Associated noncommutative vector bundles over the Vaksman-Soibelman quantum complex projective spaces
AASSOCIATED NONCOMMUTATIVE VECTOR BUNDLES OVERTHE VAKSMAN–SOIBELMAN QUANTUM COMPLEX PROJECTIVE SPACES
FRANCESCA ARICI, PIOTR M. HAJAC, AND MARIUSZ TOBOLSKI
Abstract.
By a diagonal embedding of U (1) in SU q ( m ), we prolongate the diagonalcircle action on the Vaksman–Soibelman quantum sphere S n +1 q to the SU q ( m )-actionon the prolongated bundle. Then we prove that the noncommutative vector bundlesassociated via the fundamental representation of SU q ( m ), for m ∈ { , . . . , n } , yieldgenerators of the even K-theory group of the C*-algebra of the Vaksman–Soibelmanquantum complex projective space C P nq . Introduction
The K-theory of complex projective spaces was unraveled by Atiyah and Todd in [5,Prop. 2.3, 3.1 and 3.3] (cf. [1, Thm. 7.2] and [18, Cor. IV.2.8]). Denoting by L n the dualtautological line bundle over C P n and settingt := [ C P n × C ] − [L n ] ∈ K ( C P n ) , one obtains K ( C P n ) = Z [t] / t n +1 . (1.1)Recently, this result was extended to the Vaksman–Soibelman quantum complex projec-tive spaces [2, Prop. 3.3 and 3.4]: K ( C ( C P nq )) = Z [ t ] /t n +1 . (1.2)Here 0 < q < t := [1] − [ L n ], and L n is the section module of the noncommutative dualtautological line bundle over C P nq .The K-theory of both the Vaksman–Soibelman and the multipullback quantum complexprojective planes [11, 21, 14] was thoroughly analysed in [9]. In this special n = 2 case,the following elements [1] , [ L ] − [1] , [ L − ⊕ L ] − K ( C ( C P q )). Here L − is the section module of the noncommutativetautological line bundle over C P q .A particular feature of the module L − ⊕ L established in [9] is that it can be ob-tained as a Milnor module [19, Thm. 2.1] constructed from the fundamental represen-tation of SU q (2). This was achieved by taking U (1) as a subgroup of SU q (2), and thenrealising L − ⊕ L as the section module of the noncommutative vector bundle associatedto the prolongation S q × U (1) SU q (2) via the fundamental representation of SU q (2). The Mathematics Subject Classification.
Key words and phrases.
K-theory, noncommutative vector bundle, compact quantum group, Peter–Weyl decomposition, principal comodule algebra. a r X i v : . [ m a t h . K T ] O c t F. ARICI, P. M. HAJAC, AND M. TOBOLSKI upshot of having [ L − ⊕ L ] − C P q .The goal of this paper is to show that the above construction works in any dimension.More precisely, except for the classes [1] and [ L n ] − [1], we prove that all other generatorsof K ( C ( C P nq )) come from the section module of the noncommutative vector bundle as-sociated to the prolongation S n +1 q × U (1) SU q ( m ), for m ∈ { , . . . , n } , via the fundamentalrepresentation of SU q ( m ).Finally, since there is the exact sequence [3]0 −→ K ( C ( S n − q )) ∂ −→ K ( C ( C P nq )) −→ K ( C ( C P n − q )) −→ , a generator of K ( C ( C P nq )) that does not come from C P n − q can always be constructed viathe Milnor connecting homomorphism ∂ from a generator of K ( C ( S n − q )). Combiningthis with the fact that S n − q is a homogeneous space of SU q ( n ), we arrive at the following: Question : Can a generator of the K ( C ( S n − q )) ∼ = Z be always expressed in terms ofrepresentations of SU q ( n )?This question has positive answer for q = 1 by the work of Harris [16] based on the workof Atiyah and Hodgkin [6, 17]. For 0 < q < n = 2, we just take the fundamentalrepresentation of SU q (2). 2. Preliminaries
The Vaksman–Soibelman odd quantum spheres [24] are defined as quantum homoge-nous spaces for Woronowicz’s quantum special unitary groups [25]: C ( S n +1 q ) := C ( SU q ( n + 1)) SU q ( n ) . Here 0 < q ≤
1, and we use the notation A G for the fixed-point subalgebra of a C*-algebra A under an action of a compact quantum group G . (Note that the q = 1 case recoversthe classical situation.) One can show that C ( S n +1 q ) is the universal C*-algebra given bythe following generators and relations: z i z j = qz j z i for i < j, z i z ∗ j = qz ∗ j z i for i (cid:54) = j,z i z ∗ i = z ∗ i z i + ( q − − n (cid:88) m = i +1 z m z ∗ m , n (cid:88) m =0 z m z ∗ m = 1 . Much like in the classical case, the Vaksman–Soibelman quantum odd sphere enjoy thediagonal circle action given on generators by( z , z , . . . , z n ) (cid:55)→ ( λz , λz , . . . , λz n ) , λ ∈ U (1) . One uses this action to define the quantum complex projective spaces [24] C ( C P nq ) := C ( S n +1 q ) U (1) . SSOCIATED VECTOR BUNDLES OVER QUANTUM PROJECTIVE SPACES 3
Recall that freeness of circle actions can be characterised in terms of their spectralsubspaces (see e.g. [20]). Given an action α : U (1) → Aut( A ) on a unital C*-algebra, foreach character m ∈ Z , one defines the m -th spectral subspace A m as A m := { a ∈ A | α λ ( a ) = λ m a for all λ ∈ U (1) } . The subspace A agrees with the fixed-point subalgebra A U (1) , and A m A n ⊆ A m + n forall m, n ∈ Z , turning A into a Z -graded algebra. Now, one can say that the action α is free if and only if the Z -grading is strong [23], i.e. A m A n = A m + n for all m, n ∈ Z . It isstraightforward to check that the bimodules A m are finitely generated projective both asleft and right A U (1) -modules. Furthermore, they are invertible and they can be interpretedas modules of sections of associated noncommutative line bundles (see e.g. [4]).Note that, using the spatial tensor product, the action α can be dualised to the coaction δ : A −→ A ⊗ min C ( U (1)) = C ( U (1) , A ) , δ ( a )( λ ) := α λ ( a ) . Denote by O ( U (1)) the dense Hopf subalgebra of C ( U (1)) consisting of Laurent polyno-mials in one variable. The Peter–Weyl O ( U (1))-comodule algebra P U (1) ( A ), defined asthe set of all a ∈ A such that δ ( a ) ∈ A ⊗ O ( U (1)) (see [7]), is the purely algebraic directsum of spectral subspaces: P U (1) ( A ) = (cid:77) m ∈ Z A m . (2.1)The P U (1) ( A ) comodule algebra is principal in the sense of [12].A centrepiece concept for our paper is the notion of a prolongation of a principalcomodule algebra. To define it, we need to recall the construction of cotensor product.For a coalgebra C , the cotensor product of a right C -comodule M with a coaction ρ R : M → M ⊗ C and a left C -comodule N with a coaction ρ L : N → C ⊗ N is defined as thedifference kernel M C (cid:3) N := ker ( ρ R ⊗ id − id ⊗ ρ L : M ⊗ N −→ M ⊗ C ⊗ N ) . Given a surjection of Hopf algebras π : H → ¯ H , we can treat H as a left ¯ H -comodule viathe coaction ( π ⊗ id) ◦ ∆. For any right ¯ H -comodule algebra P , we define its prolongation as the cotensor product P (cid:3) ¯ H H . It is easy to check that this cotensor product is a right H -comodule algebra for the coaction id ⊗ ∆. It is also straightforward to verify that, if P is principal, then so is its prolongation.Let P be a principal H -comodule algebra with a coaction ∆ R and let V be a left H -comodule. Much as in the classical case, we can form the associated left module P (cid:3) H V over the coaction-invariant subalgebra P co H := { a ∈ P | ∆ R ( a ) = a ⊗ } . We think ofthis module as the section module of the associated noncommutative vector bundle . If P is principal and V is finite dimensional, then it is known that the associated module isfinitely generated projective. F. ARICI, P. M. HAJAC, AND M. TOBOLSKI Line bundles
As explained in the introduction, the K-theory of the Vaksman–Soibelman quantumcomplex projective spaces is determined by section modules of associated noncommutativeline bundles. Therefore, we begin our considerations by putting together some factsinvolving these noncommutative line bundles.For the Vaksman–Soibelman quantum sphere, the Peter–Weyl subalgebra (2.1) becomes P U (1) ( C ( S n +1 q )) = (cid:77) m ∈ Z L nm , where L nm := { a ∈ C ( S n +1 q ) | α λ ( a ) = λ m a for all λ ∈ U (1) } . The U (1)-action on C ( S n +1 q ) is free by [22, Cor. 3], so we can think of L nm as modulesof sections of associated noncommutative line bundles. Note that our sign convention isopposite to that used in [2], i.e. L nm = L − m .Recall that there exists a U (1)-equivariant ∗ -homomorphism ϕ : C ( S n +1 q ) → C ( S q )given on standard generators by ϕ ( z i ) = z if i = 0 z if i = 10 if i ≥ (cid:0) ϕ | C ( C P nq ) (cid:1) ∗ : K (cid:0) C ( C P nq ) (cid:1) −→ K (cid:0) C ( C P q ) (cid:1) satisfies (cid:0) ϕ | C ( C P nq ) (cid:1) ∗ (cid:16) [ L nm ] (cid:17) = [ L m ]for any m ∈ Z . Since the index pairing computation of [10, Thm. 2.1] proves that[ L m ] = [ L k ] implies m = k for 0 < q <
1, and the q = 1 case is well known, we thus arriveat the following: Theorem 3.1.
The spectral subspaces L nm are pairwise stably non-isomorphic as finitelygenerated projective left modules over C ( C P nq ) . That is, for all n ∈ N \ { } and any m, k ∈ Z , we have [ L nm ] = [ L nk ] if and only if m = k. Observe that, for 0 < q <
1, this fact was already proven in [8, Prop. 4 and 5] usingan index pairing between the K-homology group K ( C ( C P nq )) and the K-theory group K ( C ( C P nq )).To use the index pairing, for the rest of this section we assume 0 < q <
1. Denote by[ µ ] , . . . , [ µ n ] ∈ K ( C ( C P nq )) the K-homology generators constructed in [8, Sec. 2]. In the SSOCIATED VECTOR BUNDLES OVER QUANTUM PROJECTIVE SPACES 5 same paper, the authors proved that (cid:104) [ µ k ] , [ L nm ] (cid:105) = (cid:18) mk (cid:19) (3.1)for all m ∈ N and for all 0 ≤ k ≤ n . Here we adopt the convention that (cid:0) mk (cid:1) := 0 when k > m . Furthermore, by [2, Prop. 3.2], (cid:104) [ µ ] , [ L nm ] (cid:105) = 1 and (cid:104) [ µ ] , [ L nm ] (cid:105) = m (3.2)for all m ∈ Z . We view the pairing with [ µ ] as computing the rank, and the pairing with[ µ ] as computing the noncommutative first Chern class. In agreement with the classicalsetting, we call L n − the section module of the noncommutative tautological line bundle,and we refer to L n as the section module of the dual noncommutative tautological linebundle. The latter is also known as the noncommutative Hopf line bundle.As mentioned in the introduction, the K-theory groups of quantum projective spaceshave the same bases as their classical counterparts. This was proven in [2] by showingthat, for 0 ≤ j ≤ n and for 0 ≤ k ≤ n , (cid:104) [ µ k ] , t j (cid:105) = j (cid:54) = k ( − j for j = k . (3.3)Here t := [1] − [ L n ] is the noncommutative Euler class of L n .We conclude this section by computing the pairings of the K-homology generators withthe class of the section module of the noncommutative tautological line bundle L n − . Proposition 3.2.
For all positive integers n and for all ≤ k ≤ n , we have (cid:104) [ µ k ] , [ L n − ] (cid:105) = ( − k . Proof.
The cases k = 0 , ≤ k ≤ n , we use the identity[ L n − ] = [ L n ] − = (1 − t ) − = 1 + t + · · · + t n in Z [ t ] /t n +1 (3.4)together with the pairings in (3.3). (cid:4) Vector bundles
Although the K -groups of the Vaksman–Soibelman quantum projective spaces aredetermined by associated noncommutative line bundles, expressing the generators as as-sociated noncommutative vector bundles has its merits, as explained in the introduction.For every m = 2 , . . . , n , consider the following surjection of Hopf algebras π m : O ( SU q ( m )) −→ O ( U (1)) , π m ( U ij ) := δ ij u − i < nδ ij u m − i = n , (4.1)where U ij are the matrix coefficients of the fundamental representation of SU q ( m ). Toprove that π m is well defined, we have to verify that the determinant formulae (1.17) and(1.18) in [25] are satisfied. This can be done by a direct computation taking advantage of F. ARICI, P. M. HAJAC, AND M. TOBOLSKI the fact that u is unitary and that π m assigns zero to off-diagonal entries. With the helpof π m , we define the prolongations of principal comodule algebras: P U (1) ( C ( S n +1 q )) O ( U (1)) (cid:3) O ( SU q ( m )) . Next, taking the fundamental representation V m of SU q ( m ), we construct the associatedfinitely generated projective left C ( C P nq )-modules F nm := P U (1) ( C ( S n +1 q )) O ( U (1)) (cid:3) O ( SU q ( m )) O ( SU q ( m )) (cid:3) V m . (4.2)We are now ready to prove our main result: Theorem 4.1.
For any positive integer n and < q ≤ , the classes E n := [1] , E n := [ L n ] − [1] , E nm := [ F nm ] − m [1] , m = 2 , . . . , n, form a basis of K ( C ( C P nq )) .Proof. To begin with, plugging in the explicit formula (4.1) into the definition (4.2), wederive the decomposition F nm = ( L n − ) ⊕ ( m − ⊕ L nm − . Combining (1.1) and (1.2), we know that { , t, . . . , t n } is a basis of K ( C ( C P nq )). We nowuse (3.4) to write E nm in the above basis E n = 1 , E n = − t, E nm = n (cid:88) k =2 (cid:18) ( m −
1) + ( − k (cid:18) m − k (cid:19)(cid:19) t k , m ≥ . (4.3)It remains to show that the matrix M n implementing (4.3) is invertible over the integers.Since M n +1 always contains M n as an n × n submatrix in the upper-left corner, it isstraightforward to verify the claim using elementary row operations and induction. (cid:4) For n = 2, the above theorem yields[1] , [ L ] − [1] , [ L − ⊕ L ] − K ( C ( C P q )). This agrees with the K-theory computations done in [9] forthe quantum complex projective plane. Note also that the matrix M n used in the aboveproof agrees up to a sign with the matrix of pairings (cid:104) [ µ k ] , E nj (cid:105) that can be computedusing (3.1) and Proposition 3.2. Acknowledgements
This work is part of the project Quantum Dynamics partially supported by EU-grantH2020-MSCA-RISE-2015-691246 and Polish Government grants 3542/H2020/2016/2 and328941/PnH/2016. We acknowledge a substantial logistic support from the Max PlanckInsitute for Mathematics in the Sciences in Leipzig, where much of this research wascarried out. It is a pleasure to thank Nigel Higson for discussions and references.
SSOCIATED VECTOR BUNDLES OVER QUANTUM PROJECTIVE SPACES 7
References [1] J. F. Adams. Vector Fields on Spheres.
Ann. of Math.
75 (1962), 603–632.[2] F. Arici, S. Brain, and G. Landi. The Gysin sequence for quantum lens spaces.
J. Noncommut.Geom.
J. Noncommut. Geom.
10 (2016), 29–64.[5] M. F. Atiyah and J. A. Todd. On complex Stiefel manifolds.
Math. Proc. Cambridge Philos. Soc.
Topology
Doc. Math.
22 (2017), 825–849.[8] F. D’Andrea and G. Landi. Bounded and unbounded Fredholm modules for quantum projectivespaces.
J. K-theory
K-theory
21 (2000),141–150.[11] P. M. Hajac, A. Kaygun, B. Zieli´nski. Quantum complex projective spaces from Toeplitz cubes.
J.Noncommut. Geom.
J.Noncommut. Geom.
J. Noncommut. Geom. , in press.[14] P. M. Hajac and J. Rudnik. Noncommutative bundles over the multi-pullback quantum complexprojective plane.
New York J. Math.
23 (2017), 295–313.[15] P. M. Hajac, A. Rennie, and B. Zieli´nski. The K-theory of Heegaard quantum lens spaces.
J. Non-commut. Geom.
Transac. Amer. Math. Soc.
131 (1968),323–332.[17] L. Hodgkin. On the K-theory of Lie groups.
Topology
K-Theory: an Introduction . Grundlehren der math. Wiss. 226, Springer, 1978.[19] J. Milnor.
Introduction to algebraic K -theory . Annals of Mathematics Studies, No. 72. PrincetonUniversity Press, 1971.[20] W. L. Paschke. K -Theory for actions of the circle group on C ∗ -algebras. J. Oper. Theory K -theory of the triple-Toeplitz deformation of the complex projective plane. BanachCenter Publ.
98 (2012), 303–310.[22] W. Szyma´nski. Quantum lens spaces and principal actions on graph C ∗ -algebras. Banach CenterPubl.
61 (2003), 299–304.[23] K. H. Ulbrich. Vollgraduierte Algebren,
Abh. Math. Sem. Univ. Hamburg
51 (1981), 136–148.
F. ARICI, P. M. HAJAC, AND M. TOBOLSKI [24] L. L. Vaksman and Ya. S. Soibelman. Algebra of functions on the quantum group SU(n + 1) andodd-dimensional quantum spheres.
Algebra i Analiz N ) group. Inv. Math.
93 (1988), 35–76.(F. Arici)
Max-Planck-Institut f¨ur Mathematik in den Naturwissenschaften, Inselstr. 22,04103 Leipzig, Leipzig, Germany.
E-mail address : [email protected] (P. M. Hajac) Instytut Matematyczny, Polska Akademia Nauk, ul. ´Sniadeckich 8, Warszawa,00-656 Poland
E-mail address : [email protected] (M. Tobolski) Instytut Matematyczny, Polska Akademia Nauk, ul. ´Sniadeckich 8, Warszawa,00-656 Poland
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