The fundamental group of a noncommutative space

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 $(\mathbb Z+\theta \mathbb Z)^2$.


Introduction
The fundamental group is one of the first tools used in algebraic topology to collect information about the shape of a topological space or of a manifold. Given the simplicity of its definition, it is quite surprising that no analogue of this group has been found yet for noncommutative spaces, especially in the context of noncommutative differential geometry [5]. This is in contrast to other structures on topological spaces that have found their counterparts in terms of noncommutative (C * )-algebras, such as de Rham cohomology (in terms of cyclic cohomology), topological K-theory (as K-theory for C * -algebras), Riemannian metrics (as spectral triples [5,Ch. VI]) and measures (as von Neumann algebras [5, Ch.V]).
This paper aims for a definition of the fundamental group of a noncommutative space. Of course, in view of Gelfand duality one could try to dualise based loops in terms of *homomorphisms from a C * -algebra A to C([0, 1]). However, such a homomorphism ϕ would send any commutator ab − ba to 0, which for many noncommutative spaces already means that ϕ is trivial. Instead, in the spirit of [15,14] we adopt a Tannakian approach to our definition of a fundamental group. In fact, in the case of a differentiable manifold without boundary this becomes very concrete given the classical result that there is an equivalence between the category of representations of the fundamental group and the category of flat connections on vector bundles over that manifold (see for instance [19,Proposition I.2.5]). One may then reconstruct (the pro-algebraic completion of) the fundamental group of that manifold as the automorphism group of the corresponding Tannakian category (cf. [30] and [9]).
The noncommutative generalisation of the fundamental group that we propose here is based on the construction in Section 2 of a category of finitely generated projective bimodules over a differential graded algebra (dga) together with a connection. The difference with previous approaches to connections on bimodules (such as for instance [11]) is that we demand compatibility of the bimodule connection with all elements in the dga, which also solves the apparent incompatibility recorded in [18,Example 2.13].
Our main result (Section 3) is then that under some analytical assumptions on the dga the subcategory where the connections are flat is a Tannakian category. These assumptions are completely natural in the context of noncommutative differential geometry and are for instance fulfilled for quantum metric spaces in the sense of Rieffel [29]. It leads us to define the fundamental group π 1 of the dga as the group corresponding to this Tannakian category. We establish the appropriate functorial properties of the fundamental group, as well as homotopy and Morita invariance in Section 4. Note that an alternative way to arrive at a fundamental group for noncommutative spaces would be to develop a theory of noncommutative covering spaces and consider the corresponding automorphisms. This is for instance the approach taken in [3,16,32].
In Section 5 we then illustrate our construction by considering examples, including all noncommutative tori and, more generally, we consider toric noncommutative manifolds.
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.

Categories of connections
In this section we will define connections over a noncommutative space. These noncommutative spaces are described by differential graded algebras and, in fact, the results in this section apply to any (noncommutative) dga. We consider the category of all connections on finitely generated modules for a dga (ΩA, d), and we will prove that it is a rigid tensor category. Of course, the same then holds for the subcategory of flat connections. We also show that the category only depends on the (graded commutative) centre of the dga.
Some words on notation: in this paper we will write a dga as (ΩA, d). Even though this might be considered unnatural as dga's are usually written simply as (A, d), we use this notation to stress that it furnishes a differential geometric structure on the noncommutative space described by A (and a C * -algebra A). This becomes even more apparent in Section 3 where we impose some analytical conditions on A. For the same reason, we will write (ΩE, ∇) for a graded bimodule with a connection, which in the flat case is then of course nothing but a differential graded bimodule.
2.1. Connections and bimodules over a dga. There are multiple ways to generalise vector bundles to a noncommutative space. By Serre-Swan Theorem the vector bundles over a manifold correspond to finitely generated projective modules over the algebra C ∞ (M ) of smooth functions. Over a noncommutative algebra we can look at right modules, but there is no suitable tensor product of right modules. We can look at bimodules, but here we have to remember that over a commutative space, only very special bimodules are allowed: the multiplication on the left is the same as on the right. We can look at bimodules that satisfy this condition for the centre of the algebra. Instead we look at a more restrictive class of bimodules, namely those that are a direct summand of a free finite bimodule ( [12] defines diagonal bimodules as modules that are a summand of a free module). This is very natural when one considers bimodule connections. Namely, connections are usually defined as maps E → E ⊗ Ω 1 A that satisfy the appropriate Leibniz rules. However to satisfy the left Leibniz rule the image should actually be Ω 1 A ⊗ E. This is solved in [11] by using an isomorphism σ : E ⊗ Ω 1 A → Ω 1 A ⊗ E. However, then the tensor product of two flat connections is not necessarily flat. Instead if one considers graded bimodules ΩE = ⊕ k≥0 Ω k E over a graded algebra ΩA = ⊕ k≥0 Ω k A that are a summand of a free finite module, we automatically get isomorphisms Lemma 2.4). Moreover, now the tensor product of flat connections is always flat (see Lemma 2.18).
Let us now proceed and describe the precise algebraic setup.
Definition 2.1. A differential graded algebra or dga is a graded algebra ΩA together with a C-linear map d : ΩA → ΩA of degree +1 satisfying the Leibniz rule for ω, ν ∈ ΩA, and d 2 = 0.
The following lemma shows why it is convenient to work with these graded fgp ΩAbimodules instead of just fgp bimodules over A. Lemma 2.4. Let ΩE be a graded fgp ΩA-bimodule. Write Ω 0 A = A and Ω 0 E = E. Then for all k ≥ 0 the multiplication induces isomorphisms Proof. Let ΩF be another graded ΩA-bimodule satisfying ΩE ⊕ ΩF ∼ = ΩA n . Then we have the following commuting diagram: Here the top arrow is the direct sum of the maps E ⊗ A Ω k A → Ω k E and F ⊗ A Ω k A → Ω k F. All the other maps are isomorphisms, so these maps are isomorphisms as well. This shows that the first map in the lemma is an isomorphism and the second one follows analogously.
From this we see that if ΩA is graded commutative, then ΩE is completely determined by just E: it is the module ΩA ⊗ A E. Definition 2.5. Let ΩE be a graded fgp ΩA-bimodule. A connection on ΩE is a C-linear map ∇ : ΩE → ΩE of degree +1 satisfying the following equations for ε ∈ ΩE, α ∈ ΩA: Remark 2.6. The equations in this definition are called the right Leibniz rule and the left Leibniz rule, respectively. They are meaningful because the elements of ΩE can be multiplied with elements of ΩA both on the left and the right.
Remark 2.8. The curvature is an ΩA-bilinear map: for ε ∈ ΩE and α ∈ ΩA we have Definition 2.9. A connection ∇ is called flat if the curvature ∇ 2 is zero. Now we can define the category of flat connections. Definition 2.10. We define C(ΩA) to be the category whose objects are graded fgp ΩAbimodules with a connection, and whose morphisms are graded fgp ΩA-bimodule morphisms of degree 0 which commute with the connections. Let C flat (ΩA) be the full subcategory where the connections are required to be flat.
Example 2.11. Let M be a manifold without boundary. The corresponding dga is the de Rham differential algebra of the manifold ΩA = ΩM . Any fgp ΩA-bimodule ΩE is determined by the fgp A-bimodule E by ΩE = ΩA ⊗ A E. This corresponds to a vector bundle over M by the Serre-Swan Theorem. A flat connection on E corresponds to a (usual) flat connection on this vector bundle. So C(ΩM ) is equivalent to the category of vector bundles over M with a flat connection, which in turn is equivalent to the category of representations of π 1 (M ) by [19,Proposition I.2.5].

2.2.
Connections over the graded centre. We will now show that each graded fgp ΩAbimodule is determined by a graded fgp bimodule over a graded commutative subalgebra. For this we need the following definitions: Definition 2.12. Let ΩA be a dga. We define the graded commutative centre Z g (ΩA) as If ΩE is an fgp ΩA-bimodule we define Z g (ΩE) as Lemma 2.13. With notations as in the previous definition we have the following, part of which is shown in [11]: (i) The graded commutative centre of the algebra Z g (ΩA) is a dga.
So all graded fgp ΩA-bimodules are determined by a graded fgp Z g (ΩA)-bimodule. This is in turn determined by an fgp Z g (ΩA)-module. Note that Z g (A) may be smaller than the centre of the algebra A.
Since this holds for all α ∈ ΩA we conclude that ∇(ε) ∈ Z g (ΩE). So ∇ restricts to a function Z g (ΩE) → Z g (ΩE). Then (Z g (ΩE), ∇ |Zg(ΩE) ) is an object of C(Z g (ΩA)). It is easy to see that this is a functorial construction. Conversely, let (ΩF, ∇) be an object of C(Z g (ΩA)). Then we can define the graded fgp ΩA-bimodule ΩF ⊗ Zg(ΩA) ΩA, and the connection ∇ given by for ζ ∈ ΩF, α ∈ ΩA. This gives an object of C(ΩA). It is then easy to show that this construction is also functorial, and that the two functors thus defined are inverse to each other.
Finally, we observe that the functor and its inverse preserve flatness.
Example 2.16. We consider the dga corresponding to the noncommutative space of two points at finite distance (see [8] or [22, pp 116-118]). If is given by for all k ≥ 0. The differential is given by We have Z g (Ω 2k A) = C and Z g (Ω 2k+1 A) = 0 so that in view of Remark 2.15 the fgp bimodules over Z g (ΩA) are simply determined by a vector space over C. We conclude that C(ΩA) ∼ = C(Z g (ΩA)) is equivalent to the category of vector spaces.
2.3. Tensor products. In this subsection we will construct the tensor product for the category C(ΩA).
Proof. The left action of ΩA on ΩE and the right action of ΩA on ΩF make ΩG into a ΩAbimodule. A grading on the tensor product is given as follows: for any r ≥ 0 the degree r subspace Ω r G is the linear span of elements ε ⊗ ζ, with ε ∈ Ω k E, ζ ∈ Ω l F, k + l = r. To show that ΩG is fgp, suppose that ΩE ⊕ ΩE ′ ∼ = ΩA m and ΩF ⊕ ΩF ′ ∼ = ΩA n . Then So ΩG is a graded fgp ΩA-bimodule. We can then define the connection ∇ G by Equation (3) on pure tensors, and extend Clinearly. To show that it is well-defined, let α ∈ ΩA. Then by the above definition we have Since these are the same, ∇ G is well-defined.
Lastly, ∇ G satisfies the Leibniz rules: for ε ∈ ΩE, ζ ∈ ΩF, α ∈ ΩA we have and The curvature on the tensor product is easily calculated: Lemma 2.18. In the notation of the previous lemma, we have In particular, the tensor product of flat connections is again flat.
Proof. For ε ∈ ΩE, ζ ∈ ΩF we have Remark 2.19. The above lemma does not apply for some other definitions of connections on bimodules, see for instance [18,Example 2.13].
It is easy to see that this tensor product is associative. The tensor product commutes with the equivalence of categories C(ΩA) → C(Z g (ΩA)) from Theorem 2.14. In the commutative case it is easy to see that the tensor product is also commutative; so we have a commutativity constraint in the general case as well.
There is a unit in C(ΩA): it is the bimodule ΩA with the connection d. It is easy to see that the isomorphism ΩE → ΩE ⊗ ΩA ΩA intertwines the connection ∇ E with the tensor product connection on ΩE ⊗ ΩA ΩA.

Duals.
We will now construct dual objects in the category C(ΩA).

A Tannakian category and the fundamental group
In this section we will show that under some analytical conditions on the dga the category C flat (ΩA) constructed above is actually a neutral Tannakian category. In particular, if the algebra A = Ω 0 A is a dense * -subalgebra of a quantum metric space in the sense of Rieffel [29] these conditions are met, so that our results apply to a broad class of noncommutative differential spaces. We then define the fundamental group of the pertinent noncommutative space as the automorphism group of the fiber functor in this Tannakian category (we refer to [9] for more details).

3.1.
Abelianness of the category. In this subsection we will study when the category C(ΩA) is abelian. Using Theorem 2.14 we can always reduce to a graded commutative dga and this allows for the description of C(ΩA) simply as the category of fgp A-modules equipped with a connection (cf. Remark 2.15). We will assume that A is a unital * -algebra that is dense in a unital C * -algebra A. This will be necessary for some of the constructions below. Moreover we also need the star operation on ΩA.
We also assume that the elements in A that are invertible in A are also invertible in A, so A ∩ A × = A × . This is in particular the case if A is stable under holomorphic functional calculus (see [13, p.134]).
The category C(ΩA) is always an additive category: given two objects (ΩE, ∇ E ) and (ΩF, ∇ F ) the morphisms from ΩE to ΩF form an additive group, and there is an object ΩE ⊕ ΩF where the connection is simply given by ∇ E⊕F = ∇ E ⊕ ∇ F . In general, C(ΩA) is not an abelian category. For example, if Ω k A = 0 for all k ≥ 1, then C(ΩA) is simply the category of fgp modules over A, which is generally not an abelian category. In fact we can easily prove a necessary condition on a graded commutative dga ΩA if C(ΩA) is abelian.
We will call a differential graded commutative algebra ΩA connected if A is connected (i.e. contains no non-trivial projections).
Lemma 3.2. Let ΩA be a connected graded commutative dga and suppose that C(ΩA) is abelian. Let a ∈ A and suppose that da = aω for some ω ∈ Ω 1 A. Then a is either 0 or invertible.
Proof. Consider the two objects (A, d + ω) and (A, d) of C(ΩA). Since da = aω we have a commuting diagram where a denotes the multiplication by a. So multiplication by a is a morphism between these objects. We then get a short exact sequence 0 → ker(a) → A a − → im(a) → 0. This is a short exact sequence of A-modules, and since im(a) is an fgp A-module, it is split, and we get A ∼ = ker(a) ⊕ im(a). Since A is connected this means that either im(a) = 0, which means that a = 0, or im(a) = A, which means that a is invertible.
We will now define a slightly stronger condition on A, and we will show later that this is a sufficient condition for the category C(ΩA) to be abelian. Definition 3.3. Let ΩA be a * -dga with A ⊆ A, densely. We say that ΩA satisfies property Q if it satisfies the following condition: for all a ∈ A with a ≥ 0 and all a 1 , . . . , a s ∈ A with all |a i | ≤ a, and all ω 1 , . . . , ω s ∈ Ω 1 A: if da = s i=1 a i ω i , then either a = 0 or a is invertible. If ΩA is graded commutative, this easily implies the conclusion in Lemma 3.2: if da = aω, we have a * a ≥ 0 and d(a * a) = d(a) * a + a * da = aa * (ω + ω * ), so aa * is 0 or invertible, hence a is 0 or invertible. It also implies that A is connected, in the sense that there are no nontrivial projections: if p ∈ A is a projection, then dp = d(p 2 ) = 2pdp, so (1 − 2p)dp = 0 and multiplying by 1 − 2p gives dp = 0. Then p should be 0 or invertible, so any projection is 0 or 1.
3.1.1. Quantum metric differential graded algebras. We will now show that property Q holds for quantum metric differential graded algebras. First we recall the notion of a compact quantum metric space, introduced by Rieffel [29]. Let A be a C * -algebra and let L be a seminorm on A that takes finite values on a dense subalgebra A. We think of L as a Lipschitz norm. This defines a metric on the state space S(A) by Connes' distance formula [5, Ch.
This metric then defines a topology on the state space. We already had the weak- * topology, so it is natural to make the following definition: Definition 3.4. Let A be a unital C * -algebra and let L be a seminorm on A taking finite values on a dense subalgebra. Then A is called a compact quantum metric space if the topology on S(A) induced by the metric d L coincides with the weak- * topology. Now we go back to the case that we have a * -dga ΩA and A is a dense subset of a unital C *algebra A. Suppose that a norm · is given on Ω 1 A, satisfying the inequality aω ≤ a · ω for a ∈ A, ω ∈ Ω 1 A. This defines a seminorm L on A by L(a) = da . The space ΩA is called a quantum metric dga if A is a compact quantum metric space with this seminorm.
Remark 3.5. Note that any compact quantum metric space gives rise to a quantum metric dga. Indeed, it was realised in [29] that the Lipschitz norm can be obtained as L(a) = [D, a] for a suitable operator D on a Hilbert space. The space of Connes' differential forms [5,Ch. VI] is then a quantum metric dga.
Also note that if ΩA is a quantum metric dga, the same holds for Z g (ΩA), by [29, Lemma 3.6. Let ΩA be a graded commutative quantum metric dga and suppose that A∩A × = A × . Then ΩA satisfies property Q.
Proof. Let a ∈ A with a ≥ 0, and let a 1 , . . . , a s ∈ A with all |a i | ≤ a, and ω 1 , . . . , ω s ∈ Ω 1 A satisfying da = s i=1 a i ω i . By scaling we may assume that 0 ≤ a ≤ 1. Define the polynomial in particular the norm of dp n (a) is bounded as n → ∞.
If a is neither 0 nor invertible in A, there are points χ, ψ in the Gelfand spectrum of A satisfying χ(a) = 0 and ψ(a) = t > 0. Then But the metric d should give the weak- * topology on the spectrum, and the spectrum is connected, so this is a contradiction. So either a = 0 or a ∈ A × , and in the second case a ∈ A ∩ A × = A × .
3.1.2. Proof of abelianness. In the rest of this section, we will show that if a graded commutative dga ΩA satisfies property Q, then C(ΩA) is an abelian category. Suppose we have a morphism ϕ : E → F in the category C(ΩA). We have to show that ker(ϕ), im(ϕ), coker(ϕ) are also in the category. The most difficult part is to show that these are finitely generated projective modules.
In the case that ker(ϕ), im(ϕ), coker(ϕ) are finitely generated projective it is easy to construct connections on these modules. Lemma 3.9. Let ϕ : E → F be a morphism in C(ΩA) and suppose that ker(ϕ), im(ϕ) and coker(ϕ) are finitely generated projective. Then there are natural induced connections on ker(ϕ) and coker(ϕ). There are then also natural induced connections on coim(ϕ) = coker(ker(ϕ)) and im(ϕ) = ker(coker(ϕ)) and these are compatible with the natural isomorphism of A-bimodules coim(ϕ) Proof. We have the isomorphisms E ∼ = ker(ϕ) ⊕ im(ϕ) and F ∼ = im(ϕ) ⊕ coker(ϕ) as in the proof of Lemma 3.7. This makes the commuting diagram The lower horizontal map sends an element (a, b) to (b, 0). Tensoring this diagram with Ω 1 A gives Now we see that ker(ϕ) ⊗ Ω 1 A is the kernel of the lower horizontal map, so it is also the kernel of ϕ ⊗ Ω 1 A. We also know that coker(ϕ) ⊗ Ω 1 A is the cokernel of the map ϕ ⊗ Ω 1 A because tensoring with Ω 1 A is right exact. Now consider the diagram It is easy to check that these satisfy the Leibniz rules.
These are the connections we want. Now consider the diagram The connections ∇ coim(ϕ) and ∇ im(ϕ) are constructed as above as connections on the cokernel of the morphism ker(ϕ) → E and the kernel of the morphism F → coker(ϕ), respectively. We know that the left and the right square of the diagram commute and simple diagram chasing shows that the middle square commutes as well. So the isomorphism coim(ϕ) We need the inequality below involving D m . Its proof is an easy calculation after diagonalising M * M , and not very interesting. Its proof can be found in the appendix. The term 2 Re(M * [M, K]) will appear in the proof of Theorem 3.13.  We are now ready to prove that C(ΩA) is an abelian category. We need to show that any ϕ : E → F has a finitely generated projective kernel, image and cokernel. In the first part of the proof we reduce to the case ϕ : A n → A n . In the second part we prove that each term in the characteristic polynomial of ϕ * ϕ is either zero or invertible. Lastly we use this to prove that ϕ has a pseudo-inverse (as in Lemma 3.7). Theorem 3.13. Let ΩA be a graded commutative * -dga satisfying property Q. Then the category C(ΩA) is abelian.
Proof. Let ϕ : E → F be a morphism in C(ΩA). We will show that ker(ϕ), im(ϕ), coker(ϕ) are finitely generated projective A-modules, and then we are done by Lemma 3.9.
There is a projective module G with E ⊕F ⊕G ∼ = A n . We can write G ∼ = pA n for a projection p ∈ End A (A n ). Then we can define a connection ∇ G : pA n → p(Ω 1 A) n by ∇ G (g) = pdg. It is easy to check that this defines a connection on G (it is called the Grassmannian connection). This makes (G, ∇ G ) an object of C(ΩA) and it also defines a connection on the direct sum module E ⊕ F ⊕ G.
is a morphism in C(ΩA). Its kernel is ker(ϕ)⊕F ⊕G, its image is 0⊕im(ϕ)⊕0 and its cokernel is E ⊕coker(ϕ)⊕G. So it is enough to show that these are finitely generated projective. Therefore it is enough to prove: for a connection ∇ : A n → (Ω 1 A) n and a morphism ϕ : A n → A n that commutes with ∇, the kernel, image and cokernel of ϕ are fgp modules. The connection ∇ : A n → (Ω 1 A) n can be written as ∇ = d + κ, where κ : A n → (Ω 1 A) n is an A-linear function. We can view κ as an n × n matrix with coefficients in Ω 1 A. The induced connection on Hom A (A n , A n ), which we still call ∇, satisfies for f ∈ Hom A (A n , A n ) and e ∈ A n . So d( f, e ) + κ f, e = ∇(f ), e + f, de + f, κe and this gives ∇(f ) = df + [κ, f ]. Since ϕ : A n → A n commutes with the connection, we know that ∇(ϕ) = 0 so we conclude that dϕ = [ϕ, κ] where ϕ is viewed as an element of M n (A). We get Now let a m = D m (ϕ * ϕ) ∈ A be the m-th term of the characteristic polynomial of ϕ * ϕ (up to sign).

3.2.
Definition of the fundamental group. In this section we will define the fundamental group of a dga satisfying suitable analytical conditions. We will first complete the proof that C flat (ΩA) is a Tannakian category, after which the fundamental group is defined as the group of automorphisms of the fibre functor. Since, we have already proven that C flat (ΩA) is a rigid tensor category, and under some conditions on A, that it is abelian, what is left to show is that End(ΩA) = C and constructing a fiber functor ω : C flat (ΩA) → Vec, where Vec is the category of finite-dimensional vector spaces over C. Proof. Let θ : ΩA → ΩA be an isomorphism. Since θ is bilinear, for all α ∈ ΩA we have θ(α) = αθ(1). So θ is determined by a = θ(1) ∈ A. Since θ has to commute with the connection we get da = d(θ(1)) = θ(d(1)) = 0. Let λ be a complex number in the spectrum of a. Then we have d(a − λ) = 0, but a − λ is not invertible. Since ΩA satisfies property Q it follows that a = λ ∈ C.
For the fibre functor, pick a point p in the Gelfand spectrum A of a commutative A (that contains A densely). Then our fibre functor is given by sending a bimodule E to the localisation of its centre at p. This is defined as E ⊗ A C, where the A-module structure on C is given by p. Note that this depends on a choice of a point in the Gelfand spectrum. This point plays a similar role as the base point of the usual fundamental group. Lemma 3.17. Let ΩA be a graded commutative dga that satisfies property Q and let p ∈ A. There is a faithful exact fibre functor ω : C flat (ΩA) → Vec sending E to E p .
Proof. Let (E, ∇ E ) and (F, ∇ F ) be objects of C flat (ΩA) and let ϕ : E → F be a morphism commuting with the connections. Since ϕ is A-linear, this induces a map E p → F p , showing that ω is functorial.
To show that ω is faithful, suppose that ϕ p = 0. Since C flat (ΩA) is abelian we know that im(ϕ) is an fgp module. Now look at im(ϕ) ⊗ A A. This is an fgp module over the C * -algebra A, which corresponds to a vector bundle on A. It is zero at p, and the rank is locally constant, and A is connected, so im(ϕ) ⊗ A A = 0. Since im(ϕ) is projective it is flat, and im(ϕ) ֒→ im(ϕ) ⊗ A A is an injection, so also im(ϕ) = 0. We conclude that ϕ = 0.
The fibre functor is exact because a localisation is always exact.
Theorem 3.18. Let ΩA be a * -dga such that Z g (ΩA) satisfies property Q. Then the category C flat (ΩA) can be equipped with the structure of a neutral Tannakian category.
Proof. We already know that the category C flat (ΩA) is an abelian rigid tensor category. The reduction to the graded commutative case (Theorem 2.14) then allows us to apply the previous two Lemma's to complete the proof.
The fibre functor ω : C flat (ΩA) → Vec is then of course given as the composition of the functor defined in Theorem 2.14 and the fibre functor in Lemma 3.17.
We thus derive from eg. [9, Theorem 2.11] that the category C flat (ΩA) is equivalent to the category of representations of an algebraic group scheme, which allows us to make the following definition.
Definition 3.19. Let ΩA be a dga such that Z g (ΩA) satisfies property Q. Let p ∈ Z g (A). Then we define π 1 (ΩA, p) to be the group scheme of automorphisms of the fibre functor ω : C flat (ΩA) → Vec at p.
Note that in practice we will simply recognise the category C flat (ΩA) as being equivalent to the category of representations of some (topological) group whose pro-algebraic completion is the fundamental group.
Example 3.20. We have seen in Example 2.11 that for a connected manifold M without boundary we have C flat (ΩM ) ∼ = Rep(π 1 (M )). Hence π 1 (ΩM ) is the pro-algebraic completion of π 1 (M ). Let W ⊗ C ∞ [0, 1] be a module over C ∞ [0, 1] where W is a vector space, and let ∇ : Hence we are looking for an invertible element α ∈ End(W ) ⊗ C ∞ [0, 1] satisfying d(α) = αω. The solution of this equation is given by a path-ordered exponential, namely, we set α = ∞ n=0 α n where α 0 = 1 and where recursively α n+1 (t) = t 0 α n ω for n ≥ 0. It follows easily by induction that α n (t) ≤ t n ω n n! , so the series converges. Since dα n+1 = α n ω we get dα = αω. In a similar way we can construct α ′ satisfying dα ′ = −ωα ′ , and it is easy to see that this is the inverse of α.
, d) and we conclude that π 1 (Ω[0, 1]) = 0. The graded centre of this dga is just Z g (ΩA) = C ⊕ 0 ⊕ C ⊕ 0 ⊕ · · · where C is embedded diagonally in M 2 (C). Then C flat (Z g (ΩA)) is just equivalent to the category of vector spaces. The fundamental group is trivial. Example 3.24. Let ΩB be the dga from the previous example and let ΩA be any graded commutative dga satisfying property Q, and let p ∈ A. Consider the graded tensor product ΩA ′ = ΩA ⊗ ΩB. This is a dga, its modules are given by (Ω 0 A ′ ) = ΩA and Ω n A ′ = Ω n A ⊕ Ω n−1 A for n ≥ 1. Let E be an fgp A-bimodule. A connection over (ΩA ′ ) is given by a map ∇ : is a connection over ΩA and ∇ B : ΩE → ΩE is an A-linear map. The curvature of the connection So ∇ is flat if and only if ∇ A is flat and commutes with the endomorphism ∇ B . We get a series of equivalences C flat (ΩA ′ ) ≃{objects of C flat (ΩA) with an endomorphism} ≃{objects of Rep(π 1 (ΩA)) with an endomorphism} ≃ Rep(π 1 (ΩA) × R).
This leads to the following useful general result. Proposition 3.25. Let V be an n-dimensional vector space and consider the graded algebra V with differential d = 0. Then π 1 ( V ) is the algebraic hull of R n . More generally, if ΩA is any graded commutative dga which satisfies property Q and p ∈ A, then the fundamental group of ΩA ⊗ V at p is the algebraic hull of π 1 (ΩA, p) × R n .
Proof. The dga V is isomorphic to the n-fold tensor product of the dga in Example 3.23, so by Example 3.24 we see that π 1 ( V ) is the algebraic hull of R n . Similar reasoning leads to the second statement.

Some properties of the fundamental group
In this section we will establish some of the crucial properties that one would like a fundamental group to possess. This includes base point invariance, functoriality, homotopy invariance and Morita invariance. 4.1. Base point invariance. As one might expect, different base points give rise to isomorphic fundamental groups, at least provided that they are joined by a smooth path. In this case we will also simply write π 1 (Ω A), omitting the base point from the notation. Let Ω A be a * -dga such that Z g (Ω A) satisfies property Q. Let p, q ∈ Z g (A) be base points. Suppose we have a * -homomorphism γ : Z g (Ω A) → Ω[0, 1] satisfying ev 0 •γ = p, ev 1 •γ = q where ev t denotes evaluation at t. Then there exists an isomorphism π 1 (Ω A, p) ∼ = π 1 (Ω A, q).
Proof. We may assume without loss of generality that Ω A is graded commutative. Consider the following diagram: Here the horizontal map γ denotes, by abuse of notation, the functor that sends E to E ⊗ γ C ∞ [0, 1], and similar for the maps p and q. The functor v 0 sends E to E 0 and similar for v 1 . Let n : C flat (Ω[0, 1]) → Vec be the functor sending (E, ∇) to the vector space ker(∇). By Example 3.21, we have ker(∇) ⊗ C ∞ [0, 1] ∼ − → E. Therefore the map ker(∇) → E 0 given by localisation at 0 is an isomorphism. This gives a natural isomorphism between n and v 0 . Similarly we have a natural isomorphism from n to v 1 . So the functors v 0 and v 1 are naturally isomorphic, and it follows that the fibre functors from p and q are also naturally isomorphic. Then their group schemes of automorphisms are isomorphic as well, giving the isomorphism π 1 (Ω A, p) ∼ = π 1 (Ω A, q).
It is not known whether base point invariance holds without the additional assumption in Proposition 4.1 4.2. Functoriality of the fundamental group. For graded commutative spaces there is a good notion of functoriality for the fundamental group. In this section we will address the question for which maps between dga's there is an induced map between the fundamental groups.
We start by observing that the fundamental group π 1 (ΩA, p) is defined in terms of the dga ΩA and a character p on the center Z g (ΩA), interpreted as the base point. Now, if ϕ : ΩA → ΩB is a map of dga's one can only expect functoriality on the corresponding fundamental groups to have any meaning at all if base points are mapped to base points. In other words, characters of the center should be mapped to characters of the center. In other words, it is crucial to demand that the map ϕ maps the center to the center. Under such conditions, we can establish the following functorial property of the fundamental group of dga's. Proposition 4.2. Let ΩA and ΩB be * -dga's such that their graded centers satisfy property Q. Let ϕ : ΩA → ΩB be a degree 0 algebra morphism satisfying ϕ(dα) = d(ϕ(α)) for all α ∈ ΩA and that ϕ(Z g (ΩA)) ⊆ Z g (ΩB). Let q ∈ Z g (B) and p = ϕ * (q) ∈ Z g (A). Then ϕ induces a map π 1 ϕ : π 1 (ΩB, q) → π 1 (ΩA, p).
Proof. Since C(ΩA) is equivalent to C(Z g (ΩA)) (and the same for ΩB instead of ΩA) and ϕ induces a map of dga's from Z g (ΩA) → Z g (ΩB) we may assume without loss of generality that our dga's are graded commutative. Hence, if E is an fgp A-module then E ⊗ A B is an fgp Bmodule. A flat connection ∇ : E → E ⊗ Ω 1 A gives a flat connection ∇ : E ⊗ A B → E ⊗ A Ω 1 B, given by ∇(e ⊗ b) = ∇(e)b + e ⊗ db. So we get a map C(ΩA) → C(ΩB). It is easy to see that it is functorial and also that it commutes with the fibre functors. Then every automorphism of the fibre functor C(ΩB) → Vec can be pulled back to an automorphism of the fibre functor C(ΩA) → Vec. So we get a map π 1 ϕ : π 1 (ΩB, q) → π 1 (ΩA, p).
We can use this lemma to prove a version of homotopy invariance for π 1 . First we give the definition of homotopy for morphisms of dga's.

Definition 4.5.
Let Ω A and Ω B be * -dga's whose graded centres satisfy property Q and let p ∈ Z g (A), q ∈ Z g (B) be base points. Two maps ϕ 0 , ϕ 1 : (Ω B, q) → (Ω A, p) are called homotopic if there exists a map that satisfies the following conditions: • the map H sends graded centre to graded centre; • it satisfies ev t •H = ϕ t for t = 0, 1, where ev t : Ω[0, 1] → C denotes evaluation at t; Remark 4.6. Considering the diagram above in grade 0, we see that it means that H pulls back the point p × t ∈ Z g (A) × [0, 1] to the point q for all t ∈ [0, 1]. If Z g (Ω 1 B) is generated by elements of the form b 0 db 1 , this is an equivalent condition.
Proof. Let H : Ω B → Ω A ⊗ Ω[0, 1] be as in Definition 4.5. We may assume without loss of generality that Ω A and Ω B are graded commutative. Let E be an fgp module over A ⊗ C ∞ [0, 1] with a flat connection ∇. We can write ∇ = ∇ 1 ⊕∇ 2 : is a connection over Ω[0, 1] that is A-linear. By Lemma 4.4 we know that ker(∇ 2 ) is an fgp Ω A-module, and it is easy to check that ∇ 1|ker(∇ 2 ) is a connection on this module. So we can consider the functor given by n(E, ∇) = (ker(∇ 2 ), ∇ 1|ker(∇ 2 ) ). Consider also the functor Taking Ω A = C in the above we get similar maps n ′ , v ′ 0 : C flat (Ω[0, 1]) → Vec, and a natural isomorphism η ′ : n ′ → v ′ 0 . Now consider the diagram Here p ⊗ Ω[0, 1] denotes by abuse of notation the functor sending E to E ⊗ p ⊗ Ω[0,1] Ω[0, 1], and similar for p. It is easy to see that p Here we used similar abuse of notation as in the diagram above, and F is the functor sending a vector space W to (W ⊗ C ∞ [0, 1], d). This diagram commutes because H is a homotopy. So we have p * H * η = H * (p ⊗ Ω[0, 1]) * η ′ = q * F * η ′ . Now F * η ′ is a natural isomorphism between n ′ • F = id : Vec → Vec and v ′ 0 • F = id : Vec → Vec, and it is in fact the identity. So H * η is a natural isomorphism from n • H to ϕ 0 , and p * H * η : q → q is the identity. Of course we get a similar natural isomorphism for evaluation at 1 instead of evaluation at 0, and composing these we get a natural isomorphism µ : ϕ 0 → ϕ 1 satisfying p * µ = id : q → q. Then the maps ϕ 0 , ϕ 1 induce the same map π 1 ϕ 0 = π 1 ϕ 1 = π 1 (Ω A, p) → π 1 (Ω B, q).
It follows directly that π 1 is an invariant for homotopy equivalence: Let Ω A and Ω B be * -dga's whose graded centres satisfy property Q and let p ∈ Z g (A), q ∈ Z g (B) be base points. Let ϕ : (Ω A, p) → (Ω B, q) and ψ : (Ω B, q) → (Ω A, p) be morphisms such that ϕ•ψ and ψ•ϕ are homotopic to the identity on Ω B and Ω A respectively. Then π 1 (Ω A, p) is isomorphic to π 1 (Ω B, q).
Proof. It follows from the theorem that π 1 (ϕ) and π 1 (ψ) are inverse to each other.

4.4.
Invariance under Morita equivalence. We now address the question whether π 1 is invariant under Morita equivalence of the underlying dga's. Since we work with differential graded algebras as well as with C * -algebras, for both of which there exist notions of Morita equivalence, let us make more precise what we mean.
Let (R, d) be a differential graded ring. We denote by Mod dg R the category of all differential graded right R-modules (M, d) and with morphisms all graded module morphisms, not only those of degree 0. We write d instead of ∇ here to distinguish these differentials from the flat bimodule connections considered before. They satisfy the right Leibniz rule (cf. Equation (1)): d(mr) = d(m)r + (−1) |m| mdr; (m ∈ M, r ∈ R).
It then follows that the morphisms Hom((M, d), (N, d)) become differential graded modules (over the differential graded ring End((M, d))), with Thus, the category Mod dg R is a so-called differential graded category, or dg-category. Definition 4.9. Two differential graded rings R and S are called dg-Morita equivalent if the categories Mod dg R and Mod dg S are equivalent. It is a classical result in Morita theory that Morita equivalent rings have isomorphic centers (see for instance [21,Remark 18.43]). We prove an analogue for dg-Morita equivalent differential graded rings. Recall that the graded center of an additive category is given by all graded natural transformations η from the identity functor to itself (cf. [24,Section 4] and references therein), i.e. for all dg-modules M, N and f ∈ Hom((M, d), Proposition 4.10. The center of the category Mod dg R is a (graded commutative) differential graded ring which is isomorphic to the graded center Z g (R) of R.
Proof. Since η M ∈ Hom((M, d), (M, d)) we can define (dη) M = d(η M ) using Equation (5). This turns the graded center C(Mod dg R ) of Mod dg R into a dg ring. Let us then show that C(Mod dg R ) is isomorphic to Z g (R). As in [21,Remark 18.43] we define a map where η (r) is given by right multiplication by r, that is to say, η It is also clearly injective. To prove that ρ is surjective take any η ∈ C(Mod dg R ) and consider first η R : R → R. This map satisfies η R (s) = η R (1)s = rs where we have set r := η R (1) ∈ R. On the other hand, multiplication on the left on R by an element in R is a morphism in Mod dg R so that by graded naturality we also have η R (s) = (−1) |s||η| sη R (1) = (−1) |r||s| sr.
Hence r ∈ Z g (R) and η R = η (r) R . For an arbitrary module M in Mod dg R consider the following (graded) commuting diagram: where for m ∈ M we have defined f (r) = mr, a morphism of graded right R-modules of degree |f | = |m|. Then M for all dg-modules M and hence that ρ is surjective.
Remark 4.12. It is an interesting question to see whether π 1 is invariant under derived Morita equivalence as well. For instance, in [25,Prop. 9.2] or [20,Prop. 6.3.2] it is shown that the center of a ring is invariant under derived Morita equivalence. For the generalization to differential graded rings we refer to the notes [17,31], see also [24,Remark 5.6].

5.
Examples: toric noncommutative manifolds 5.1. Noncommutative tori. In this section we will consider the noncommutative torus, also called the rotation algebra. Let A θ be the rotation algebra, as studied by Rieffel [26] and Connes [4], and described in [13,Ch.12]. For any real number θ we define it to be the following * -algebra where u, v are unitaries that satisfy uv = λvu where λ = e 2πiθ . Note that the algebra A θ has a natural Z 2 -grading where u m v n has degree (m, n). In fact, this degree is related to the action α of a 2-dimensional torus T 2 by automorphisms on A θ given by α t (u m v n ) = e imt 1 +nt 2 u m v n . Let us now introduce the dga for the noncommutative torus, given by noncommutative differential forms ΩA θ . The elements of Ω 1 (A θ ) are of the form adu + bdv with a, b ∈ A θ and they satisfy udu = du · u, udv = λdv · u, vdu = λdu · v, vdv = dv · v and du · dv = −λdv · du. The elements of Ω 2 A θ are of the form adudv with a ∈ A θ (see [13,Sect. 12.2]). The action α extends to the dga as a graded automorphism by demanding that it commutes with the differential.
Note that an integer value of θ gives back the algebra C ∞ (T 2 ) of the usual torus. The noncommutative torus looks rather differently for θ irrational and θ rational. In both cases we will compute the graded centre of ΩA θ and from there the fundamental group.
Proposition 5.1. Let θ be irrational. Then the center of ΩA θ is trivial and the fundamental group of ΩA θ is isomorphic to (the algebraic hull of ) R 2 .
Proof. It is well-known that Z(A θ ) is trivial, see for instance [13,Corl. 12.12]). In fact, this result extends to the differential forms for which we have Z g (ΩA θ ) = (ΩA θ ) T 2 , the subalgebra of invariant vectors for the action α of T 2 . We see that From Proposition 3.25 we then conclude that the fundamental group of ΩA θ is isomorphic to (the algebraic hull of) R 2 .
Apparently the flat connections on fgp ΩA θ -bimodules correspond to continuous representations of R 2 . We can give the correspondence explicitly. Any continuous representation of R 2 on a vector space W is given by (t 1 , t 2 ) → exp(t 1 α + t 2 β) with α, β ∈ End(W ) commuting endomorphisms of the vector space. The corresponding module is ΩE = W ⊗ ΩA θ , and the connection is given by Note that all fgp A θ -bimodules are free by Lemma 2.13, as Z g (A θ ) = C.
Remark 5.2. This should be compared to [23], which considered the irrational rotation algebra with a holomorphic structure as a noncommutative elliptic curve, with resulting fundamental group equal to (the algebraic hull of) Z. Proposition 5.3. Let θ be rational. Then the fundamental group of ΩA θ coincides with that of the classical manifold T 2 , i.e. it is (the algebraic hull of ) Z 2 .
Proof. We write θ = p q with p, q coprime integers. Then the centre of A θ is given by power series in the commuting unitaries u q and v q with coefficients of radid decay, and thus isomorphic to the algebra C ∞ (T 2 ) (see also [13,Corl. 12.3]). Furthermore, we have . This is also generated by u q and v q and their derivations, as We see that Z g (ΩA θ ) ∼ = ΩT 2 and hence that the fundamental group of ΩA θ is the same as that of the classical manifold T 2 .
Remark 5.4. For any θ we have an inclusion map Z g (ΩA θ ) ֒→ ΩA 0 . This gives a map π 1 (ΩA 0 ) → π 1 (Z g (ΩA θ )) = π 1 (ΩA θ ). In the case that θ is irrational, this comes from the inclusion Z 2 → R 2 . In the case that θ = p q it comes from the multiplication Z 2 ·q − → Z 2 . With this map we can distinguish the rational rotation algebras for different values of q.
For any θ the flat connections over ΩA θ correspond to the continuous representations of the group (Z + θZ) 2 : for irrational θ, this is a dense subgroup of R 2 which has the same continuous representations as R 2 (see Lemma A.2 in the appendix), and for rational θ we have (Z + θZ) 2 ∼ = Z 2 . The map π 1 (ΩA 0 ) → π 1 (ΩA θ ) is then always given by the inclusion Z 2 ֒→ (Z + θZ) 2 .

5.2.
Higher-dimensional noncommutative tori. The irrational tori can be generalised to higher dimensions, as in [27]. Let Θ be a skew-symmetric n × n matrix with coefficients in R. We use the notation of [13,Sect. 12.2]. The algebra A Θ is generated by unitaries u 1 , . . . , u n satisfying u k u l = e 2πiΘ kl u l u k . We can also write this as u k u l = τ (e k , e l ) 2 u l u k where τ : (Z n ) 2 → C is the two-cocycle defined by τ (r, s) = exp(πir t Θs) and e k ∈ Z n denotes the k-th unit vector. A general term in A Θ is a power series expansion in the u k with coefficients in the Schwartz space S(Z n ). It is the noncommutative interpretation of the quotient of R n by Z n + ΘZ n , which is simply R n /Z n if Θ = 0. Note that we get the two-dimensional noncommutative torus back by taking n = 2 and the matrix Θ = 0 θ −θ 0 . For any r ∈ Z we define the Weyl element These are linearly independent and generate the algebra A Θ , and they satisfy u r u s = τ (r, s)u r+s .
The one-form module Ω 1 A Θ is free with generators {u −1 k du k }. We use these generators because they are in the centre of Ω 1 A Θ . The two-form module Ω 2 A Θ is free with generators {u −1 k du k · u −1 l du l , k < l}, et cetera. Proposition 5.5. The fundamental group π 1 (ΩA Θ ) of the n-dimensional noncommutative torus is the algebraic hull of Z n + ΘZ n .
Proof. Define the lattice (6) Λ = {r ∈ Z n | Θr ∈ Z n } (it is reciprocal to the lattice used in [13,Sect. 12.2]). Let r 1 , . . . , r m be a basis of Λ and let r m+1 , . . . , r n be elements of Z n such that r 1 , r 2 , . . . , r n are linearly independent. Then Z g (A Θ ) is generated by the m independent unitaries u r k , so it is isomorphic to C ∞ (T m ), the algebra of smooth functions on the (commutative) m-torus. Let V ⊆ Ω 1 A Θ be the m-dimensional vector space spanned by u −r k du r k , 1 ≤ k ≤ m. Then ΩT m is isomorphic to the sub-dga Z g (A Θ ) ⊗ V of ΩA Θ . Also, let W ⊆ Ω 1 ΩA Θ be the (n − m)-dimensional vector space spanned by u −r k du r k , m + 1 ≤ k ≤ n. Thus Since all basis elements of V and W are in the graded centre we get With this we can compute the fundamental group of ΩA Θ . The fundamental group of the torus T m is Z m . Then we see from Proposition 3.25 that the fundamental group of ΩA Θ is the algebraic hull of Z m × R n−m . The subgroup Z n + ΘZ n ⊆ R n is dense in {u ∈ R n | r t u ∈ Z for all r ∈ Λ}, and this is isomorphic to Z m × R n−m . So for any Θ the fundamental group of ΩA Θ equals the algebraic hull of Z n + ΘZ n .
Remark 5.6. If G is a (discrete) group that acts freely and properly on R n , then R n → R n /G is the universal cover of R n /G and it is a G-principal bundle. In the example above we show that the fundamental group corresponding to the noncommutative realisation of the quotient of R n by Z n + ΘZ n can be identified with Z n + ΘZ n , which is exactly as we would expect.
Of course, since we get the fundamental group from its representations we cannot actually distinguish between Z n + ΘZ n and Z m × R n−m as fundamental group of A Θ . 5.3. Toric noncommutative manifolds. It is possible to deform any manifold M that carries an action of a torus in a similar manner to the deformation of the tori described above. This is described in full detail in [7,6], and the differential graded algebra that we use is described in [6,Sect. 12].
Let M be a manifold and let σ : T n → Aut(M ) be a smooth effective action of the ndimensional torus. This defines an action of T n on ΩM , still denoted by σ. The deformation of C ∞ (M ) is conveniently described as the following fixed-point subalgebra: where ⊗ denotes the (projective) tensor product of Fréchet algebras and α is the action of T n on A Θ given by α t (u r ) = e irt u r . Similarly, a dga is defined by The differential in ΩM Θ is given by d ⊗ 1.
In fact, we may write any element ω ∈ ΩM as a series expansion in the Weyl elements u r ∈ ΩA Θ : ω = r∈Z n ω r ⊗ u r in terms of homogeneous elements ω r ∈ ΩM for the torus action, i.e. σ t (ω r ) = e irt ω r . This series expansion is convergent with respect to the Fréchet topology on ΩM (see [28,Ch. 2]). Similar to [1] we introduce the following subgroup of T n dual to the lattice Λ ⊂ Z n defined by equation (6): Γ = {t ∈ T n | t · r = 0 mod Z for all r ∈ Λ} Proposition 5.7. The graded center of ΩM Θ is given by Consequently, the fundamental group is π 1 (ΩM Θ ) = π 1 ((ΩM ) Γ ).
Proof. The graded center of ΩM Θ is given by elements of the form ω = r∈Λ ω r ⊗ u r as for r ∈ Λ we have τ (r, s) = 1 for any s ∈ Z n . We can use the subgroup Γ < T n to select these vectors by setting σ t (ω) = ω for all t ∈ Γ.
Example 5.8. Let M = T n and let the torus T n act on itself by addition. The algebra C(M Θ ) is then the same as the noncommutative torus ΩA Θ , and according to the above result we have π 1 (ΩM Θ ) = π 1 (Ω(T n ) Γ ). Let us confront this with Proposition 5.5. We may write where V, W are defined as in the proof of Proposition 5.5 (but with Θ = 0). But then Accordingly, we have π 1 (Ω(T n ) Γ ) = Z m × R n−m , as desired.

Conclusion and outlook
We have defined a notion of connections on finitely generated projective bimodules over a differential graded algebra ΩA. We have defined the category C flat (ΩA) of these bimodules with flat connections, and shown that it is equal to the category of flat connections over the graded centre Z g (ΩA). We have constructed a tensor product in this category and shown that it admits dual objects. The category is also abelian for a large class of noncommutative spaces. This was used to define an affine algebraic group scheme, which we called the fundamental group π 1 (ΩA) of the dga. After having established some crucial properties for π 1 , we computed the fundamental group for noncommutative tori, where we realised that it depends on the deformation parameter.
The structure we introduced suggests the following some natural, still open problems: a) Does the fundamental group respect products: is it true that π 1 (ΩA ⊗ ΩB, p ⊗ q) = π 1 (ΩA, p) × π 1 (ΩB, q)? b) Is the fundamental group π 1 a derived Morita invariant?
Appendix A. Representations of dense subgroups of 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 R.
We consider representations of R and of dense subgroups. Let V be a finite-dimensional vector space and α ∈ End(V ), and view R as additive topological group. There is a continuous representation π : R → GL(V ) given by π(t) = exp(tα). In fact every continuous representation has this form.
Lemma A.1. Let V be a finite-dimensional vector space and π : R → GL(V ) a continuous representation. Then there is a unique α ∈ End(V ) such that π(t) = exp(tα) for all t ∈ R.
This result also holds for continuous representations of dense subgroups of R. This is used in the computation of the fundamental group for noncommutative tori in section 4.
Lemma A.2. Let G ⊆ R be a dense subgroup of R. Let V be a finite-dimensional vector space. Each continuous representation π : G → GL(V ) extends to a representation of R, and is therefore of the form π(t) = exp(tα) 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 R → End(V ), and it follows easily that this is still a representation.