}$ that works over arbitrary commutative ground rings. Working over the integers, a theorem by Mandell implies that $A^{\mathcal {I}}(X)$ determines the homotopy type of $X$ when $X$ is a nilpotent space of finite type.

scholarly journals MODELS OF TORSORS AND THE FUNDAMENTAL GROUP SCHEME

2016 ◽  
Vol 230 ◽  
pp. 18-34 ◽  
Author(s):  
MARCO ANTEI ◽  
MICHEL EMSALEM
Keyword(s):  
Finite Number ◽  
Fundamental Group ◽  
Galois Group ◽  
Closed Subset ◽  
Valuation Ring ◽  
Group Scheme ◽  
Discrete Valuation Ring ◽  
Generic Fiber ◽  
Natural Morphism ◽  
Tannakian Category

Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring $X\rightarrow S$, this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber $X_{\unicode[STIX]{x1D702}}$ to the generic fiber of the fundamental group scheme of $X$. Given a torsor $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under an affine group scheme $G$ over the generic fiber of $X$, we address the question of finding a model of this torsor over $X$, focusing in particular on the case where $G$ is finite. We provide several answers to this question, showing for instance that, when $X$ is integral and regular of relative dimension 1, such a model exists on some model $X^{\prime }$ of $X_{\unicode[STIX]{x1D702}}$ obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of $X$. Furthermore, we show that when $G$ is étale, then we can find a model of $T\rightarrow X_{\unicode[STIX]{x1D702}}$ under the action of some smooth group scheme. In the first part of the paper, we show that the relative fundamental group scheme of $X$ has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor.

scholarly journals Higgs bundles and fundamental group schemes

2019 ◽  
Vol 19 (3) ◽  
pp. 381-388
Author(s):  
Indranil Biswas ◽  
Ugo Bruzzo ◽  
Sudarshan Gurjar
Keyword(s):  
Fundamental Group ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Higgs Bundles ◽  
Chern Classes ◽  
Group Schemes ◽  
Tannakian Category

Abstract Relying on a notion of “numerical effectiveness” for Higgs bundles, we show that the category of “numerically flat” Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the “Higgs fundamental group scheme of X,” and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

Differential Graded Algebras

2020 ◽  
pp. 145-150
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Algebraic Model ◽  
Lie Derivative ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Rham Complex ◽  
De Rham

This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex. If the Lie group G acts smoothly on M, then the de Rham complex Ω‎(M) is more than a differential graded algebra. It has in addition two actions of the Lie algebra: interior multiplication and the Lie derivative. A differential graded algebra Ω‎ with an interior multiplication and a Lie derivative satisfying Cartan's homotopy formula is called a g-differential graded algebra. To construct an algebraic model for equivariant cohomology, the chapter first constructs an algebraic model for the total space EG of the universal G-bundle. It is a g-differential graded algebra called the Weil algebra.

scholarly journals (q,σ,τ)-Differential Graded Algebras

Universe ◽  
2018 ◽  
Vol 4 (12) ◽  
pp. 138 ◽  
Author(s):  
Viktor Abramov ◽  
Olga Liivapuu ◽  
Abdenacer Makhlouf
Keyword(s):  
Clifford Algebra ◽  
Quantum Plane ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Generalized Clifford Algebra

We propose the notion of ( q , σ , τ ) -differential graded algebra, which generalizes the notions of ( σ , τ ) -differential graded algebra and q-differential graded algebra. We construct two examples of ( q , σ , τ ) -differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a ( σ , τ ) -twisted graded q-commutator. In order to construct the second example, we introduce the notion of ( σ , τ ) -pre-cosimplicial algebra.

scholarly journals $(q,\sigma,\tau)$-Differential Graded Algebras

2018 ◽  
Author(s):  
Viktor Abramov ◽  
Olga Liivapuu ◽  
Abdenacer Makhlouf
Keyword(s):  
Clifford Algebra ◽  
Quantum Plane ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Generalized Clifford Algebra

We propose a notion of $(q,\sigma,\tau)$-differential graded algebra, which generalizes the notions of $(\sigma,\tau)$-differential graded algebra and $q$-differential graded algebra. We construct two examples of $(q,\sigma,\tau)$-differential graded algebra, where the first one is constructed by means of generalized Clifford algebra with two generators (reduced quantum plane), where we use a $(\sigma,\tau)$-twisted graded $q$-commutator. In order to construct the second example, we introduce a notion of $(\sigma,\tau)$-pre-cosimplicial algebra.

scholarly journals Generalized Euler classes, differential forms and commutative DGAs

2019 ◽  
Vol 11 (01) ◽  
pp. 109-118
Author(s):  
Alexander Gorokhovsky ◽  
Dennis Sullivan ◽  
Zhizhang Xie
Keyword(s):  
Differential Forms ◽  
Cohomological Dimension ◽  
Total Space ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Differential Graded Algebra ◽  
Differential Graded Algebras ◽  
Odd Degree

In the context of commutative differential graded algebras over [Formula: see text], we show that an iteration of “odd spherical fibration” creates a “total space” commutative differential graded algebra with only odd degree cohomology. Then we show for such a commutative differential graded algebra that, for any of its “fibrations” with “fiber” of finite cohomological dimension, the induced map on cohomology is injective.

scholarly journals Homological epimorphisms, homotopy epimorphisms and acyclic maps

2020 ◽  
Vol 32 (6) ◽  
pp. 1395-1406
Author(s):  
Joseph Chuang ◽  
Andrey Lazarev
Keyword(s):  
Wide Class ◽  
Topological Spaces ◽  
Loop Spaces ◽  
Graded Algebras ◽  
Differential Graded Algebras ◽  
Algebraic Description ◽  
Acyclic Map ◽  
Induced Maps

AbstractWe show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

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