Chern numbers for two families of noncommutative Hopf fibrations

Abstract

Abstract.We consider noncommutative line bundles associated with the Hopf fibrations of SUq(2) over all Podleś spheres and with a locally trivial Hopf fibration of S 3 pq . These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U (1) with Galois-type extensions encoding the principal fibrations of SUq(2) and S 3 pq . We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them. Abridged versionIn this paper, we combine the algebraic tool of Galois-type extensions with the analytic tool of the noncommutative index formula to study two kinds of examples of quantum fibrations. Our main result is that the line bundles associated to these principal fibrations are mutually non-isomorphic. This gives an estimate of the positive cones of the algebraic K 0 -groups of the base-space quantum spheres.Let B ⊆ P be an inclusion of algebras such that B is the coinvariant subalgebra for some coalgebra C coaction ∆ R : P → P ⊗ C. Using the framework of Galois-type extensions, one can say when such an extension of algebras is principal. (The definition is tuned in such a way that for commutative algebras it coincides with the concept of affine group scheme torsors -the principal bundles of algebraic geometry.) Every principal C-extension B ⊆ P allows one to assign to any finite-dimensional corepresentation of C a finitely generated projective left B-module of colinear homomorphisms Hom C (V ϕ , P ) [2]. Taking its class in K 0 (B) and composing it with the Chern character defines the Chern-Galois character from the space of all finite-dimensional corepresentations of C to the even cyclic homology of B. On the other hand, the K-homology Chern character assigns to finitely summable Fredholm modules cyclic cocycles. In the 1-summable case it takes a particularly simple form, notably it turns a pair of bounded * -representations (ρ 1 , ρ 2 ) into a trace (cyclic 0-cocycle) on B via the formula tr ρ = Tr • (ρ 1 − ρ 2 ). The evaluation of this trace on the Chern-Galois character applied to a corepresentation gives a numerical invariant of the K 0 -class of the module defined by this corepresentation. Moreover, for our examples, the integrality of these invariants (guaranteed by the noncommutative index formula) makes them computable. Here {e k } k≥0 is an orthonormal basis of a separable Hilbert space and f 0 , f 1 are generators of the * -algebra O(S 2 pq ). These representations give a desired trace, and bring us to our first main result: Theorem 0.1 For all µ ∈ Z, the pairing between the cyclic 0-cocycle tr ρ and the K 0 -class of O(S 3 pq ) µ (Chern number) coincides with the winding number µ, i.e., tr ρ , [O(S 3 pq ) µ ] = µ. Our second example is a family of noncommutative Hopf fibrations of SU q (2) over all Podleś quantum spheres S 2 q,s , s ∈ [0, 1]. As handling the generic Podleś spheres requires going beyond the Hopf-Galois theory, they were among main motivating examples driving the development of the theory of pr...

Bijectivity of the canonical map for the non-commutative instanton bundle

“…k ≤ −1 (15) It is a straightforward computation to verify the induction hypothesis for N = 1 and that (12), (13), (14), (15) imply the good definition of τ k,m,n for |k| + m + n = N + 1.…”

“…and the following ones for |k| + m + n ≤ N: q m+n z 2 τ k−1,m,n + q m+n+1 τ k−1,m+1,n z 1 = τ k,m,n z 2 , z 3 τ k−1,m,n − q −k τ k−1,m+1,n z 4 = τ k,m,n z 3 , q m+n z * 1 τ k−1,m,n − q m+n τ k−1,m+1,n z * 2 = τ k,m,n z * 1 , z * 4 τ k−1,m,n + q 1−k τ k−1,m+1,n z * 3 = τ k,m,n z * 4 , k ≥ 1 (14) q m+n z 4 τ k+1,m,n − q m+n+1 τ k+1,m,n+1 z 3 = τ k,m,n z 4 , z 1 τ k+1,m,n + q k τ k+1,m,n+1 z 2 = τ k,m,n z 1 , q m+n z * 3 τ k+1,m,n + q m+n τ k+1,m,n+1 z * 4 = τ k,m,n z * 3 , z * 2 τ k+1,m,n − q 1+k τ k+1,m,n+1 z * 1 = τ k,m,n z * 2 .…”

Noncommutative index theory for mirror quantum spheres

“…Although the index computations for noncommutative line bundles over the generic Podleś and mirror quantum spheres proceed along the same lines (cf. [10]), it is in the structure of Fredholm modules where differences between these spheres are crucial. Indeed, one can take appropriate infinite-dimensional irreducible representations ρ 1 and ρ 2 of C(S 2 pq+ ), and they yield a 1-summable Fredholm module over O(S 2 pq+ ), the universal * -algebra for the defining relations of C(S 2 pq+ ) [16,10].…”

“…[10]), it is in the structure of Fredholm modules where differences between these spheres are crucial. Indeed, one can take appropriate infinite-dimensional irreducible representations ρ 1 and ρ 2 of C(S 2 pq+ ), and they yield a 1-summable Fredholm module over O(S 2 pq+ ), the universal * -algebra for the defining relations of C(S 2 pq+ ) [16,10]. This is in contrast with the situation for mirror quantum spheres, where (ρ + , ρ − ) is not a Fredholm module.…”

A Locally Trivial Quantum Hopf Fibration

“…The pairing of tr and E µ has been computed in [HMS03b,Theorem 3.3] (see [BHMS] for details) for any µ ∈ Z. Since the special case µ = −1 computation is straightforward, we enclose it here for the convenience of the reader.…”