Pullbacks and nontriviality of associated noncommutative vector bundles

Abstract

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finitedimensional representation of the structural quantum group. On the level of K 0 -groups, we realize the induced map by the pullback of explicit matrix idempotents. We also show how to extend our result to the case when the quantum-group representation is infinite dimensional, and then apply it to the Ehresmann-Schauenburg quantum groupoid. Finally, using noncommutative Milnor's join construction, we define quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles. Contents 1. Pushing forward modules associated with Galois-type coactions 4 1.1. Faithfully flat coalgebra-Galois extensions 5 1.2. Principal coactions 8 1.3. The Hopf-algebraic case revisited 10 2. Pulling back noncommutative vector bundles associated with free actions of compact quantum groups 11 2.1. Iterated equivariant noncommutative join construction 12 2.2. Noncommutative tautological quaternionic line bundles and their duals 13Acknowledgments 16 References 16Our result is motivated by the search of K 0 -invariants. The main idea is to use equivariant homomorphisms to facilitate computations of such invariants by moving them from more complicated to simpler algebras. This strategy was recently successfully applied in [19] to distinguish the K 0 -classes of noncommutative line bundles over two different types of quantum complex projective spaces. Herein we 1 1 Keywords: K-theory of C*-algebras, free actions of compact quantum groups, Hopf algebras and faithful flatness, Chern-Galois character, noncommutative join construction and Borsuk-Ulam-type conjecture, quantum quaternionic projective spaces.

Rank-two Milnor idempotents for the multipullback quantum complex projective plane

“…))) −→ K 0 (C(S 2 q )), with q 1 defined in Diagram (4.24). Next, using [10] and [12], we can easily see that [1] and [ L1 ] − [1] generate t(K 0 (C(S 2 q ))). Moreover, taking advantage of Theorem 3.2, we compute that [ L1 ⊕ L−1 ] − 2 [1] generates ∂ 10 (K 1 (C(SU q (2))).…”

“…In this section, we will homotopy the Milnor idempotent p U in the simplified form p U given in Equation (5.22). Let Therefore, by replacing (x, y) by (x t , y t ) in Equation (5.17), we obtain a homotopy t → V t of invertible 2 × 2 matrices with entries in T ⊗ T , with V 0 = V and T )p to (T ⊗ T )f (p), by [12], we obtain a positive-rank free module over (T ⊗ T ). It follows from (5.41) that the rank of C(CP 2 T )p is one, so it is a spectral subspace L n for some integer n. Now we use another U( 1 ).…”

“…). Next, we take advantage of the "pushforwards commute with associations" theorem in [12] to conclude that [1] is the third generator of K 0 (C(CP 2 T )). As before, L 1 ⊕ L −1 can be easily identified with the section module E of the rank-two noncommutative vector bundle associated via the fundamental representation of SU q (2) with a natural SU q (2)prolongation of S 5…”

“…)-equivariant *-homomorphism g : C(S 5 H ) → C(S 3 H ) ⊗ T from Diagram 4.18 to pushforward all spectral subspaces in (5.41) to P 1 :(5.42) [(C(S 3 H ) ⊗ T ) 1 ⊕ (C(S 3 H ) ⊗ T ) −1 ] − 2[1] = ±([(C(S 3 H ) ⊗ T ) n ] − [1]). Then, taking again advantage of[12], we use the equivariant map Ω in Diagram 4.18 to view any spectral subspace (C(S 3 H ) ⊗ T ) k as the image of C(SU q (2)) k . Now, since (by Lemma 4.5) Ω * is an isomorphism, we apply Ω−1 * to (5.43) to obtain(5.43) [C(SU q (2)) 1 ⊕ C(SU q (2)) −1 ] − 2[1] = ±([C(SU q (2)) n ] − [1]).Finally, we compute an index pairing as in [10, Theorem 2.1] to conclude that n = 0.…”

Triviality of equivariant maps in crossed products and matrix algebras

“…More specifically, ΣA is the (non-equivariant) noncommutative join A ⊛ C(Z/2Z), and the action presented in [4] is the diagonal action of Z/2Z. [3,10,1,6]. In [17], we proposed a different type of join (and similarly, unreduced suspension) for C * -algebras with free actions of Z/kZ, replacing the tensor product with a crossed product.…”