Noncommutative Balls and Mirror Quantum Spheres

Noncommutative analogues of n-dimensional balls are defined by repeated application of the quantum double suspension to the classical low-dimensional spaces. In the ‘even-dimensional’ case they correspond to the Twisted Canonical Commutation Relations of Pusz and Woronowicz. Then quantum spheres are constructed as double manifolds of noncommutative balls. Both C-algebras and polynomial algebras of the objects in question are defined and analyzed, and their relations with previously known examples are presented. Our construction generalizes that of Hajac, Matthes and Szymański for ‘dimension 2’, and leads to a new class of quantum spheres (already on the C-algebra level) in all ‘even-dimensions’. 0. Introduction Just as classical spheres appear in variety of contexts, their quantum analogues may be studied from many a different perspective. One of the most common strategies is to view them as homogeneous spaces of compact quantum groups [21, 9, 29, 14]. In addition to quantum symmetry considerations, homological approach in the spirit of Connes noncommutative geometry has recently become prominent. Indeed, examples of quantum spheres have been constructed via Chern character techniques [6]. We refer the reader to [7] for an overview of various constructions of quantum spheres. Other noncommutative analogues of classical topological methods have also been used in the study of quantum manifolds, and quantum spheres in particular. Among them, noncommutative analogues of the classical suspension were used explicitly or implicitly by several authors. Quantum double suspension was applied systematically in [15, 3], and noncommutative Heegaard splitting was used in [20, 4, 12, 2]. The main purpose of the present article is to relate quantum spheres to noncommutative balls, and to examine them from two other natural topological perspectives. Firstly, we realize quantum spheres as boundaries of noncommutative balls. Secondly, we construct quantum spheres by gluing as ‘double manifolds’ of noncommutative balls. Even though the latter technique goes back to [27], only recently has it been used to produce new examples of ‘two-dimensional’ mirror quantum spheres [13], and we generalize this approach to ‘higher dimensions’. In Section 2, working with arbitrary unital C-algebras and their generators, we show how to perform the quantum double suspension operation not only on the C-algebra level as in [15] but also on the level of a dense ∗-subalgebra (of polynomial functions). In Sections 3 and 4, we use this procedure to construct noncommutative balls in all ‘dimensions’ via repeated application of the quantum double suspension to a point (‘even dimensions’) and to a closed interval (‘odd dimensions’). In Theorems 3.1 and 4.2, we Date: 18 August, 2006. † Supported by the Korea Research Foundation Grant (KRF-2004-041-C00024). ‡ Partially supported by the KBN grant 2 P03A 013 24, and the European Commission grant MKTDCT-2004-509794. 1 2 HONG AND SZYMAŃSKI present the resulting algebras in terms of convenient generators and relations. Remarkably, it turns out that in the ‘even-dimensional’ case our relations are essentially identical with the Twisted Canonical Commutation Relations of Pusz and Woronowicz [23]. The C-algebra C(B q ) of the noncommutative 2n-ball is generated by n elements z1, . . . , zn. Their commutation relations imply that ∑n i=1 ziz ∗ i ≤ 1. Thus, it is natural to consider the quotient of C(B q ) by the ideal generated by 1 − ∑n i=1 ziz ∗ i as the algebra of functions on the boundary ∂B q of this noncommutative ball. In fact, there is a natural identification of this boundary with the quantum unitary sphere S μ . Similar considerations apply in the ‘odd-dimensional’ case as well, with the boundary ∂B q identified with the Euclidean quantum sphere S μ . In Sections 5 and 6, we construct the noncommutative double manifold S q,β of B n q , by gluing two copies of B q along their common boundary ∂B n q . On the C -algebra level, C(S q,β) is defined by the pull-back C(B n q ) ⊕β C(B n q ) over C(∂B n q ). This construction involves the choice of an automorphism β of C(∂B q ), responsible for the identification of the boundaries of the two noncommutative balls. Polynomial algebras Ø(S q,β) are then defined by a suitable choice of generators inside C(S q,β). In the ‘odd-dimensional’ case, it turns out that the isomorphism class of the C-algebras C(S q,β ) does not depend on the choice of β. Moreover, these glued quantum spheres S q,β can be naturally identified with the unitary quantum spheres (Proposition 6.1). The situation is quite different in the ‘even-dimensional’ case. Indeed, we find an automorphism β of C(∂B q ) such that the C -algebra C(S q,β) is not even stably isomorphic to C(S q,id) (Corollary 5.4). This happens in spite of the fact that these two C -algebras (of type I) have homeomorphic primitive ideal spaces and isomorphic (classical) K-groups (Theorem 5.3). For such a β, we call S q,β mirror quantum sphere. Our construction generalizes that of [13] carried for ‘dimension 2’. While S q,id may be naturally identified with the Euclidean quantum spheres (Proposition 5.1), the mirror quantum spheres S q,β are new (already on the C-algebra level). Finally, in Section 7, irreducible representations of the C-algebras of noncommutative balls B q and of the mirror quantum spheres S 2n q,β are presented. Acknowledgements. The second named author would like to thank Piotr Hajac and the entire team of the program in Noncommutative Geometry and Quantum Groups for their warm hospitality during his stay in Warsaw in March–May 2006. 1. The double of a noncommutative space Let X be a compact manifold with non-empty boundary ∂X. Given a homeomorphism f : ∂X → ∂X of the boundary, the classical topological gluing construction yields a double X ∪f X of X. To translate this picture into the language of C -algebras, let C(X) be the commutative C-algebra of continuous complex-valued functions on X, and let C∂X(X) denote the continuous complex-valued functions on X vanishing on ∂X. Then C∂X(X) is an essential ideal of C(X). If π : C(X) → C(∂X) is the surjection given by restriction then we have an exact sequence of commutative C-algebras (1) 0 −→ C∂X(X) −→ C(X) π −→ C(∂X) −→ 0. The C-algebra C(X∪fX) is isomorphic to the pull-back of C(∂X) along two surjections π : C(X) → C(∂X) and f∗ ◦ π : C(X) → C(∂X), where f∗ : C(∂X) → C(∂X) is the automorphism dual to f . NONCOMMUTATIVE BALLS AND MIRROR QUANTUM SPHERES 3 Remark 1.1. An imbedding of X into a Euclidean space (R or C) gives rise to a dense ∗-subalgebra Ø(X) (the polynomial algebra) of C(X), generated by the restrictions of the coordinate functions to X. Clearly, we have π(Ø(X)) = Ø(∂X). Furthermore, ∂X is the intersection of X with an affine variety if and only if Ø(X)∩C∂X(X) is dense in C∂X(X) in the sup norm topology. In the present article, we are concerned with noncommutative analogues of the aforementioned classical setting. Let A be a unital C-algebra (not necessarily commutative), J be an essential proper ideal of A, and π : A → B = A/J be the natural surjection. Thus, we have an essential extension (2) 0 −→ J −→ A π −→ B −→ 0. Suppose that β is an automorphism of B. Then we define A⊕βA, the double of A, as (3) A⊕β A = {(x, y) ∈ A⊕A : π(x) = (β ◦ π)(y)}. That is, A⊕β A is the C -algebra defined by the pull-back diagram A⊕β A pr2 −−−→ A pr1 