aa r X i v : . [ m a t h . OA ] J a n Finite Nocommutative CoveringsandFlat Connections
September 21, 2018
Petr R. Ivankov* e-mail: * [email protected] flat connection on a principal fibre bundle comes from a linear representation ofthe fundamental group. The noncommutative analog of this fact is discussed here.
Definition 1.1. [10] Let e π : e X → X be a continuous map. An open subset
U ⊂ X issaid to be evenly covered by e π if e π − ( U ) is the disjoint union of open subsets of e X eachof which is mapped homeomorphically onto U by e π . A continuous map e π : e X → X iscalled a covering projection if each point x ∈ X has an open neighborhood evenly coveredby e π . e X is called the covering space and X the base space of the covering. Definition 1.2. [10] Let p : e X → X be a covering. A self-equivalence is a homeomorphism f : e X → e X such that p ◦ f = p . This group of such homeomorphisms is said to be the group of covering transformations of p or the covering group . Denote by G (cid:16) e X | X (cid:17) thisgroup.
Remark 1.3.
Above results are copied from [10]. Below the covering projection word isreplaced with covering . 1 .2 Flat connections in the differential geometry
Here I follow to [8]. Let M be a manifold and G a Lie group. A ( differentiable ) principalbundle over M with group G consists of a manifolfd P and an action of G on P satisfying thefollowing conditions:(a) G acts freely on P on the right: ( u , a ) ∈ P × G ua = R a u ∈ P ;(b) M is the quotient space of P by the equivalence relation induced by G , i.e. M = P / G ,and the canonical projection π : P → M is differentiable;(c) P is locally trivial, that is, every point x of M has an open neighborhood U suchthat π − ( U ) is isomophic to U × G in the sense that there is a diffeomorphism ψ : π − ( U ) → U × G such that ψ ( u ) = ( π ( u ) , ϕ ( u )) where ϕ is a mapping of π − ( U ) into G satisfying ψ ( ua ) = ( ψ ( u )) a for all u ∈ π − ( U ) and a ∈ G .A principal fibre bundle will be denoted by P ( M , G , π ) , P ( M , G ) or simply P .Let P ( M , G ) be a principal fibre bundle over a manifold with group G . For each u ∈ P let T u ( P ) be a tangent space of P at u and G u the subspace of T u ( P ) consisting of vectorstangent to the fibre through u . A connection Γ in P is an assignment of a subspace Q u of T u ( P ) to each u ∈ P such that(a) T u ( P ) = G u ⊕ Q u (direct sum);(b) Q ua = ( R a ) ∗ Q u for every u ∈ P and a ∈ G , where R a is a transformation of P induced by a ∈ G , R a u = ua .Let P = M × G be a trivial principal bundle. For each a ∈ G , the set M × { a } is asubmanifold of P . The canonical flat connection in P is defined by taking the tangentspace to M × { a } at u = ( x , a ) as the horizontal tangent subspace at u . A connection inany principal bundle is called flat if every point has a neighborhood such that the inducedconnection in P | U = π − ( U ) is isomorphic with the canonical flat connection. Corollary 1.4. (Corollary II 9.2 [8]) Let Γ be a connection in P ( M , G ) such that the curvaturevanishes identically. If M is paracompact and simply connected, then P is isomorphic to the trivialbundle and Γ is isomorphic to the canonical flat connection in M × G. If e π : e M → M is a covering then the e π - lift of P is a principal e P (cid:16) e M , G (cid:17) bundle, given by e P = n ( u , e x ) ∈ P × e M | π ( u ) = e π ( e x ) o .If Γ is a connection on P ( M , G ) and e M → M is a covering then is a canonical connection e Γ on e P (cid:16) e M , G (cid:17) which is the lift of Γ , that is, for any e u ∈ e P the horizontal space e Q e u isisomorphically mapped onto the horizontal space Q e π ( e u ) associated with the connection2 . If Γ is flat then from the Proposition (II 9.3 [8]) it turns out that there is a covering e M → M such that e P (cid:16) e M , G (cid:17) (which is the lift of P ( M , G ) ) is a trivial bundle, so the lift e Γ of Γ is a canonical flat connection (cf. Corollary 1.4). From the the Proposition (II 9.3 [8])it follows that for any flat connection Γ on P ( M , G ) there is a group homomorphism ϕ : G (cid:16) e M | M (cid:17) → G such that(a) There is an action G (cid:16) e M | M (cid:17) × e P → e P ≈ e M × G given by g ( e x , a ) = ( g e x , ϕ ( g ) a ) ; ∀ e x ∈ e M , a ∈ G ,(b) There is the canonical diffeomorphism P = e P / G (cid:16) e M | M (cid:17) ,(c) The lift ˜ Γ of Γ is a canonical flat connection. Definition 1.5.
In the above situation we say that the flat connection Γ is induced by thecovering e M → M and the homomorphism G (cid:16) e M | M (cid:17) → G , or we say that Γ comes fromG (cid:16) e M | M (cid:17) → G . Remark 1.6.
The Proposition (II 9.3 [8]) assumes that e M → M is the universal coveringhowever it is not always necessary requirement. Remark 1.7. If π ( M , x ) is the fundamental group [10] then there is the canonical surjec-tive homomorphism π ( M , x ) → G (cid:16) e M | M (cid:17) . So there exist the composition π ( M , x ) → G (cid:16) e M | M (cid:17) → G . It follows that any flat connection comes from the homomorphisms π ( M , x ) → G .Suppose that there is the right action of G on P and suppose that F is a manifold withthe left action of G . There is an action of G on P × F given by a ( u , ξ ) = (cid:0) ua , a − ξ (cid:1) forany a ∈ G and ( u , ξ ) ∈ P × F . The quotient space P × G F = ( P × F ) / G has the naturalstructure of a manifold and if E = P × G F then E ( M , F , G , P ) is said to be the fibre bundleover the base M, with (standard) fibre F, and (structure) group G which is associated with theprincipal bundle P (cf. [8]). If P = M × G is the trivial bundle then E is also trivial, that is, E = M × F . If F = C n is a vector space and the action of G on C n is a linear representationof the group then E is the linear bundle. Denote by T ( M ) (resp. T ∗ ( M ) ) the tangent (resp.contangent) bundle, and denote by Γ ( E ) , Γ ( T ( M )) , Γ ( T ∗ ( M )) the spaces of sections of E , T ( M ) , T ∗ ( M ) respectively. Any connection Γ on P gives a covariant derivative on E ,that is, for any section X ∈ Γ ( T ( M )) and any section ξ ∈ Γ ( E ) there is the derivativegiven by ∇ X ( ξ ) ∈ Γ ( E ) .If E = M × C n , Γ is the canonical flat connection and ξ is a trivial section, that is, ξ = M × { x } then ∇ X ξ = ∀ X ∈ T ( M ) . (1.1)3or any connection there is the unique map ∇ : Γ ( E ) → Γ ( E ⊗ T ∗ ( M )) (1.2)such that ∇ X ξ = ( ∇ ξ , X ) where the pairing ( · , · ) : Γ ( E ⊗ T ∗ ( M )) × Γ ( T ( M )) → Γ ( E ) is induced by the pairing Γ ( T ∗ ( M )) × Γ ( T ( M )) → C ∞ ( M ) . The noncommutative analog of manifold is a spectral triple and there is the noncom-mutative analog of connections. [3](a) A cycle of dimension n is a triple ( Ω , d , R ) where Ω = L nj = Ω j is a graded algebraover C , d is a graded derivation of degree 1 such that d =
0, and R : Ω n → C is aclosed graded trace on Ω ,(b) Let A be an algebra over C . Then a cycle over A is given by a cycle ( Ω , d , R ) and ahomomorphism A → Ω . Definition 1.9. [3] Let A ρ −→ Ω be a cycle over A , and E a finite projective module over A . Then a connection ∇ on E is a linear map ∇ : E → E ⊗ A Ω such that ∇ ( ξ x ) = ∇ ( ξ ) x = ξ ⊗ d ρ ( x ) ; ∀ ξ ∈ E , ∀ x ∈ A . (1.3)Here E is a right module over A and Ω is considered as a bimodule over A . Remark 1.10.
The map ∇ : E → E ⊗ A Ω is an algebraic analog of the map ∇ : Γ ( E ) → Γ ( E ⊗ T ∗ ( M )) given by (1.2). Proposition 1.11. [3] Following conditions hold:(a) Let e ∈ End A ( E ) be an idempotent and ∇ is a connection on E ; then ξ ( e ⊗ ) ∇ ξ (1.4) is a connection on e E ,(b) Any finite projective module E admits a connection,(c) The space of connections is an affine space over the vector space Hom A (cid:0) E , E ⊗ A Ω (cid:1) , d) Any connection ∇ extends uniquely up to a linear map of e E = E ⊗ A Ω into itself such that ∇ ( ξ ⊗ ω ) = ∇ ( ξ ) ω + ξ ⊗ d ω ; ∀ ξ ∈ E , ω ∈ Ω . (1.5)A curvature of a connection ∇ is a (right A -linear) map F ∇ : E → E ⊗ A Ω (1.6)defined as a restriction of ∇ ◦ ∇ to E , that is, F ∇ = ∇ ◦ ∇| E . A connection is said to be flat if its curvature is identically equal to 0 (cf. [1]). Remark 1.12.
Above algebraic notions of curvature and flat connection are generalizationsof corresponding geometrical notions explained in [8] and the Section 1.2.For any projective A module E there is a trivial connection ∇ : E ⊗ A Ω → E ⊗ A Ω , ∇ = Id E ⊗ d .From d = d ◦ d = ( Id E ⊗ d ) ◦ ( Id E ⊗ d ) = 0, i.e. any trivial connection isflat. Lemma 1.13. If ∇ : E → E ⊗ A Ω is a trivial connection and e ∈ End A ( E ) is an idempotentthen the given by (1.4) ξ ( e ⊗ ) ∇ ξ connection ∇ e : e E → e E ⊗ Ω on e E is flat.Proof. From ( e ⊗ ) ( Id E ⊗ d ) ◦ ( e ⊗ ) ( Id E ⊗ d ) = e ⊗ d = ∇ e ◦ ∇ e =
0, i.e. ∇ e is flat. Remark 1.14.
The notion of the trivial connection is an algebraic version of geometricalcanonical connection explained in the Section 1.2.
This section contains citations of [7].
Definition of spectral triplesDefinition 1.15. [7] A (unital) spectral triple ( A , H , D ) consists of: • a pre- C ∗ -algebra A with an involution a a ∗ , equipped with a faithful representa-tion on: • a Hilbert space H ; and also • a selfadjoint operator D on H , with dense domain Dom D ⊂ H , such that a ( Dom D ) ⊆ Dom D for all a ∈ A .There is a set of axioms for spectral triples described in [7, 11].5 oncommutative differential forms Any spectral triple naturally defines a cycle ρ : A → Ω D (cf. Definition 1.9). In par-ticular for any spectral triple there is an A -module Ω D ⊂ B ( H ) of order-one differentialforms which is a linear span of operators given by a [ D , b ] ; a , b ∈ A . (1.7)There is the differential map d : A → Ω D , a [ D , a ] . (1.8) C ∗ -algebras Definition 2.1. If A is a C ∗ - algebra then an action of a group G is said to be involutive if ga ∗ = ( ga ) ∗ for any a ∈ A and g ∈ G . The action is said to be non-degenerated if for anynontrivial g ∈ G there is a ∈ A such that ga = a . Definition 2.2.
Let A ֒ → e A be an injective *-homomorphism of unital C ∗ -algebras. Sup-pose that there is a non-degenerated involutive action G × e A → e A of a finite group G , suchthat A = e A G def = n a ∈ e A | a = ga ; ∀ g ∈ G o . There is an A -valued product on e A given by h a , b i e A = ∑ g ∈ G g ( a ∗ b ) (2.1)and e A is an A -Hilbert module. We say that a triple (cid:16) A , e A , G (cid:17) is an unital noncommutativefinite-fold covering if e A is a finitely generated projective A -Hilbert module. Remark 2.3.
Above definition is motivated by the Theorem 2.4.
Theorem 2.4. [9]. Suppose X and Y are compact Hausdorff connected spaces and p : Y → X isa continuous surjection. If C ( Y ) is a projective finitely generated Hilbert module over C ( X ) withrespect to the action ( f ξ )( y ) = f ( y ) ξ ( p ( y )) , f ∈ C ( Y ) , ξ ∈ C ( X ) , then p is a finite-fold covering. Definition 2.5.
Let ( A , H , D ) be a spectral triple, and let A be the C ∗ -norm completionof A . Let (cid:16) A , e A , G (cid:17) be an unital noncommutative finite-fold covering such that there is6he dense inclusion A ֒ → A . Let e H def = e A ⊗ A H is a Hilbert space such that the Hilbertproduct ( · , · ) e H is given by ( a ⊗ ξ , b ⊗ η ) e H = | G | ξ , ∑ g ∈ G g (cid:16)e a ∗ e b (cid:17)! η ! H ; ∀ e a , e b ∈ e A , ξ , η ∈ H where ( · , · ) H is the Hilbert product on H . There is the natural representation e A → B (cid:16) e H (cid:17) .A spectral triple (cid:16) e A , e H , e D (cid:17) is said to be a (cid:16) A , e A , G (cid:17) - lift of ( A , H , D ) if following condi-tions hold:(a) e A is a C ∗ -norm completion of e A ,(b) e D (cid:0) e A ⊗ A ξ (cid:1) = e A ⊗ A D ξ ; ∀ ξ ∈ Dom D ,(c) e D (cid:16) g e ξ (cid:17) = g (cid:16) e D e ξ (cid:17) for any e ξ ∈ Dom e D , g ∈ G . Remark 2.6.
It is proven in [5] that for any spectral triple ( A , H , D ) and any unital non-commutative finite-fold covering (cid:16) A , e A , G (cid:17) there is the unique (cid:16) A , e A , G (cid:17) -lift (cid:16) e A , e H , e D (cid:17) of ( A , H , D ) . Remark 2.7.
It is known that if M is a Riemannian manifold and e M → M is a covering,then e M has the natural structure of Riemannian manifold (cf. [8]). The existence of lifts ofspectral triples is a noncommutative generalization of this fact (cf. [5]) Let ( A , H , D ) be a spectral triple, let (cid:16) e A , e H , e D (cid:17) is the (cid:16) A , e A , G (cid:17) -lift of ( A , H , D ) . Let V = C n and with left action of G , i.e. there is a linear representation ρ : G → GL ( C , n ) .Let e E = A ⊗ C n ≈ e A n be a free module over e A , so e E is a projective finitely generated A -module (because e A is a finitely generated projective A -module). Let e ∇ : e E → e E ⊗ e A Ω e D be the trivial flat connection. In [5] it is proven that Ω e D = e A ⊗ A Ω D it follows that theconnection e ∇ : e E → e E ⊗ e A Ω e D can be regarded as a map ∇ ′ : e E → e E ⊗ e A e A ⊗ A Ω e D = e E ⊗ A Ω D , i.e. one has a connection ∇ ′ : e E → e E ⊗ A Ω D .From e ∇ ◦ e ∇| E = ∇ ′ ◦ ∇ ′ | E =
0, i.e. ∇ ′ is flat. There is the action of G on e E = e A ⊗ C n given by g ( e a ⊗ x ) = g e a ⊗ gx ; ∀ g ∈ G , e a ∈ e A , x ∈ C n . (3.1)Denote by E = e E G = ne ξ ∈ e E | G e ξ = e ξ o (3.2)7learly E is an A - A -bimodule. For any e ξ ∈ e E there is the unique decomposition e ξ = ξ + ξ ⊥ , ξ = | G | ∑ g ∈ G g e ξ , ξ ⊥ = e ξ − ξ . (3.3)From the above decomposition it turns out the direct sum e E = e E G L E ⊥ of A -modules. So E = e E G is a projective finitely generated A -module, it follows that there is an idempotent e ∈ End A e E such that E = e e E . The Proposition 1.11 gives the canonical connection ∇ : E → E ⊗ A Ω D (3.4)which is defined by the connection ∇ ′ : e E → e E ⊗ A Ω D and the idempotent e . From theLemma 1.13 it turns out that ∇ is flat. Definition 3.1.
We say that ∇ is a flat connection induced by noncommutative covering (cid:16) A , e A , G (cid:17) and the linear representation ρ : G → GL ( C , n ) , or we say the ∇ comes from therepresentation ρ : G → GL ( C , n ) . 8 Mapping between geometric and algebraic constructions
The geometric (resp. algebraic) construction of flat connection is explained in the Sec-tion 1.2 (resp. 3). Following table gives a mapping between these constructions.Geometry Agebra1 Riemannian manifold M . Spectral triple (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) .2 Topological covering e M → M . Noncommutative covering, (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M | M (cid:17)(cid:17) ,given by the Theorem 2.4.3 Natural structure of Reimannian Triple (cid:16) C ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e / D (cid:17) is themanifold on the covering space e M . (cid:16) C ( M ) , C (cid:16) e M (cid:17) , G (cid:16) e M | M (cid:17)(cid:17) -liftof (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) .4 Group homomorphism Action G (cid:16) e M | M (cid:17) × C n → C n G (cid:16) e M | M (cid:17) → GL ( n , C ) e M × C n . Free module C ∞ (cid:16) e M (cid:17) ⊗ C n .6 Canonical flat connection on e M × C n Trivial flat connection on C ∞ (cid:16) e M (cid:17) ⊗ C n G (cid:16) e M | M (cid:17) on e M × C n Action of G (cid:16) e M | M (cid:17) on C ∞ (cid:16) e M (cid:17) ⊗ C n P = (cid:16) e M × C n (cid:17) / G (cid:16) e M | M (cid:17) . E = (cid:16) C ∞ (cid:16) e M (cid:17) ⊗ C n (cid:17) G ( e M | M ) P Algebraic flat connection on E . Following text is the citation of [5]. If Θ be a real skew-symmetric n × n matrix. Thereis a C ∗ - algebra C (cid:0) T n Θ (cid:1) which is said to be the noncommutative torus (cf. [5]). There is apre- C ∗ -algebra C ∞ (cid:0) T n Θ (cid:1) and the spectral triple (cid:0) C ∞ (cid:0) T n Θ (cid:1) , H , D (cid:1) such that it is the dense9nclusion C ∞ (cid:0) T n Θ (cid:1) ֒ → C (cid:0) T n Θ (cid:1) . If k = ( k , ..., k n ) ∈ N n and e Θ = e θ . . . e θ n e θ e θ n ... ... . . . ... e θ n e θ n . . . 0 is a skew-symmetric matrix such that e − π i θ rs = e − π i e θ rs k r k s then one has a following theorem. Theorem 5.1. [6] The triple (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z k × ... × Z k n (cid:17) is an unital noncommutativefinite-fold covering. There is (cid:16) C (cid:0) T n Θ (cid:1) , C (cid:16) T n e Θ (cid:17) , Z k × ... × Z k n (cid:17) -lift (cid:16) C ∞ (cid:16) T n e Θ (cid:17) , e H , e D (cid:17) of (cid:0) C ∞ (cid:0) T n Θ (cid:1) , H , D (cid:1) .From the construction of the Section 3 it follows that for any representation ρ : Z k × ... × Z k n → GL ( N , C ) there is a finitely generated C ∞ (cid:0) T n Θ (cid:1) -module E and a flat connection E → E ⊗ C ∞ ( T n Θ ) ⊗ Ω D which comes from ρ . A very general construction of isospectral deformations of noncommutative geometriesis described in [4]. The construction implies in particular that any compact Spin-manifold M whose isometry group has rank ≥ M θ . We let (cid:0) C ∞ ( M ) , L ( M , S ) , / D (cid:1) be the canon-ical spectral triple associated with a compact spin-manifold M . We recall that C ∞ ( M ) isthe algebra of smooth functions on M , S is the spinor bundle and / D is the Dirac operator.Let us assume that the group Isom ( M ) of isometries of M has rank r ≥
2. Then, we havean inclusion T ⊂ Isom ( M ) ,with T = R /2 π Z the usual torus, and we let U ( s ) , s ∈ T , be the correspondingunitary operators in H = L ( M , S ) so that by construction U ( s ) / D = / D U ( s ) .Also, U ( s ) a U ( s ) − = α s ( a ) , ∀ a ∈ A , (5.1)where α s ∈ Aut ( A ) is the action by isometries on the algebra of functions on M . In[4] is constructed a spectral triple (cid:0) lC ∞ ( M ) , L ( M , S ) , / D (cid:1) such that lC ∞ ( M ) is a non-commutative algebra which is said to be an isospectral deformation of C ∞ ( M ) . For any10nite-fold topological covering e M → M there is the finite-fold noncommutative covering (cid:16) lC (cid:16) e M (cid:17) , l ( M ) , G (cid:16) e M | M (cid:17)(cid:17) (cf. [6]). So there is the (cid:16) lC (cid:16) e M (cid:17) , l ( M ) , G (cid:16) e M | M (cid:17)(cid:17) -lift (cid:16) lC ∞ (cid:16) e M (cid:17) , L (cid:16) e M , e S (cid:17) , e / D (cid:17) of (cid:0) lC ∞ ( M ) , L ( M , S ) , / D (cid:1) . From the construction of the Section 3 it follows that for anyrepresentation ρ : G (cid:16) e M | M (cid:17) → GL ( N , C ) there is a finitely generated lC ∞ ( M ) -module E and a flat connection E → E ⊗ lC ∞ ( M ) ⊗ Ω D which comes from ρ . References [1] Tomasz Brzezinski
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