Abstract
We 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 \((\mathbb Z+\theta \mathbb Z)^{2}\).
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Acknowledgements
We would like to thank Ben Moonen and Steffen Sagave for inspiring discussions and valuable suggestions, as well as Alain Connes for several useful remarks. We also thank the anonymous referee for useful commnents.
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Presented by: Michel Brion
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Appendices
Appendix A: Representations of dense subgroups of \(\mathbb {R}\)
We have exploited Tannaka duality to reconstruct a group from the category of its representations, however, this only gives access to the pro-algebraic completion of the group. In other words, via this procedure many groups give rise to the same algebraic group. We will illustrate this phenomenon for dense subgroups of \(\mathbb {R}\).
We consider representations of \(\mathbb {R}\) and of dense subgroups. Let V be a finite-dimensional vector space and α ∈End(V ), and view \(\mathbb {R}\) as additive topological group. There is a continuous representation \(\pi :\mathbb {R}\to \text {GL}(V)\) given by \(\pi (t)=\exp (t\alpha )\). In fact every continuous representation has this form.
Lemma A.1
Let V be a finite-dimensional vector space and \(\pi :\mathbb {R}\to \text {GL}(V)\) a continuous representation. Then there is a unique α ∈End(V ) such that \(\pi (t)=\exp (t\alpha )\) for all \(t\in \mathbb {R}\).
Proof
Since π is continuous there is a δ > 0 such that for all \(t\in \mathbb {R}\) with |t| < δ we have ∥π(t) − 1∥ < 1. Let 0 < t < δ. The formal power series \(\log (x+1)\) has radius of convergence 1, so \(\alpha =\frac 1t\log (\pi (t))\) is well-defined. Then \(\pi (t)=\exp (t\alpha )\). Also \(\alpha ^{\prime }=\frac 2t\log (\pi (t/2))\) is well-defined. Since \(\log (x^{2})=2\log (x)\) whenever |x − 1| < 1 and |x2 − 1| < 1, we get \(\alpha ^{\prime }=\alpha \). So \(\pi (t/2)=\exp (t/2\alpha )\). By induction we get \(\pi (t2^{-n})=\exp (t2^{-n}\alpha )\) for positive integers n. For integers m it also follows that \(\pi (tm2^{-n})=\pi (t2^{-n})^{m}=\exp (tm2^{-n}\alpha )\). Since the set \(\{tm2^{-n}\mid m,n\in \mathbb Z\}\) is already dense in \(\mathbb {R}\) it follows that \(\pi (x)=\exp (x\alpha )\) for all \(x\in \mathbb {R}\). □
This result also holds for continuous representations of dense subgroups of \(\mathbb {R}\). This is used in the computation of the fundamental group for noncommutative tori in section 4.
Lemma A.2
Let \(G\subseteq \mathbb {R}\) be a dense subgroup of \(\mathbb {R}\). Let V be a finite-dimensional vector space. Each continuous representation π : G →GL(V ) extends to a representation of \(\mathbb {R}\), and is therefore of the form \(\pi (t)=\exp (t\alpha )\) for some α ∈End(V ).
Proof
Since the representation is continuous there is a δ > 0 such that for t ∈ [−δ,δ] ∩ G we have ∥π(t) − 1∥≤ 1, so ∥π(t)∥≤ 2. For all positive integers n we have π(nt) = π(t)n, and it follows that π is bounded on bounded intervals.
Now we show from this that π is uniformly continuous on bounded intervals. Let 𝜖 > 0. There is δ > 0 such that for all t ∈ G ∩ (−δ,δ) we have ∥π(t) − 1∥ < 𝜖. Let M > 0 and let x,y ∈ G ∩ [−M,M] with |x − y| < δ. Then ∥π(x) − π(y)∥≤∥π(x)∥⋅∥1 − π(y − x)∥≤∥π(x)∥⋅ 𝜖. Since π is bounded on G ∩ [−M,M] we see that π is uniformly continuous on bounded intervals. Then it can be extended uniquely to a function \(\mathbb {R}\to \text {End}(V)\), and it follows easily that this is still a representation. □
Appendix B: Proof of Lemma 3.10
We have put here the proof of Lemma 3.10 that is a bit technical and would otherwise have disrupted the flow of the argument.
Lemma B.1
For \(M,K\in M_{n}(\mathbb C)\) we have
Here 2Re(M∗[M,K]) = M∗[M,K] + (M∗[M,K])∗ and ∥K∥HS denotes the Hilbert-Schmidt norm of K.
Proof
The inequality is invariant under a unitary change of basis of M, and M∗M is self-adjoint so we may choose a basis in which M∗M is diagonal, with eigenvalues \(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\in \mathbb C\). Now
where S runs over the m-element subsets of {1, 2,…,n}. Let
The t-coefficient in the polynomial \(P(t)=D_{m}(M(t))\in \mathbb C[t]\) is \(\frac d{dt}_{|t=0}D_{m}(M(t))\). The matrix M(t) only has multiples of t outside the diagonal. The determinant of an m × m submatrix is then modulo t2 equal to the product of the values on its diagonal. So
We have
The t-coefficient in Pm(t) is then
For all i,j,l we have \((M^{*})_{ij}M_{li}\leq \frac 12(|M_{ji}|^{2}+|M_{li}|^{2})\leq {\sum }_{k=1}^{n}|M_{ki}|^{2}=\lambda _{i}\). So we get
Now we conclude
□
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van Suijlekom, W.D., Winkel, J. The Fundamental Group of a Noncommutative Space. Algebr Represent Theor 25, 1003–1035 (2022). https://doi.org/10.1007/s10468-021-10057-7
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DOI: https://doi.org/10.1007/s10468-021-10057-7