An Atiyah Sequence for Noncommutative Principal Bundles

We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way, we treat the problem of deciding when a given *-automorphism on the quantum base space lifts to a *-automorphism on the quantum total space that commutes with the underlying structure group.

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2022
Main Authors:	Schwieger, Kay, Wagner, Stefan
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2022
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/211530
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Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	An Atiyah Sequence for Noncommutative Principal Bundles. Kay Schwieger and Stefan Wagner. SIGMA 18 (2022), 015, 22 pages

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Schwieger, Kay Wagner, Stefan
author_facet	Schwieger, Kay Wagner, Stefan
citation_txt	An Atiyah Sequence for Noncommutative Principal Bundles. Kay Schwieger and Stefan Wagner. SIGMA 18 (2022), 015, 22 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way, we treat the problem of deciding when a given -automorphism on the quantum base space lifts to a -automorphism on the quantum total space that commutes with the underlying structure group.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 015, 22 pages An Atiyah Sequence for Noncommutative Principal Bundles Kay SCHWIEGER a and Stefan WAGNER b a) iteratec GmbH, Zettachring 6, 70567 Stuttgart, Germany E-mail: kay.schwieger@gmail.com URL: https://www.xing.com/profile/Kay_Schwieger/cv b) Blekinge Tekniska Högskola, SE-371 79 Karlskrona, Sweden E-mail: stefan.wagner@bth.se URL: https://www.bth.se/eng/staff/stefan-wagner-stw/ Received July 26, 2021, in final form February 21, 2022; Published online March 07, 2022 https://doi.org/10.3842/SIGMA.2022.015 Abstract. We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way we treat the problem of deciding when a given ∗-automorphism on the quantum base space lifts to a ∗-automorphism on the quantum total space that commutes with the underlying structure group. Key words: Atiyah sequence; noncommutative principal bundle; freeness; factor system 2020 Mathematics Subject Classification: 46L87; 46L85; 55R10 1 Introduction Chern–Weil theory is an important tool for many disciplines such as analysis, geometry, and mathematical physics. For instance, it provides invariants of principal bundles and vector bun- dles by means of connections and curvature, and thus a way to measure their non-triviality. The Chern–Weil homomorphism of a smooth principal bundle q : P → M with structure group G is an algebra homomorphism from the algebra of polynomials that are invariant under the ad- joint action of G on its Lie algebra, into the even de Rham cohomology H2• dR(M,K). This map is achieved by evaluating an invariant polynomial on the curvature of a connection 1-form ω on P . The latter procedure involves lifting vector fields on M horizontally with respect to ω to G-equivariant vector fields on P . An important remark in this context is that connection 1-forms on P are in a bijective correspondence with C∞(M)-linear sections of the associated Lie algebra extension 0 −→ gau(P ) −→ V(P )G −→ V(M) −→ 0, (1.1) which is well-known as the so-called Atiyah sequence of the principal bundle P (see, e.g., [4, 24]). About three decades after the seminal works of Chern, Weil, and Atiyah, Lecomte described in [26] a cohomological construction which generalizes the classical Chern–Weil homomorphism: Lecomte’s construction associates characteristic classes to each Lie algebra extension, and the classical construction of Chern and Weil arises in this context from the Atiyah sequence above. The work presented here is an attempt towards a derivation-based Chern–Weil theory for noncommutative principal bundles. More precisely, our main objective is to provide a derivation- based generalization of the classical Atiyah sequence in equation (1.1) to the setting of noncom- mutative principal bundles. For this purpose, we focus on free C∗-dynamical systems, which provide a natural framework for noncommutative principal bundles (see, e.g., [6, 19, 31, 36] and references therein). Their structure theory and their relation to K-theory (see, e.g., [15, mailto:kay.schwieger@gmail.com https://www.xing.com/profile/Kay_Schwieger/cv mailto:stefan.wagner@bth.se https://www.bth.se/eng/staff/stefan-wagner-stw/ https://doi.org/10.3842/SIGMA.2022.015 2 K. Schwieger and S. Wagner 20, 34, 35, 36] and references therein) certainly appeal to operator algebraists and functional analysts. Additionally, noncommutative principal bundles are becoming increasingly prevalent in various applications of geometry (cf. [22, 23, 28, 37, 38]) and mathematical physics (see, e.g., [7, 10, 13, 14, 18, 21, 25, 41] and references therein). For the sake of completeness, we bring to mind that the algebraic setting of Hopf-Galois ex- tensions already comprises abstract notions of connections, curvature and characteristic classes in terms of a universal differential calculus (see, e.g., [8, 9, 12] and references therein). Fur- thermore, we recall that Neeb [29] associated Lie group extensions with projective modules that generalize the classical Atiyah sequence for vector bundles. Finally, we wish to mention the recent theory of pseudo-Riemannian calculus introduced by Arnlind and Wilson in [3], which constitutes a derivation-based computational framework for Riemannian geometry over noncommutative algebras (see also [1, 2]). It is our hope that this work will advance to the development of pseudo-Riemannian calculus. Organization of the article Let (A, G, α) be a free C∗-dynamical system with fixed point algebra B. After some preliminaries, we review in Section 3 the notion of a factor system of (A, G, α), which is the key feature of our investigation. In fact, factor systems provide us with a natural framework for doing computations and constitute invariants for (A, G, α). In Section 4 we utilize factor systems to give a characterization of when a ∗-automorphism on B can be lifted to a ∗-automorphism on A that commutes with α (Theorem 4.1). Moreover, we use our findings to examine an “integrated” version of the Atiyah sequence for noncommutative principal bundles (equation (4.2)). Section 5 is devoted to the study of the special case when G is compact Abelian. Most notably, we get a characterization in terms of second group cohomology on the dual group of G with values in the unitary group of the center of B (Theorem 5.5). In addition, we are able to show that if A is commutative, then every ∗-automorphism on B lifts to A provided it leaves the class of A invariant (Corollary 5.7). In Section 6 we make use of the results of Section 4 to establish a lifting result for 1-parameter groups (Theorem 6.4). This is of particular interest in Section 7, where we finally present a generalization of the classical Atiyah sequence in equation (1.1) to the setting of free C∗-dynamical systems. Last but not least, we discuss infinitesimal objects such as connections and curvature (Section 7.3). Finally, we would like to mention that with little effort the arguments and the results in Sections 3, 4, and 6 extend to free actions of quantum groups (see [36]). 2 Preliminaries and notation This preliminary section exhibits the most fundamental definitions and notations used in this article. About groups. Let G be a compact group. We write Irr(G) for the set of equivalence classes of irreducible representations of G and denote by 1 ∈ Irr(G) the class of the trivial representation. About Hilbert spaces. Let G be a compact group. Furthermore, let Hσ, σ ∈ Irr(G), be a family of Hilbert spaces, and for each σ ∈ Irr(G) let Tσ be an operators on Hσ. Throughout this article, we shall freely utilize the fact that σ 7→ Hσ and σ 7→ Tσ can be extended to arbitrary finite-dimensional representations ofG by taking direct sums with respect to irreducible subrepresentations. About C∗-dynamical systems. Let A be a unital C∗-algebra and let G be a compact group that acts on A by ∗-automorphisms αg : A → A, g ∈ G, such that G ×A → A, (g, x) 7→ αg(x) is continuous. Throughout this article, we call such data a C∗-dynamical system and denote it An Atiyah Sequence for Noncommutative Principal Bundles 3 briefly by (A, G, α). Moreover, we typically write B := AG for the corresponding fixed point algebra. The conditional expectation P1 onto B allows us to define a definite right B-valued inner product on A by ⟨x, y⟩B := P1(x ∗y) = ∫ G αg(x ∗y) dg, x, y ∈ A. The completion of A with respect to the induced norm yields a right Hilbert B-module, which we denote by L2(A). The algebra A admits a faithful ∗-representation on L2(A) by adjointable operators given by λ : A → L ( L2(A) ) , λ(x)y := x · y, and consequently we may identify A with λ(A) ⊆ L ( L2(A) ) . Furthermore, for each g ∈ G we have a unitary operator Ug on L2(A) defined for x ∈ A ⊆ L2(A) by Ugx := αg(x). The map G ∋ g 7→ Ug ∈ U ( L2(A) ) is strongly continuous and implements the ∗-automorphisms αg, g ∈ G, via λ(αg(x)) = Ugλ(x)U ∗ g for all x ∈ A. Like every representation of G, the algebra A can be decomposed into its isotypic components, let us say, A(σ), σ ∈ Irr(G), which amounts to saying that their algebraic sum Af := alg⊕ σ∈Irr(G) A(σ) is a dense ∗-subalgebra of A. Furthermore, the isotypic components are pairwise orthogonal, right Hilbert B-submodules of L2(A) such that L2(A) = ⊕ π∈ĜA(π). About freeness. A C∗-dynamical system (A, G, α) is called free if the Ellwood map Φ: A⊗alg A → C(G,A), Φ(x⊗ y)(g) := xαg(y) has dense range with respect to the canonical C∗-norm on C(G,A). This condition was originally introduced for actions of quantum groups on C∗-algebras by Ellwood [19] and is known to be equivalent to Rieffel’s saturatedness [32] and the Peter–Weyl–Galois condition [6]. One of the key tools used in this article is a characterization of freeness that we provided in [36, Lemma 3.3], namely that a C∗-dynamical system (A, G, α) is free if and only if for each irreducible representation (σ, Vσ) of G there is a finite-dimensional Hilbert space Hσ and an isometry s(σ) ∈ A ⊗ L(Vσ,Hσ) satisfying αg(s(σ)) = s(σ) · σg for all g ∈ G. A rich class of free actions is given by so-called cleft actions, which are characterized as follows: For each irreducible representation (σ, Vσ) of G there is a unitary element u(σ) ∈ A⊗ L(Vσ) such that αg(u(σ)) = u(σ) · σg for all g ∈ G (cf. [35, Definition 4.1]). About 1-parameter groups. Let A be a unital C∗-algebra and let (φt)t∈R be a 1-parameter group of ∗-automorphisms φt ∈ Aut(A). We typically use the letter A∞ to denote the smooth domain of (φt)t∈R, which is the set of elements x ∈ A such that R ∋ t 7→ φt(x) ∈ A is smooth. Moreover, we let Dφt : A∞ → A∞, Dφt(x) := lim t→0 φt(x)− x t stand for the corresponding ∗-derivation. About derivations. Let A be a unital ∗-algebra. We let der(A) stand for the Lie algebra of ∗-derivations of A. Furthermore, we write Askew ⊆ A for the subset of skew-adjoint elements, i.e., Askew := {a ∈ A : a∗ = −a}, and recall that each a ∈ Askew gives rise to a ∗-derivation defined by A ∋ x 7→ [a, x] = ax− xa ∈ A. 4 K. Schwieger and S. Wagner 3 Factor systems Let (A, G, α) be a free C∗-dynamical system. Furthermore, for each σ ∈ Irr(G) let Hσ be a finite- dimensional Hilbert space and let s(σ) ∈ A⊗L(Vσ,Hσ) an isometry satisfying αg ( s(σ) ) = s(σ)·σg for all g ∈ G (cf. [36, Lemma 3.3]). For 1 ∈ Irr(G) we choose H1 := C and let s(1) := 1A. A key feature of (A, G, α) is the factor system associated with the isometries s(σ), σ ∈ Irr(G), (see [36, Definition 4.1]), which we now recall for the convenience of the reader. First, we put B := AG. Second, for expediency, we naturally extend σ 7→ Hσ and σ 7→ s(σ) to arbitrary finite-dimensional representations σ of G by taking the direct sum with respect to irreducible subrepresentations. For each finite-dimensional representation σ of G we may then define the ∗-homomorphism γσ : B → B ⊗ L(Hσ), γσ(b) := s(σ)(b⊗ 1Vσ)s(σ) ∗, and for each pair (σ, π) of finite-dimensional representations of G the element ω(σ, π) := s(σ)s(π)s(σ ⊗ π)∗ ∈ B ⊗ L(Hσ⊗π,Hσ ⊗ Hπ). Then the following relations hold: ω(σ, π)∗ω(σ, π) = γσ⊗π(1B), ω(σ, π)ω(σ, π)∗ = γσ(γπ(1B)), (3.1) ω(σ, π)γσ⊗π(b) = γσ(γπ(b))ω(σ, π), (3.2) ω(σ, π)ω(σ ⊗ π, ρ) = γσ(ω(π, ρ))ω(σ, π ⊗ ρ) (3.3) for all finite-dimensional representations σ, π, ρ of G and b ∈ B (see [37, Lemma 4.3]). The triple (H, γ, ω) of the above families is referred to as the factor system of (A, G, α) associated with s(σ), σ ∈ Irr(G), or simply as a factor system of (A, G, α) when no explicit reference to the isometries is needed. We recall from [36] that, for σ ∈ Irr(G), the isotypic component A(σ) of A can be written as A(σ) = { Tr(ys(σ)) : y ∈ B ⊗ L(Hσ, Vσ) } . In fact, the map y 7→ Tr(ys(σ)) is bijective from B⊗L(Hσ, Vσ)γσ(1B) to A(σ). The action α on the isotypic component takes the form αg ( Tr(ys(σ)) ) = Tr ( σg · ys(σ) ) (3.4) for all g ∈ G and y ∈ B ⊗ L(Hσ, Vσ). The multiplication between isotypic components can be written as Tr ( yσs(σ) ) · Tr ( yπs(π) ) = Tr ( yσγσ(yπ)ω(σ, π)s(σ ⊗ π) ) (3.5) for all σ, π ∈ Irr(G), yσ ∈ B ⊗ L(Hσ, Vσ), and yπ ∈ B ⊗ L(Hπ, Vπ). The element on the right hand side is, in fact, a sum of elements in the isotypic components corresponding to subrepresentations of σ⊗π. Hence equation (3.5) uniquely determines the multiplication on the dense ∗-subalgebra Af of A. The involution can be phrased in terms of the factor system, too (see [36, Section 5] or [30, Lemma 2.4]). In order to see this, let us fix σ ∈ Irr(G), choose an isometric intertwiner v0 : C → Vσ̄ ⊗ Vσ, and write w0 : C → Hσ̄⊗σ for the associated isometry. Then the element pσ := (dimσ)2w0v ∗ 0 does not depend on the choice of the intertwiner. It can be shown that there are element ȳ1, . . . , ȳn ∈ B⊗L(Hσ̄, Vσ̄) and y1, . . . , yn ∈ B⊗L(Hσ, Vσ) such that pσ = n∑ k=1 ȳkγσ̄(yk)ω(σ̄, σ) (3.6) An Atiyah Sequence for Noncommutative Principal Bundles 5 and that, for each y ∈ B ⊗ L(Hσ, Vσ), the element J(y) := 1 dimσ n∑ k=1 ȳkγσ̄ ( Tr(yky ∗) ) ∈ B ⊗ L(Hσ̄, Vσ̄) does not depend on the choice of ȳ1, . . . , ȳn and y1, . . . , yn. The involution on Af then reads as Tr ( ys(σ) )∗ = Tr ( J(y)s(σ)∗ ) for all y ∈ B ⊗ L(Hσ, Vσ). Noteworthily, the notion of a factor system of (A, G, α) only depends on the fixed point algebra B and the group G, which leads to the following definition: Definition 3.1. Let B be a unital C∗-algebra and let G be a compact group. 1. A factor system for (B, G) is a triple (H, γ, ω) comprising families of Hilbert spaces Hσ, σ ∈ Irr(G), ∗-homomorphisms γσ : B → B ⊗ L(Hσ), σ ∈ Irr(G), and elements ω(σ, π) in B ⊗ L(Hσ⊗π,Hσ ⊗ Hπ), σ, π ∈ Irr(G), satisfying equations (3.1), (3.2), (3.3), as well as the normalization conditions H1 = C, γ1 = idB, and ω(1, σ) = γσ(1B) = ω(σ, 1) for all σ ∈ Irr(G). 2. Two factor systems (H, γ, ω) and (H′, γ′, ω′) for (B, G) are called conjugated if there are partial isometries v(σ) ∈ B ⊗ L(Hσ,H ′ σ), σ ∈ Irr(G), normalized to v(1) = 1B, such that Ad[v(σ)] ◦ γσ = γ′σ, Ad[v(σ)∗] ◦ γ′σ = γσ, v(σ)γσ ( v(π) ) ω(σ, π) = ω′(σ, π)v(σ ⊗ π) for all σ, π ∈ Irr(G). In this case we write (H′, γ′, ω′) = v(H, γ, ω)v∗ or simply (H, γ, ω) ∼ (H′, γ′, ω′) when no reference to the partial isometries is needed. Note that, above, we have implicitly used functorial versions of the families Hσ, γσ, v(σ), and ω(σ, π), σ, π ∈ Irr(G). By construction, each factor system of (A, G, α) is a factor system for (B, G) and, by [36, Lemma 4.3], all factor systems of (A, G, α) are conjugated. In fact, given any unital C∗-algebra B and any compact group G, we have shown in [36, Section 5] that the equivalence classes of free C∗-dynamical systems (A, G, α) with fixed point algebra B are in 1-to-1 correspondence with conjugacy classes of factor systems of (B, G). Remark 3.2. For a factor system (H, γ, ω) of (A, G, α) equations (3.1) and (3.2) suggest to look at theK-theory of B and the induce positive group homomorphismsK0(γσ) : K0(B) → K0(B) for a finite-dimensional representation σ of G. Indeed, these maps only depend on the conjugacy class of the factor system and thus amount to invariants for (A, G, α). In fact, the mapping σ 7→ K0(γσ) constitutes a nice functor: For direct sums σ ⊕ π of representation σ, π of G, we obviously have K0(γσ⊕π) = K0(γσ) +K0(γπ). Moreover, by equation (3.2), we have K0(γσ⊗π) = K0(γσ) ◦K0(γπ) for all finite-dimensional representations σ, π of G. 6 K. Schwieger and S. Wagner 4 Lifting an automorphism Let (A, G, α) be a free C∗-dynamical system with fixed point algebra B and let β be a ∗-automor- phism of B. In this section we address the question whether β can be lifted to a ∗-automorphism β̂ of A that commutes with all αg, g ∈ G. In the affirmative case we say that β lifts to A and that β̂ is a lift of β. To phrase our result we note that Aut(B) acts on the factor systems for (B, G). For β ∈ Aut(B) and a factor system (H, γ, ω) for (B, G) we may define a new factor system ( H, γβ, ωβ ) for (B, G) by putting γβσ := β ◦ γσ ◦ β−1, ωβ(σ, π) := β ( ω(σ, π) ) for all σ, π ∈ Irr(G). With this we give an answer to the above lifting problem by proving the following theorem: Theorem 4.1. Let (A, G, α) be a free C∗-dynamical system with fixed point algebra B and let β be a ∗-automorphism of B. Then the following statements are equivalent: (a) β lifts to A. (b) For any factor system (H, γ, ω) of (A, G, α) we have (H, γ, ω) ∼ ( H, γβ, ωβ ) . Remark 4.2. By the above theorem, a necessary condition for β to lift to A is that K0 ( γβσ ) = K0(γσ) for all σ ∈ Irr(G) or, equivalently, that K0(γσ) commutes with K0(β) for all σ ∈ Irr(G) (cf. Remark 3.2). In particular, K0(β) is needs to fix the characteristic classes [γσ(1B)] ∈ K0(B), σ ∈ Irr(G). We split the proof of Theorem 4.1 into a sequence of lemmas. For a start we fix, for each σ ∈ Irr(G), a finite-dimensional Hilbert space Hσ and an isometry s(σ) ∈ A⊗L(Vσ,Hσ) satisfying αg(s(σ)) = s(σ)·σg for all g ∈ G; for 1 ∈ Irr(G) we take H1 := C and s(1) := 1A. As in Section 3, we write (H, γ, ω) for the associated factor system. Our first result establishes the implication “(a)⇒(b)” of Theorem 4.1: Lemma 4.3. If β lifts to A, then (H, γ, ω) and ( H, γβ, ωβ ) are conjugated. Proof. Let β̂ be a lift of β. For each σ ∈ Irr(G) we put sβ(σ) := β̂(s(σ)) and note that sβ(σ) is an isometry in A⊗L(Vσ,Hσ) satisfying αg ( sβ(σ) ) = sβ(σ) ·σg for all g ∈ G. Clearly, sβ(1) = 1A. Furthermore, it is readily checked that the associated factor system is equal to ( H, γβ, ωβ ) . The claim thus follows from the fact that all factor systems of (A, G, α) are conjugated. ■ The task is now to prove the converse implication, “(b)⇒(a)”, of Theorem 4.1. For this purpose, we consider partial isometries v(σ) ∈ B ⊗ L(Hσ), σ ∈ Irr(G), normalized to v(1) = 1B, such that ( H, γβ, ωβ ) = v(H, γ, ω)v∗. Then for each σ ∈ Irr(G) we obtain a well-defined map on the isotypic component A(σ) by putting β̂σ ( Tr(ys(σ)) ) := Tr ( β(y)v(σ)s(σ) ) (4.1) for all y ∈ B⊗L(Hσ, Vσ). Taking direct sums gives a map β̂ on the dense ∗-subalgebra Af . Due to the normalizations v(1) = 1B and s(1) = 1A, we have β̂(b) = β(b) for all b ∈ B. That is, β̂ extends β. Furthermore, a few moments of thought show that β̂ is bijective. In fact, its inverse is given by the direct sum of the maps β̂−1 : A(σ) → A(σ), σ ∈ Irr(G), defined by β̂−1 σ ( Tr(ys(σ)) ) = Tr ( β−1(yv(σ)∗)s(σ) ) for all y ∈ B ⊗ L(Hσ, Vσ). We proceed to establish further properties. An Atiyah Sequence for Noncommutative Principal Bundles 7 Lemma 4.4. The following assertions hold for the map β̂ on Af : 1. β̂ ◦ αg = αg ◦ β̂ for all g ∈ G. 2. β̂ is multiplicative. 3. β̂ is involutive. Proof. 1. This is immediate from equations (3.4) and (4.1). 2. Let σ, π ∈ Irr(G), let xσ = Tr(yσs(σ)) ∈ A(σ) for some yσ ∈ B ⊗ L(Hσ, Vσ), and let xπ = Tr(yπs(π)) ∈ A(π) for some yπ ∈ B ⊗ L(Hπ, Vπ). Then β̂(xσ) · β̂(xπ) = β̂ ( Tr(yσs(σ)) ) · β̂ ( Tr(yπs(π)) ) = Tr ( β(yσ)v(σ)s(σ) ) · id⊗Tr ( β(yπ)v(π)s(π) ) = Tr ( β(yσ)v(σ)γσ(β(yπ)v(π))ω(σ, π)s(σ ⊗ π) ) . Using the conjugacy equations in Definition 3.1 now yields β̂(xσ) · β̂(xπ) = Tr ( β(yσ)γ β σ (β(yπ))v(σ)γσ(v(π))ω(σ, π)s(σ ⊗ π) ) = Tr ( β(yσγσ(yπ))ω β(σ, π)v(σ ⊗ π)s(σ ⊗ π) ) = Tr ( β(yσγσ(yπ)ω(σ, π))v(σ ⊗ π)s(σ ⊗ π) ) = β̂(xσ · xπ), which establishes that β̂ is multiplicative. 3. Let σ ∈ Irr(G). To deal with the involution, we choose ȳ1, . . . ȳn ∈ B ⊗ L(Hσ̄, Vσ̄) and y1, . . . , yn ∈ B ⊗ L(Hσ, Vσ) satisfying equation (3.6) (cf. Section 3). Now, let x = Tr(ys(σ)) be in A(σ) for some y ∈ B ⊗ L(Hσ, Vσ). Then β̂(x∗) = β̂ ( Tr(J(y)s(σ̄)) ) = 1 dimσ n∑ k=1 β̂ ( Tr(ȳkγσ̄(Tr(yky ∗))s(σ̄)) ) = 1 dimσ n∑ k=1 Tr ( β(ȳkγσ̄(Tr(yky ∗)))v(σ̄)s(σ̄) ) . It is easily checked that ȳβk := β(ȳk)v(σ̄) and yβk := β(yk)v(σ), 1 ≤ k ≤ n, also satisfy equa- tion (3.6). Since there is no loss of generality in assuming y = yγσ(1B), we thus get β̂(x)∗ = Tr(β(y)v(σ)s(σ))∗ = Tr ( J(β(y)v(σ))s(σ̄) ) = 1 dimσ n∑ k=1 Tr ( β(ȳk)v(σ̄)γσ̄(Tr(β(yk)v(σ)v(σ) ∗β(y∗)))s(σ̄) ) = 1 dimσ n∑ k=1 Tr ( β(ȳk)v(σ̄)γσ̄(β ⊗ Tr(yky ∗))s(σ̄) ) . Invoking the conjugacy equations in Definition 3.1 finally gives β̂(x)∗ = 1 dimσ n∑ k=1 Tr ( β(ȳk)γ β σ̄ (β ⊗ Tr(yky ∗))v(σ̄)s(σ̄) ) = 1 dimσ n∑ k=1 Tr ( β(ȳk)γσ̄(Tr(yky ∗))v(σ̄)s(σ̄) ) = β̂(x∗), which shows that β̂ is involutive. ■ 8 K. Schwieger and S. Wagner Lemma 4.5. We have ⟨β̂(x1), β̂(x2)⟩B = β ( ⟨x1, x2⟩B ) for all x1, x2 ∈ Af . Proof. Since isotypic components are pairwise orthogonal, it suffices to show ⟨β̂(x1), β̂(x2)⟩B = β ( ⟨x1, x2⟩B ) for all x1, x2 ∈ A(σ) and σ ∈ Irr(G). To this end, let σ ∈ Irr(G) and let x1, x2 ∈ A(σ). By Lemma 4.4, we obtain ⟨β̂(x1), β̂(x2)⟩B = P1 ( β̂(x1) ∗β̂(x2) ) = P1 ( β̂(x∗1x2) ) . We now decompose x∗1x2 as ∑ i∈I Tr(yis(σi)) for some mutually distinct representations σi ∈ Irr(G) and yi ∈ B ⊗ L(Hσi , Vσi) and note that there is i0 ∈ I such that σi0 = 1, which is due to the fact that σ ⊗ σ̄ contains 1 as a subrepresentation. Hence P1 ( β̂(x∗1x2) ) = P1 ( β(yi0) ) = β ( P1(yi0) ) = β ( P1(x ∗ 1x2) ) = β ( ⟨x1, x2⟩B ) . ■ By Lemma 4.5, the bijectivity of β̂, and the fact that Af is dense in L2(A), we may extend β̂ to a unitary map, let’s say, U on L2(A). Consequently, there is a ∗-automorphism on A, for which we shall use the same letter β̂ by a slight abuse of notation, such that λ ( β̂(x) ) = Uλ(x)U∗ for all x ∈ A. It is easily checked that β̂ extends β and commutes with all αg, g ∈ G. Summa- rizing, we have shown the implication “(b)⇒(a)” of Theorem 4.1, which concludes the proof of this theorem. We now turn to an application of our findings: The group AutG(A) := { φ ∈ Aut(A) : αg ◦ φ = φ ◦ αg ∀g ∈ G } admits a short exact sequence 1 −→ Gau(A) −→ AutG(A) −→ Aut(B)[A] −→ 1, (4.2) where Gau(A) := { φ ∈ AutG(A) : φ\|B = idB } is the group of gauge transformations of (A, G, α) and Aut(B)[A] ⊆ Aut(B) is the group of ∗-automorphisms that lift to A. Theorem 4.1 states that Aut(B)[A] can be characterized in terms of a factor system (H, γ, ω) of (A, G, α) as Aut(B)[A] = { β ∈ Aut(B) : (H, γ, ω) ∼ ( H, γβ, ωβ )} . Looking at equation (4.1), we easily see that different choices of v(σ), σ ∈ Irr(G), amount to different lifts. Hence the construction, in fact, shows that Gau(A) is isomorphic to the group U(H, γ, ω) := { u = (u(σ))σ∈Irr(G) : u(H, γ, ω)u ∗ = (H, γ, ω) } , which consists of all families of unitaries u(σ) ∈ γσ(1B) ( B ⊗ L(Hσ) ) γσ(1B), σ ∈ Irr(G), that lie in the commutant of γσ(B) and satisfy the equation u(σ)γσ(u(π))ω(σ, π) = ω(σ, π)u(σ ⊗ π) for all σ, π ∈ Irr(G). An Atiyah Sequence for Noncommutative Principal Bundles 9 5 The special case of a compact Abelian structure group Let G be a compact Abelian group, let (A, G, α) be a free C∗-dynamical system with fixed point algebra B, and let β be a ∗-automorphism of B. In the previous section we showed in Theorem 4.1 that β lifts to A if and only if (H, γ, ω) ∼ ( H, γβ, ωβ ) for any factor system (H, γ, ω) of (A, G, α). In this section we give another characterization of when β lifts to A in terms of second group cohomology on the dual group Ĝ := Hom(G,T) with values in the group UZ(B) of unitaries in the center of B. To begin with, let us fix, for each σ ∈ Ĝ, a finite-dimensional Hilbert space Hσ and an isome- try s(σ) ∈ A⊗L(C,Hσ) satisfying αg(s(σ)) = σ(g) · s(σ) for all g ∈ G. For the trivial character, denoted by 0, we choose H0 := C and s(0) := 1A. Just as before, we write (H, γ, ω) for the asso- ciated factor system. We shall also make use of the so-called Fröhlich map ∆: Ĝ → Aut(UZ(B)) associated with (A, G, α), which is for each σ ∈ Ĝ defined by restricting the map ∆σ : B → B, ∆σ(b) := s(σ)∗bs(σ). We point out that, since all factor systems of (A, G, α) are conjugated, the Fröhlich map does not depend on the choice of the factor system. Given a lift β̂ of β, we have seen in the proof of Lemma 4.3 that v(σ) := β̂ ( s(σ) ) s(σ)∗, σ ∈ Ĝ, are partial isometries in B ⊗ L(Hσ), σ ∈ Ĝ, normalized to v(0) = 1B, such that γβσ = Ad[v(σ)] ◦ γσ, γσ = Ad[v(σ)∗] ◦ γβσ , (5.1) v(σ)γσ(v(π))ω(σ, π) = ωβ(σ, π)v(σ + π) (5.2) for all σ, π ∈ Ĝ. A moment’s thought shows that equation (5.2) can be rewritten as s(σ + π)∗β−1 ( v(σ + π)ω(σ, π)∗γσ ( v(π)∗ ) v(σ)∗ ) s(σ)s(π) = 1B. Our objective is now to give a group cohomological interpretation of the latter equation. For this purpose, let us for a moment assume that for all σ ∈ Ĝ the ∗-homomorphisms γσ and γβσ are conjugated, i.e., there is a partial isometry v(σ) ∈ B ⊗ L(Hσ) satisfying equation (5.1). We freely use the fact that there is no loss of generality in assuming that γσ(1B) and γβσ (1B) are the initial and final projections of v(σ), respectively. Let us now consider the map u : Ĝ × Ĝ → B defined by u(σ, π) := s(σ + π)∗β−1 ( v(σ + π)ω(π, σ)∗γπ ( v(σ)∗ ) v(π)∗ ) s(π)s(σ). (5.3) Lemma 5.1. The following assertions hold: 1. u(σ, π) is central in B for all σ, π ∈ Ĝ. 2. u(σ, π) is unitary for all σ, π ∈ Ĝ. 3. u constitutes a 2-cocycle, i.e., for all σ, π, ρ ∈ Ĝ it satisfies u(σ + π, ρ)u(σ, π) = u(σ, π + ρ)∆σ(u(π, ρ)). Remark 5.2. We recall from [34, Corollary 3.11] that, for each σ ∈ Ĝ, the isotypic compo- nent A(σ) is a Morita equivalence bimodule over B and that, for all σ, π ∈ Ĝ, the canonical multiplication map mσ,π : A(σ) ⊗B A(π) → A(σ + π) is an isomorphisms of Morita equiv- alence bimodules over B. All assertions in Lemma 5.1 may be derived from the facts that mσ+π,ρ ◦ (mσ,π ⊗ id) = mσ,π+ρ ◦ (id⊗mπ,ρ) for all σ, π, ρ ∈ Ĝ and that the maps Φ(σ, π) : A(σ + π) → A(σ + π), Φ(σ, π) := β̂−1 σ+π ◦mσ,π ◦ ( β̂σ ⊗ β̂π ) ◦m∗ σ,π 10 K. Schwieger and S. Wagner for all σ, π ∈ Ĝ, with β̂σ, σ ∈ Ĝ, being defined by equation (4.1), are automorphisms of Morita equivalence bimodules over B. Instead, we decided to present basic direct proofs despite rather long computations. Proof. 1. Let σ, π ∈ Ĝ. Then for each b ∈ B we have bu(σ, π) = bs(σ + π)∗β−1 ( v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ) = s(σ + π)∗γσ+π(b)β −1 ( v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ) (5.1) = s(σ + π)∗β−1 ( v(σ + π)γσ+π(β(b))ω(π, σ) ∗γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ) (3.2) = s(σ + π)∗β−1(v(σ + π)ω(π, σ)∗γπ(γσ(β(b))v(σ) ∗)v(π)∗)s(π)s(σ) (5.1) = s(σ + π)∗β−1 ( v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ ) γπ(γσ(b))s(π)s(σ) = s(σ + π)∗β−1 ( v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ)b = u(σ, π)b. That is, u(σ, π) is central in B as claimed. 2. Let σ, π ∈ Ĝ. Then u(σ, π)∗u(σ, π) is equal to s(σ)∗s(π)∗β−1 ( v(π)γπ(v(σ))ω(π, σ)v(σ + π)∗ ) × γσ+π(1B) · β−1 ( v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ) = s(σ)∗s(π)∗β−1 ( v(π)γπ(v(σ))ω(π, σ)ω(π, σ) ∗γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ) (3.1) = s(σ)∗s(π)∗β−1 ( v(π)γπ(v(σ))γπ(γσ(1B))γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ) (5.1) = s(σ)∗s(π)∗γσ(γπ(1B))s(σ)s(π) = 1B. In other words, u(σ, π) is an isometry, and hence unitary due to part 1 above. 3. Let σ, π, ρ ∈ Ĝ. To prove that u(σ + π, ρ)u(σ, π) = u(σ, π + ρ)∆σ(u(π, ρ)), we examine both sides of the equation. Indeed, the left hand side reads as s(σ + π + ρ)∗β−1 ( v(σ + π + ρ)ω(ρ, σ + π)∗γρ(v(σ + π)∗)v(ρ)∗ ) s(ρ) × γσ+π(1B) · β−1 ( v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ) = s(σ + π + ρ)∗β−1 ( v(σ + π + ρ)ω(ρ, σ + π)∗γρ(v(σ + π)∗)v(ρ)∗ ) s(ρ) × β−1 ( v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ ) s(π)s(σ) = s(σ + π + ρ)∗β−1 ( v(σ + π + ρ)ω(ρ, σ + π)∗γρ(v(σ + π)∗)v(ρ)∗ ) × ( γρ ◦ β−1 )( v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ ) s(ρ)s(π)s(σ) (5.1) = s(σ + π + ρ)∗β−1 ( v(σ + π + ρ)ω(ρ, σ + π)∗γρ(ω(π, σ) ∗) × γρ(γπ(v(σ) ∗))γρ(v(π) ∗)v(ρ)∗ ) s(ρ)s(π)s(σ). Moreover, a similar computation shows that the right hand side is given by s(σ + π + ρ)∗β−1 ( v(σ + π + ρ)ω(π + ρ, σ)∗ω(ρ, π)∗ × γρ(γπ(v(σ) ∗))γρ(v(π) ∗)v(ρ)∗ ) s(ρ)s(π)s(σ). Comparing these two expressions, we see that the 2-cocycle equation will be established once we verify that ω(ρ, σ + π)∗γρ ( ω(π, σ)∗ ) = ω(π + ρ, σ)∗ω(ρ, π)∗. But this is immediate from equation (3.3). ■ Lemma 5.3. The cohomology class of u is independent of all choices made. More precisely, the following holds: Let (H′, γ′, ω′) be another factor system of (A, G, α) with generators s′(σ), An Atiyah Sequence for Noncommutative Principal Bundles 11 σ ∈ Ĝ. Furthermore, let v′(σ) ∈ B ⊗ L(H′ σ), σ ∈ Ĝ, be partial isometries, normalized to v′(0) = 1B, satisfying equation (5.1) for γ′σ. Write u′ : Ĝ × Ĝ → UZ(B) for the corresponding 2-cocycle defined by equation (5.3). Then u and u′ are cohomologous. Proof. As a preliminary step, let us denote by w(σ) := s(σ)s′(σ)∗, σ ∈ Ĝ, the family in B⊗L(H′ σ,Hσ) implementing the conjugation between (H′, γ′, ω′) and (H, γ, ω). We define a map u : Ĝ → B by u(σ) := s(σ)∗β−1 ( v(σ)w(σ)v′(σ)∗ ) s′(σ). For each σ ∈ Ĝ the element u(σ) is, in fact, central in B, because for each b ∈ B we have u(σ)b = s(σ)∗β−1 ( v(σ)w(σ)v′(σ)∗βγ′σ(b) ) s′(σ) (5.1) = s(σ)∗β−1 ( βγ(b)v(σ)w(σ)v′(σ)∗ ) s′(σ) = bu(σ). Furthermore, it is straightforwardly checked that β−1(w(σ)∗v(σ∗))s(σ)u(σ) = β−1(v′(σ)∗)s′(σ) (5.4) for all σ ∈ Ĝ. For each (σ, π) ∈ Ĝ× Ĝ we may thus compute u′(σ, π) = s′(σ + π)∗β−1 ( v′(σ + π)ω′(π, σ)∗γπ(v ′(σ)∗)v′(π)∗ ) s′(π)s′(σ) (5.1) = s′(σ + π)∗β−1 ( v′(σ + π)ω′(π, σ)∗v′(π)∗ ) γπβ −1(v′(σ)∗)s′(π)s′(σ) = s′(σ + π)∗β−1 ( v′(σ + π)ω′(π, σ)∗v′(π)∗ ) s′(π)β−1(v′(σ)∗)s′(σ) (5.4) = u(σ + π)∗s(σ + π)∗β−1 ( v(σ + π)w(σ + π)∗ω′(π, σ)∗w(π)∗v(π)∗ ) s(π)u(π) × β−1(w(σ)∗v(σ)∗)s(σ)u(σ). Since u(σ) and u(π) are central in B, we may collect the terms u(σ + π), u(σ), and u(π). Traversing the arguments for the first equalities further yields u′(σ, π) = ∆σ(u(π))u(σ)u(σ + π)∗ · s(σ + π)∗β−1 × ( v(σ + π)w(σ + π)ω′(π, σ)∗γπ(w(σ)) ∗w(π)∗ · γπ(v(σ)∗)v(π)∗ ) s(π)s(σ) = ∆σ(u(π))u(σ)u(σ + π)∗ · u(σ, π). This shows that u and u′ are cohomologous as asserted. ■ Lemma 5.4. Suppose that u is a 2-coboundary for some 1-cochain u : Ĝ → UZ(B), i.e., u(σ, π) = ∆σ(u(π))u(σ)u(σ + π)∗ (5.5) for all σ, π ∈ Ĝ. Then( H, γβ, ωβ ) = v′(H, γ, ω)(v′)∗ for the partial isometries v′(σ) := βγσ(u(σ))v(σ), σ ∈ Ĝ. Proof. Using that each u(σ), σ ∈ Ĝ, is unitary in B, we see at once that v′(σ), σ ∈ Ĝ, are partial isometries in B⊗L(Hσ), σ ∈ Ĝ. Since u(0) = 1B, it is also clear that v′(0) = 1B. We proceed to prove that ( H, γβ, ωβ ) = v′(H, γ, ω)(v′)∗. For this, we first observe that equation (5.3) can be rewritten as v(σ + π)ω(π, σ)∗γπ(v(σ) ∗)v(π)∗ = β(s(σ + π)u(σ, π)s(σ)∗s(π)∗) 12 K. Schwieger and S. Wagner (5.5) = β ( s(σ + π)u(σ + π)∗u(σ)∆σ(u(π))s(σ) ∗s(π)∗ ) = (ω′)β(π, σ)∗ for all σ, π ∈ Ĝ. Combining this with the fact that the partial isometries v(σ), σ ∈ Ĝ, sat- isfy equation (5.1) gives v(H, γ, ω)v∗ = ( H, γβ, (ω′)β ) . Next, for each σ ∈ Ĝ we put s′(σ) := s(σ)u(σ)∗ = γσ ( u(σ)∗ ) s(σ) and note that s′(σ) is an isometry in A ⊗ L(Vσ,Hσ) satisfying αg ( s′(σ) ) = s′(σ) · σg for all g ∈ G. Evidently, s′(0) = 1A. We write (H, γ′, ω′) for the associated factor system. As each u(σ), σ ∈ Ĝ, is central in B we see that γ′ = γ. More- over, by construction, γ(u)(H, γ, ω′)γ(u∗) = (H, γ, ω) (cf. [36, Lemma 4.3]). It follows that βγ(u) ( H, γβ, (ω′)β ) βγ(u∗) = ( H, γβ, ωβ ) . We may thus conclude that v′(H, γ, ω)(v′)∗ = βγ(u)v(H, γ, ω)v∗βγ(u∗) = ( H, γβ, ωβ ) . ■ In summary, we have thus proved: Theorem 5.5. Let G be a compact Abelian group, let (A, G, α) be a free C∗-dynamical system with fixed point algebra B, and let β be a ∗-automorphism of B. Suppose that γ and γβ and are conjugated and let u denote the corresponding 2-cocycle. Then β lifts to A if and only if the cohomology class of u in H2 ( Ĝ,UZ(B) ) ∆ vanishes. Proof. The assertion that the cohomology class of u in H2 ( Ĝ,UZ(B) ) ∆ vanishes if β lifts to A follows from the introductory considerations in this section. The converse follows from Lemma 5.4 when combined with Theorem 4.1. ■ Corollary 5.6. Let (A,T, α) be a free C∗-dynamical system with fixed point algebra B. Identify T̂ with Z and write 1 ∈ Z for its positive generator. Furthermore, let β be a ∗-automorphism of B such that γ1 and γβ1 are conjugated for some factor system (H, γ, ω) of (A,T, α). Then β lifts to A. Proof. The hypothesis does not depend on the choice of the factor system. For this reason, as a first step, we may choose the following system. Let s ∈ A ⊗ L(C,H1) be an isometry satisfying αz(s) = s(1A ⊗ z) for all z ∈ T. For each n > 1 this element yields an isometry s(n) ∈ A ⊗ L(C,H1 ⊗ · · · ⊗ H1) satisfying αz(s(n)) = s(n)(1A ⊗ zn) for all z ∈ T by putting s(n) := s1,2s1,3 · · · s1,n+1. Here and subsequently, the subindices refer to the leg numbering in the respective tensor product. It is easily checked that the coactions associated with the family s(n), n ≥ 0, satisfy γm+n = (γm ⊗ id) ◦ γn for all m,n ≥ 0. Moreover, for each n > 1 we put v(n) := v1,2s1,2(· · · (v1,ns1,n(v1,n+1)s ∗ 1,n) · · · )s∗1,2 and note that v(n) ∈ B ⊗ L(H1 ⊗ · · · ⊗ H1). A straightforward induction now proves that v(n) satisfies equation (5.1) for all n > 1. Since similar considerations apply to the partial adjoint s(−1) := (s∗)T ∈ A⊗L(C,H1), we ultimately obtain partial isometries v(n), n ∈ Z, normalized to v(0) := 1B, satisfying equation (5.1). Hence the assertion follows from combining Theorem 5.5 with the fact that H2(Z,UZ(B))∆ vanishes (cf. [27, Chapter VI.6]). ■ In the remainder of this section we additionally assume that A is commutative. In this case, the Fröhlich map ∆ is trivial. Furthermore, for each σ ∈ Ĝ we have γσ(b) = bγσ(1B) for all b ∈ B, and for each (σ, π) ∈ Ĝ × Ĝ it may be derived that ω(π, σ) = flip(ω(σ, π)), where flip stands for the tensor flip of L(Hσ) ⊗ L(Hπ). In consequence, the 2-cocycle u : Ĝ × Ĝ → UZ(B) from Lemma 5.1 above is untwisted and symmetric, and hence a 2-coboundary by [5, Lemma 3.6]. Corollary 5.7. Suppose that A is commutative. Let G be a compact Abelian group, let (A, G, α) be a free C∗-dynamical system with fixed point algebra B, and let β be a ∗-automorphism of B. Then β lifts to A if and only if βγσ(1B) is Murray–von Neumann equivalent to γσ(1B) in B ⊗ L(Hσ) for all σ ∈ Ĝ. An Atiyah Sequence for Noncommutative Principal Bundles 13 Remark 5.8. 1. Turning to K0(B), we see that the “if” condition in Corollary 5.7 is equivalent to the condition that K0(β) fixes the elements [γσ(1B)] ∈ K0(B), σ ∈ Ĝ, which form a subgroup of K0(B) by Remark 3.2. 2. Let A = C(P ) and B = C(X) for some compact spaces P and X, respectively, and con- sider P as a topological principal G-bundle over X. Let h : X → X be the homeomorphism such that β(f) = f ◦ h for all f ∈ C(X). Then the “if” condition in Corollary 5.7 states that, for each σ ∈ Ĝ, the vector bundles determined by γσ(1B) and γσ(1B)◦h, respectively, are equivalent. Proof. The “only if” direction is clear from Theorem 4.1. For the converse direction let, for each σ ∈ Ĝ, v(σ) ∈ B ⊗ L(Hσ) be a partial isometry such that v(σ)∗v(σ) = γσ(1B) and v(σ)v(σ)∗ = βγσ(1B). By Theorem 5.5 and the discussion prior to the corollary, it remains to prove that equation (5.1) holds for all σ ∈ Ĝ. Thus, let σ ∈ Ĝ. Then for each b ∈ B we have γβσ (b) = β ( β−1(b)γσ(1B) ) = bβ(γσ(1B)) = bv(σ)v(σ)∗ = v(σ)bv(σ)∗ = v(σ) · bγσ(1B) · v(σ)∗ = v(σ) · γσ(b) · v(σ)∗ = (Ad[v(σ)] ◦ γσ)(b). By conjugating with v(σ)∗, we also get γσ = Ad[v(σ)∗] ◦ γβσ . ■ 6 Lifting 1-parameter groups Let (βt)t∈R be a smooth 1-parameter group of ∗-automorphisms βt ∈ Aut(B) and let δ := Dβt denote the corresponding ∗-derivation on its smooth domain B∞ (cf. Section 2). In this section we investigate whether there is a smooth 1-parameter group ( β̂t ) t∈R of ∗-automorphisms β̂t ∈ Aut(A) such that, for each t ∈ R, β̂t is a lift of βt. In the affirmative case we say that (βt)t∈R lifts smoothly to A and that (β̂t)t∈R is a smooth lift of (βt)t∈R. Example 6.1. 1. We recall from the classical theory of smooth principal bundles that every smooth 1-pa- rameter group on the base manifold lifts (smoothly) to the total space of the principal bundle. For a compact Abelian group G this follows from Corollary 5.7, because, for each σ ∈ Ĝ, the projections γσ(1B) and βγσ(1B) are obviously homotopic, and hence Murray– von Neumann equivalent. 2. The example in [5, Section 4] shows that not all 1-parameter groups lift. Let us fix, for each σ ∈ Irr(G), a finite-dimensional Hilbert space Hσ and an isometry s(σ) ∈ A⊗ L(Vσ,Hσ) satisfying αg ( s(σ) ) = s(σ) · σg for all g ∈ G; for 1 ∈ Irr(G) we take H1 := C and s(1) := 1A. Throughout the following, we make the standing assumption that the associated factor system (H, γ, ω) is smooth in the sense that γσ(B∞) ⊆ B∞ ⊗ L(Hσ) and ω(σ, π) ∈ B∞ ⊗ L(Hσ⊗π,Hσ ⊗ Hπ) for all σ, π ∈ Irr(G). Lemma 6.2. Let ( β̂t ) t∈R be a smooth lift of (βt)t∈R and let δ̂ := Dβ̂t denote the corresponding ∗-derivation on its smooth domain A∞. Suppose that, for each σ ∈ Irr(G), the isometry s(σ) is smooth for ( β̂t ) t∈R, i.e., s(σ) ∈ A∞ ∈ L(Vσ,Hσ). Then the element H(σ) := δ̂(s(σ))s(σ)∗. 14 K. Schwieger and S. Wagner lies in B∞ ⊗ L(Hσ) and for all σ, π ∈ Irr(G) we have δγσ(b) = γσδ(b) +H(σ)γσ(b) + γσ(b)H(σ)∗ ∀b ∈ B∞, (6.1) δ(ω(σ, π)) = H(σ)ω(σ, π) + γσ(H(π))ω(σ, π) + ω(σ, π)H(σ ⊗ π)∗. (6.2) Proof. Let σ ∈ Irr(G). We first note that A∞ is a ∗-subalgebra of A satisfying δ̂(A∞) ⊆ A∞ and (A∞)G = B∞, the latter being due to the fact that ( β̂t ) t∈R is a lift of (βt)t∈R. Therefore the assumption on s(σ) implies that H(σ) = δ̂(s(σ))s(σ)∗ ∈ B∞⊗L(Hσ). Furthermore, for each t ∈ R and b ∈ B it follows from the lifting property that βt(γσ(b)) = βt(s(σ)bs(σ) ∗) = β̂t(s(σ))βt(b)β̂t(s(σ)) ∗. Taking the derivative at t = 0, for each b ∈ B∞ we get equation (6.1): δγσ(b) = δ̂(s(σ))bs(σ)∗ + s(σ)δ(b)s(σ)∗ + s(σ)bδ̂(s(σ))∗ = H(σ)γσ(b) + γσδ(b) + γσ(b)H(σ)∗. Now, let σ, π ∈ Irr(G). Just as above, for each t ∈ R we find that βt(ω(σ, π)) = β̂t(s(σ))β̂t(s(π))β̂t(s(σ ⊗ π))∗. Taking the derivative at t = 0 yields equation (6.2): δ(ω(σ, π)) = δ̂(s(σ))s(π)s(σ ⊗ π)∗ + s(σ)δ̂(s(π))s(σ ⊗ π)∗ + s(σ)s(π)δ̂(s(σ ⊗ π))∗ = H(σ)ω(σ, π) + γσ(H(π))ω(σ, π) + ω(σ, π)H(σ ⊗ π)∗. ■ Lemma 6.3. Let H(σ) ∈ B∞ ⊗ L(Hσ), σ ∈ Irr(G), be a family, normalized to H(1) = 0, satisfying equations (6.1) and (6.2). Then there is a smooth lift ( β̂t ) t∈R of (βt)t∈R with Dβ̂t ( Tr(ys(σ)) ) = Tr ( Dβ(y)s(σ) + xH(σ)s(σ) ) for all y ∈ B∞ ⊗ L(Hσ, Vσ) and σ ∈ Irr(G). Proof. Without loss of generality we may assume that H(σ) = H(σ)γσ(1B) for all σ ∈ Irr(G), because replacing H(σ) by H(σ)γσ(1B) does neither change equation (6.1) nor equation (6.2). The task is now to prove that ( H, γβt , ωβt ) ∼ (H, γ, ω) for all t ∈ R. For this purpose, let σ ∈ Irr(G). To handle the first conjugacy condition, we examine the differential equation v̇t(σ) = βt(H(σ))vt(σ) in B ⊗ L(Hσ)γσ(1B) with initial condition v0(σ) = γσ(1B). Indeed, by [11, Section 3], it admits a unique solution vt(σ) ∈ B ⊗L(Hσ)γσ(1B) satisfying the β-cocycle equations vs+t(σ) = βt(vs(σ))vt(σ) and v−t(σ) = β−1 t (vt(σ) ∗) (6.3) for all s, t ∈ R. In particular, each vt(σ), t ∈ R, is a partial isometry with initial and final pro- jection given by vt(σ) ∗vt(σ) = γσ(1B) and vt(σ)vt(σ) ∗ = βt(γσ(1B)), respectively. Furthermore, for each b ∈ B∞ we consider the function t 7→ bt := γβt σ (b) = βtγσβ −1 t (b) which is clearly smooth and satisfies the differential equation ḃt = βtδγσβ −1 t (b)− βtγσδβ −1 t (b) = βt ( δγσβ −1 t (b)− γσδβ −1 t (b) ) (6.1) = βt ( H(σ)γσβ −1 t (b) + γσβ −1 t (b)H(σ)∗ ) = βt(H(σ))bt + btβt(H(σ)) for all t ∈ R with initial condition b0 = γσ(b). As the same equation and initial condition are satisfied by the function t 7→ vt(σ)γσ(b)vt(σ) ∗ we must have γβt σ (b) = vt(σ)γσ(b)vt(σ) ∗ (6.4) An Atiyah Sequence for Noncommutative Principal Bundles 15 for all t ∈ R and b ∈ B. By conjugating with vt(σ) ∗, we also obtain γσ = Ad[vt(σ) ∗] ◦ γβt σ . Next, let σ, π ∈ Irr(G). To deal with the second conjugacy condition, we look at the function t 7→ ωt := vt(σ)γσ(v(π))ω(σ, π)vt(σ ⊗ π)∗. Evidently, this function is smooth. Moreover, it satisfies the differential equation ω̇t = v̇t(σ)γσ(vt(π))ωvt(σ ⊗ π)∗ + vt(σ)γσ(v̇t(π))ωvt(σ ⊗ π)∗ + vt(σ)γσ(vt(π))ωv̇t(σ ⊗ π)∗ = βt(H(σ))ωt + vt(σ)γσβt(H(π))vt(σ) ∗ωt + ωtH(σ ⊗ π)∗ (6.4) = βt(H(σ))ωt + βtγσ((H(π))ωt + ωtH(σ ⊗ π)∗ for all t ∈ R with initial condition ω0 = ω(σ, π). The same equation and initial condition are satisfied by the function t 7→ ωβt(σ, π) = β̂t(s(σ)s(π)s(σ ⊗ π)∗). It follows that ωβt(σ, π) = vt(σ)γσ(v(π))ω(σ, π)vt(σ ⊗ π)∗ for all t ∈ R. Summarizing, we have shown that (H, γβt , ωβt) = vt(H, γ, ω)v ∗ t for all t ∈ R. Hence, for each t ∈ R, Theorem 4.1 provides us with a lift β̂t of βt satisfying β̂t ( Tr(ys(σ)) ) = Tr ( βt(y)vt(σ)s(σ) ) (6.5) for all y ∈ B ⊗ L(Hσ, Vσ) and σ ∈ Irr(G) (cf. equation (4.1)). In addition, equations (6.3) imply that (β̂t)t∈R constitutes a 1-parameter group as required. Finally, it is immediate from equation (6.5) that, for each y ∈ B∞ ⊗ L(Hσ, Vσ), the element Tr ( ys(σ) ) is smooth and that Dβ̂ ( Tr(ys(σ) ) = Tr ( Dβ(y)v0(σ)s(σ) ) +Tr ( yv̇0(σ)s(σ) ) (3.1) = Tr ( Dβ(y)s(σ) ) +Tr ( yH(σ)s(σ) ) for all σ ∈ Irr(G) and y ∈ B∞ ⊗ L(Hσ, Vσ). ■ Combining Lemmas 6.2 and 6.3, we have established: Theorem 6.4. Let (A, G, α) be a free C∗-dynamical system with fixed point algebra B. Further- more, let (βt)t∈R be a smooth 1-parameter group of ∗-automorphisms βt ∈ Aut(B). Then the following statements are equivalent: (a) (βt)t∈R lifts smoothly to A. (b) There is a family H(σ) ∈ B∞ ⊗L(Hσ, Vσ), σ ∈ Irr(G), normalized to H(1) = 0, satisfying equations (6.1) and (6.2) for all σ, π ∈ Irr(G) and b ∈ B∞. 7 An Atiyah sequence for noncommutative principal bundles In this section we generalize the classical Atiyah sequence in equation (1.1) to the setting of free C∗-dynamical systems. In addition, we explain how this can be used to produce characteristic classes. To this end, we consider a free C∗-dynamical system (A, G, α) with fixed point algebra B and we fix a dense unital ∗-subalgebra B0 ⊆ B. Again, for each σ ∈ Irr(G) we choose a finite- dimensional Hilbert space Hσ and an isometry s(σ) in A ⊗ L(Vσ,Hσ) satisfying αg(s(σ)) = s(σ)(1A ⊗ σg) for all g ∈ G; for 1 ∈ Irr(G), we take H1 := C and s(1) := 1A. We denote by (H, γ, ω) the associated factor system and assume that γσ(B0) ⊆ B0 ⊗ L(Hσ) and ω(σ, π) ∈ B0 ⊗ L(Hσ⊗π,Hσ ⊗ Hπ) 16 K. Schwieger and S. Wagner for all σ, π ∈ Irr(G). Similar arguments as in [37, Section 5.1] establish that A0 := span { Tr(ys(σ)) : σ ∈ Irr(G), y ∈ B0 ⊗ L(Hσ, Vσ) } is a dense, α-invariant, and unital ∗-subalgebra of A such that AG 0 = B0 (cf. Section 3). The algebra A0 depends on the choice of the isometries s(σ), σ ∈ Irr(G). However, any other choice of isometries s′(σ), σ ∈ Irr(G), such that s′(σ) ∈ A0 ⊗ L(Vσ,H ′ σ) yields the same subalgebra. In fact, A0 is the smallest ∗-subalgebra of A such that A0 ⊇ B0 and s(σ) ∈ A0 ⊗ L(Vσ,Hσ) for all σ ∈ Irr(G). 7.1 The associated Atiyah sequence For a ∗-derivation δ on B0 we say that δ lifts to A0 if there is a ∗-derivation δ̂ on A0 that extends δ and commutes with all αg, g ∈ G. In this case we call δ̂ a lift of δ. As an immediate consequence of Theorem 6.4 we obtain: Corollary 7.1. Let δ ∈ der(B0). Then the following statements are equivalent: (a) δ lifts to A0. (b) There is a family H(σ) ∈ B0 ⊗ L(Hσ, Vσ), σ ∈ Irr(G), normalized to H(1) = 0, satisfying equations (6.1) and (6.2) for all σ, π ∈ Irr(G) and b ∈ B0. Moreover, under the assumptions of (b) a lift of δ is given by linearly extending δ̂ ( Tr(xs(σ)) ) := Tr ( δ(x)s(σ) + xH(σ)s(σ) ) for σ ∈ Irr(G) and x ∈ B0 ⊗ L(Hσ, Vσ). Remark 7.2. Suppose that G is compact Abelian and that (A, G, α) is cleft. This is essentially the setting studied by Batty, Carey, Evans, and Robinson in [5], but without the factor system terminology. Combining Corollary 7.1 with the results from Section 5, we obtain a generalization of the main results in [5] to the setting of free actions of compact groups. To formulate a generalized Atiyah sequence, we consider the Lie algebra derG(A0) := {δ ∈ der(A0) : αg ◦ δ = δ ◦ αg ∀g ∈ G}. This Lie algebra admits the short exact sequence 0 −→ gau(A0) −→ derG(A0) −→ der(B0)(γ,ω) −→ 0, where gau(A0) := { δ ∈ derG(A0) : δ\|B0 = 0 } is the Lie algebra of infinitesimal gauge transformations of A0 and der(B0)(γ,ω) is the Lie sub- algebra of all ∗-derivations of B0 that lift to A0. By Corollary 7.1, we have der(B0)(γ,ω) = {δ ∈ der(B0) : δ satisfies the condition in Corollary 7.1(b)}. Furthermore, it follows from Corollary 7.1 that gau(A0) can be identified with the Lie algebra, let us say, H(H, γ, ω), consisting of all families of skew-symmetric elements H(σ) ∈ B0 ⊗L(Hσ), σ ∈ Irr(G), satisfying the equations H(σ)γσ(b) + γσ(b)H(σ)∗ = 0 ∀b ∈ B0, H(σ)ω(σ, π) + γσ ( H(π) ) ω(σ, π) + ω(σ, π)H(σ ⊗ π)∗ = 0 for all σ, π ∈ Irr(G). An Atiyah Sequence for Noncommutative Principal Bundles 17 Remark 7.3. Suppose that (A, G, α) is cleft with compact Abelian G and that ω(σ, π) lies in C · 1B for all σ, π ∈ Ĝ = Hom(G,T). Then H(H, γ, ω) may be realized as the Lie algebra of all crossed homomorphisms Ĝ → Bskew 0 : Z1 ( Ĝ,Bskew 0 ) := { H : Ĝ → Bskew 0 : H(σ + π) = H(σ) + γσ(Hπ) ∀σ, π ∈ Ĝ } . Definition 7.4. A linear section χ : der(B0)(γ,ω) → derG(A0) of the Atiyah sequence in equa- tion (4.2) is called a connection of A0. Remark 7.5. If q : P → M is a smooth principal bundle with structure group G, then the previous definition reproduces, up to a suitable completion, the classical setting of connection 1-forms when restricted to sections V(M) → V(P )G that are C∞(M)-linear (see, e.g., [24, Chapter XII]). Given a connection χ : der(B0)(γ,ω) → derG(A0) of A0, we may utilize Lecomte’s Chern–Weil homomorphism to associate characteristic classes with (A, G, α). More precisely, for each k ∈ N0 we get a natural map Ck : Symk(gau(A0),B0) derG(A0) → H2k ( der(B0)(γ,ω),B0 ) , f 7→ 1 k! [fχ], which, as a matter of fact, is independent of the choice of χ. Here, Symk(gau(A0),B0) stands for the space of symmetric k-linear maps gau(A0) k → B0 and fχ is the 2k-cocycle in C2k ( der(B0)(γ,ω),B0 ) associated with χ (cf. [26, 40]). 7.2 An example: quantum 3-tori Let θ be a real skew-symmetric 3 × 3-matrix and, for 1 ≤ k, ℓ ≤ 3, put λk,ℓ := exp(2πıθk,ℓ) for short. In the following we consider the quantum 3-torus A3 θ, which is the universal C∗-algebra with unitary generators u1, u2, u3 satisfying the relation ukuℓ = λk,ℓuℓuk for all 1 ≤ k, ℓ ≤ 3. The classical torus T3 acts naturally on A3 θ via the ∗-automorphisms given by τ(uk) = zk ·uk for all z = (z1, z1, z3) ∈ T3 and 1 ≤ k ≤ 3. This is the so-called gauge action, whose generators are the ∗-derivations δk(uℓ) = 2πıδk,ℓ · uℓ, 1 ≤ k, ℓ ≤ 3, where δk,ℓ denotes the Kronecker delta. Our study revolves around the restricted gauge action α : T → Aut ( A3 θ ) defined by αz(u1) := u1, αz(u2) := u2, αz(u3) := z · u3 for all z ∈ T. Its fixed point algebra is the quantum 2-torus A2 θ′ generated by the unitaries u1 and u2, where θ′ denotes the real skew-symmetric 2 × 2-matrix with upper right off-diagonal entry θ12. More generally, for each k ∈ Z, the corresponding isotypic component is A3 θ(k) takes the form uk3A2 θ′ . In particular, the C∗-dynamical system ( A3 θ,T, α ) is cleft and therefore free. The factor system associated with the unitaries u(k) := uk3, k ∈ Z, is given by the following data: For k ∈ Z we have γk = τz with z = ( λk 31, λ k 32, 1 ) , and for k, l ∈ Z the cocycle ω(k, l) computes as 1A2 θ′ . Next, we look at the dense unital ∗-subalgebra B0 of A2 θ′ generated by all noncommutative polynomials in u1 and u2. Then A0 is given by the dense unital ∗-algebra of A3 θ generated by all noncommutative polynomials in u1, u2, and u3, as is easily seen. Furthermore, it follows from [17, Section 3.4] that the ∗-derivations of B0 split as a semidirect product of inner and outer ∗-derivations, i.e., der(B0) ∼= Inn(der(B0)) ⋊ Out(der(B0)). If moreover θ12 is irrational, then Out(der(B0)) is linearly generated by δ1\|B0 and δ2\|B0 . Obviously, δ1\|B0 and δ2\|B0 may be lifted to δ1\|A0 and δ2\|B0 , respectively. It is also clear that each inner ∗-derivation of B0 lifts to A0. In consequence, der(B0)(γ,ω) = der(B0). Since we also have Z1 ( Z,Bskew 0 ) ∼= Bskew 0 via the evaluation map f 7→ f(1), it follows that the associated Atiyah sequence reads as 0 −→ Bskew 0 −→ derG(A0) −→ der(B0) −→ 0 18 K. Schwieger and S. Wagner (cf. Remark 7.3). Finally, a moment’s thought shows that this sequence is split, and so it does, unfortunately, only give trivial Chern–Weil–Lecomte classes. However, if needed, one may associate secondary characteristic classes as described in [40]. 7.3 Associating connections and curvature Connection 1-forms are the fundamental tool in the theory of smooth principal bundles and give rise to the notion of connections on associated vector bundles. Such a connection or, more precisely, its induced covariant derivative is an operator that can differentiate sections of each associated vector bundle along tangent directions in the base manifold. In this section we discuss suitable generalizations of theses notions to the C∗-algebraic setting of noncommutative principal bundles. In particular, we provide explicit formulas for connections and curvature on associated noncommutative vector bundles. To do this, we proceed as follows: For each finite-dimensional representation (σ, Vσ) of G we consider the associated natural inner product A0 ⊗ L(Vσ)− B0-bimodule ΓA0(Vσ) := ΓA0(σ, Vσ) := {x ∈ A0 ⊗ Vσ : (αg ⊗ σg)(x) = x ∀g ∈ G} with left A0⊗L(Vσ)-valued inner product A0⊗L(Vσ) ⟨·, ·⟩ and right B0-valued inner product ⟨·, ·⟩B0 defined on simple tensors by A0⊗L(Vσ) ⟨a⊗ v, b⊗ w⟩ := ab∗ ⊗ \|v⟩⟨w\| and ⟨a⊗ v, b⊗ w⟩B0 := ⟨v, w⟩a∗b respectively. Notably, the bimodule structure and the inner products are related by the com- patibility condition A0⊗L(Vσ) ⟨x, y⟩ . z = x . ⟨y, z⟩B0 for all x, y, z ∈ ΓA0(Vσ). Remark 7.6. The space ΓA0(Vσ) may be interpreted as associated noncommutative vector bundles (cf. [39]). The following result provides a criterion for ensuring that the space ΓA0(Vσ) admits a so-called standard right-module frame (see, e.g., [33, Section 2]). Lemma 7.7. Let (σ, Vσ) be a finite-dimensional representation of G. Then there are elements s1, . . . , sd ∈ ΓA0(Vσ) such that d∑ k=1 A0⊗L(Vσ) ⟨sk, sk⟩ = 1A⊗L(Vσ). In particular, the reproducing formula x = ∑d k=1 sk . ⟨sk, x⟩B0 holds for all x ∈ ΓA0(Vσ). If (A, G, α) is cleft, we find such elements with ⟨sk, sl⟩B0 = δk,l · 1B for all 1 ≤ k, l ≤ d. Proof. Let e1, . . . , ed be an orthonormal bases of Hσ and let sk ∈ A0 ⊗ Vσ, 1 ≤ k ≤ d, be the columns of s(σ)∗ ∈ A0 ⊗ L(Hσ, Vσ). Then sk ∈ ΓA0(Vσ) for all 1 ≤ k ≤ d, which is due to the fact that αg(s(σ)) = s(σ)(1A ⊗ σg) for all g ∈ G. In addition, a moment’s thought reveals that d∑ k=1 A0⊗L(Vσ) ⟨sk, sk⟩ = s(σ)∗s(σ) = 1A⊗L(Vσ). If (A, G, α) is cleft, then ⟨sk, sl⟩B0 = δk,l ·1B for all 1 ≤ k, l ≤ d follows from the fact that s(σ) is unitary. The verification of the reproducing formula poses no trouble and is left to the reader. ■ In what follows, we consider a fixed finite-dimensional representation (σ, Vσ) ofG and elements s1, . . . , sd ∈ ΓA0(Vσ) as in Lemma 7.7. An Atiyah Sequence for Noncommutative Principal Bundles 19 Lemma 7.8. Let (σ, Vσ) be a finite-dimensional representation of G. If δ : B0 → B0 is a ∗-de- rivation, then the linear map ∇σ δ : ΓA0(σ) → ΓA0(σ), ∇σ δ (x) := d∑ k=1 sk . δ (⟨sk, x⟩B0) satisfies the following equations ∇σ δ (x . b) = ∇σ δ (x) . b+ x . δ(b), (7.1) δ (⟨x, y⟩B0) = ⟨∇σ δ (x), y⟩B0 + ⟨x,∇σ δ (y)⟩B0 (7.2) for all x, y ∈ ΓA0(σ) and b ∈ B0. Proof. Let x, y ∈ ΓA0(σ) and b ∈ B0. Using first the right B0-linearity of ⟨·, ·⟩B0 , second the derivation property of δ, and finally the reproducing formula for x, we obtain ∇σ δ (x . b) = d∑ k=1 sk . δ (⟨sk, x . b⟩B0) = d∑ k=1 sk . δ (⟨sk, x⟩B0 · b) = d∑ k=1 sk . (δ (⟨sk, x⟩B0) · b+ ⟨sk, x⟩B0 · δ(b)) = ∇σ δ (x) . b+ x . δ(b). Likewise, it may be concluded that ⟨∇σ δ (x), y⟩B0 + ⟨x,∇σ δ (y)⟩B0 = d∑ k=1 (⟨sk . δ (⟨sk, x⟩B0) , y⟩B0 + ⟨x, sk . δ (⟨sk, y⟩B0)⟩B0) = d∑ k=1 (δ (⟨x, sk⟩B0) · ⟨sk, y⟩B0 + ⟨x, sk⟩B0 · δ (⟨sk, y⟩B0)) = d∑ k=1 δ (⟨x, sk⟩B0 · ⟨sk, y⟩B0) = d∑ k=1 δ (⟨x, sk . ⟨sk, y⟩B0⟩B0) = δ ( ⟨x, d∑ k=1 sk . ⟨sk, y⟩B0⟩B0 ) = δ (⟨x, y⟩B0) , where for the second equality we have exploited the fact that δ is a ∗-derivation. ■ We refer to equation (7.1) as the Leibniz rule and to equation (7.2) as the metric compatibility between the inner product and the map ∇σ δ . In addition, we draw attention to the fact that Lemma 7.8 entails that the map ∇σ : der(B0)× ΓA0(σ) → ΓA0(σ), ∇σ(δ, x) := ∇σ δ (x) is a so-called metric connection over ΓA0(σ) (see, e.g., [3, 16]). Corollary 7.9. For each finite-dimensional representation (σ, Vσ) of G the associated vector bundle ΓA0(σ) admits a metric connection. Next, let χ : der(B0)(γ,ω) → derG(A0) be a connection of A0. It is straightforward to check that, for each δ ∈ der(B0)(γ,ω), the map (∇χ)σδ : ΓA0(σ) → ΓA0(σ), (∇χ)σδ (x) := χ(δ)⊗ idVσ(x) is well-defined, linear, and satisfies the Leibniz rule. Moreover, since (∇χ)σδ (x) = d∑ k=1 χ(δ)⊗ idVσ(sk) . ⟨sk, x⟩B0 +∇σ δ (x) ∀x ∈ ΓA0(σ), similar computations as in the proof of Lemma 7.8 yield the following result: 20 K. Schwieger and S. Wagner Corollary 7.10. Let χ : der(B0)(γ,ω) → derG(A0) be a connection of A0. Furthermore, let (σ, Vσ) be a finite-dimensional representation of G. Then ∇χ,σ : der(B0)(γ,ω) × ΓA0(σ) → ΓA0(σ), ∇χ,σ(δ, x) := (∇χ)σδ (x) is a metric connection if and only if ⟨χ(δ)⊗ idVσ(x), y⟩B0 + ⟨x, χ(δ)⊗ idVσ(y)⟩B0 = 0 for all δ ∈ der(B0)(γ,ω) and x, y ∈ ΓA0(σ). To proceed, we bring to mind that, given a finitely generated projective right B0-module E together with a connection ∇ : der(B0) × E → E, its curvature R := R∇ with respect to ∇ is the map defined by R : der(B0)× der(B0)× E → E, R(δ1, δ2, e) := [∇δ1 ,∇δ2 ](e)−∇[δ1,δ2](e). Lemma 7.11. Let (σ, Vσ) be a finite-dimensional representation of G. Then the curvature Rσ := R∇σ of ΓA0(σ) with respect to ∇σ takes the form Rσ(δ1, δ2, x) = d∑ k,l=1 sk . ( δ1(⟨sk, sl⟩B0) · δ2(⟨sl, x⟩B0)− sk . δ2(⟨sk, sl⟩B0) · δ1(⟨sl, x⟩B0) ) for all δ1, δ2 ∈ der(B0) and x ∈ ΓA0(σ). Proof. Let δ1, δ2 ∈ der(B0) and x ∈ ΓA0(σ). Then a straightforward computation gives Rσ(δ1, δ2, x) = [∇σ δ1 ,∇ σ δ2 ](x)−∇σ [δ1,δ2] (x) = ∇σ δ1(∇ σ δ2(x))−∇σ δ2(∇ σ δ1(x))−∇σ [δ1,δ2] (x) = d∑ k,l=1 sk . ( δ1(⟨sk, sl⟩B0 · δ2(⟨sl, x⟩B0))− δ2(⟨sk, sl⟩B0 · δ1(⟨sl, x⟩B0)) ) − d∑ k=1 sk . [δ1, δ2](⟨sk, x⟩B0) = d∑ k,l=1 sk . ( δ1(⟨sk, sl⟩B0) · δ2(⟨sl, x⟩B0) + ⟨sk, sl⟩B0 · δ1(δ2(⟨sl, x⟩B0)) − δ2(⟨sk, sl⟩B0) · δ1(⟨sl, x⟩B0)− ⟨sk, sl⟩B0 · δ2(δ1(⟨sl, x⟩B0)) ) − d∑ l=1 sl . (δ1(δ2(⟨sl, x⟩B0))− δ2(δ1(⟨sl, x⟩B0))) = d∑ k,l=1 sk . ( δ1(⟨sk, sl⟩B0) · δ2(⟨sl, x⟩B0)− sk . δ2(⟨sk, sl⟩B0) · δ1(⟨sl, x⟩B0) ) . ■ As a corollary we can conclude that cleft actions have vanishing curvature: Corollary 7.12. Let (A, G, α) be cleft. For a finite-dimensional representation (σ, Vσ) of G let ∇σ denote the metric connection. Then Rσ(δ1, δ2, x) = 0 for all δ1, δ2 ∈ der(B0) and x ∈ ΓA0(σ), i.e., the curvature vanishes identically. Remark 7.13. With a little more effort one can also find a similar formula for the curvature associated with the metric connection ∇χ,σ. Remark 7.14. Corollary 7.12 implies that cleft C∗-dynamical systems yield trivial (primary) characteristic classes. An Atiyah Sequence for Noncommutative Principal Bundles 21 Acknowledgement We gratefully acknowledge the Centre International de Rencontres Mathématiques, REB 2177, as well as Blekinge Tekniska Högskola for supporting this research. The first name author expresses his gratitude to iteratec GmbH. Last but not least, we wish to thank the anonymous referees for providing fruitful criticism that helped to improve the manuscript. 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Operator Theory 63 (2010), 217–242, arXiv:0707.3207. https://arxiv.org/abs/1702.07918 https://doi.org/10.1007/s00220-005-1377-7 https://arxiv.org/abs/math.QA/0410077 https://doi.org/10.24033/bsmf.2032 https://doi.org/10.1007/978-3-642-62029-4 https://doi.org/10.1007/s00209-019-02326-5 https://arxiv.org/abs/1804.00289 https://arxiv.org/abs/0802.2993 https://doi.org/10.17879/58269764511 https://arxiv.org/abs/1303.6207 https://doi.org/10.1007/BFb0078657 https://doi.org/10.1007/978-1-4612-0453-4_6 https://doi.org/10.1090/conm/449/08721 https://arxiv.org/abs/math.DG/0703496 https://doi.org/10.1016/j.aim.2017.06.036 https://arxiv.org/abs/1505.00688 https://doi.org/10.4171/JNCG/11-2-6 https://arxiv.org/abs/#2 https://doi.org/10.3842/SIGMA.2017.062 https://arxiv.org/abs/1701.05895 https://doi.org/10.7146/math.scand.a-116147 https://arxiv.org/abs/1710.09396 https://doi.org/10.1016/j.aim.2021.108160 https://arxiv.org/abs/2012.15364 https://arxiv.org/abs/1111.5562 https://doi.org/10.24033/bsmf.278 https://arxiv.org/abs/1707.08412 https://arxiv.org/abs/0707.3207 1 Introduction 2 Preliminaries and notation 3 Factor systems 4 Lifting an automorphism 5 The special case of a compact Abelian structure group 6 Lifting 1-parameter groups 7 An Atiyah sequence for noncommutative principal bundles 7.1 The associated Atiyah sequence 7.2 An example: quantum 3-tori 7.3 Associating connections and curvature References
id	nasplib_isofts_kiev_ua-123456789-211530
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-15T11:14:57Z
publishDate	2022
publisher	Інститут математики НАН України
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spelling	Schwieger, Kay Wagner, Stefan 2026-01-05T12:26:56Z 2022 An Atiyah Sequence for Noncommutative Principal Bundles. Kay Schwieger and Stefan Wagner. SIGMA 18 (2022), 015, 22 pages 1815-0659 2020 Mathematics Subject Classification: 46L87; 46L85; 55R10 arXiv:2107.04653 https://nasplib.isofts.kiev.ua/handle/123456789/211530 https://doi.org/10.3842/SIGMA.2022.015 We present a derivation-based Atiyah sequence for noncommutative principal bundles. Along the way, we treat the problem of deciding when a given -automorphism on the quantum base space lifts to a -automorphism on the quantum total space that commutes with the underlying structure group. We gratefully acknowledge the Centre International de Rencontres Mathématiques, REB 2177, as well as Blekinge Tekniska Högskola for supporting this research. The first-name author expresses his gratitude to iteratec GmbH. Last but not least, we wish to thank the anonymous referees for providing fruitful criticism that helped to improve the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications An Atiyah Sequence for Noncommutative Principal Bundles Article published earlier
spellingShingle	An Atiyah Sequence for Noncommutative Principal Bundles Schwieger, Kay Wagner, Stefan
title	An Atiyah Sequence for Noncommutative Principal Bundles
title_full	An Atiyah Sequence for Noncommutative Principal Bundles
title_fullStr	An Atiyah Sequence for Noncommutative Principal Bundles
title_full_unstemmed	An Atiyah Sequence for Noncommutative Principal Bundles
title_short	An Atiyah Sequence for Noncommutative Principal Bundles
title_sort	atiyah sequence for noncommutative principal bundles
url	https://nasplib.isofts.kiev.ua/handle/123456789/211530
work_keys_str_mv	AT schwiegerkay anatiyahsequencefornoncommutativeprincipalbundles AT wagnerstefan anatiyahsequencefornoncommutativeprincipalbundles AT schwiegerkay atiyahsequencefornoncommutativeprincipalbundles AT wagnerstefan atiyahsequencefornoncommutativeprincipalbundles

An Atiyah Sequence for Noncommutative Principal Bundles

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