Replacing ordinary tensor products by braided tensor products, the notion of Poincaré duality in KK-theory to the setting of quantum group action is presented. As an example, the standard Podleś sphere is shown to be equivariantly Poincaré dual to itself with respect to the natural action of \(SU_q(2)\) (Th. 6.5). The Drinfeld double of \(SU_q(2)\), appearing as the symmtry group in this case, is the quantum Lorentz group [P. Podleś and S. L. Woronowicz, Commun. Math. Phys. 130, No. 2, 381–431 (1990; Zbl 0703.22018)], a noncompact quantum group built up out of a compact and a discrete part. The authors remark that the additional symmetry of the Podleś sphere which is encoded in the discrete part of the quantum group is not visible classically.
Precisely, equivariant Poincaré duality is defined as follows: Let \(G\) be a regular locally compact quantum group, \(D(G)\) is its Drinfeld double. The two \(G\)-Yetter-Drinfeld algebras (\(G\)-YD-algebras, Def. 3.1) \(P\) and \(Q\) are called \(G\)-equivariantly Poincaré dual to each other if there exists a natural isomorphism
\[ KK^{D(G)}_*(P\boxtimes_G A,B)\cong KK^{D(G)}_*(A, Q\boxtimes_G B) \]
for all \(G\)-YD-algebras \(A\) and \(B\) (Def. 6.1). Here, \(A\boxtimes_GB\) means the braided tensor product of a \(G\)-YD-algebra with a \(G\)-\(C^*\)-algebra \(B\) (Def. 3.3).
The outline of the paper is as follows: In §2, basic definitions and results from the theory of locally compact quantum groups are reviewed and notations in this paper are fixed [cf. J. Kustermans, Int. J. Math. 12, No. 3, 289–338 (2001; Zbl 1111.46311) and J. Kustermans and S. Vaes, Math. Scand. 92, No. 1, 68–92 (2003; Zbl 1034.46067)]. Then, the \(H\)-\(C^*\)-algebra \(\text{res}^G_H(B)\), where \(H\to G\) is a morphism of quantum groups and \(B\) a \(G\)-\(C^*\)-algebra with coaction \(\beta: B\to M(C^r_0(G)\otimes B)\), and \(G\)-\(C^*\)-algebra \(\text{ind}^G_H(B)\), where \(G\) is a strongly regular quantum group, \(H\subset G\) is a closed quantum subgroup and \(B\) is an \(H\)-\(C^*\)-algebra are introduced. \(\text{ind}^G_H\) satisfies
\[ \text{ind}^G_H(B)\cong \text{ind}^G_K\text{ind}^K_H(B),\qquad H\subset K\subset G, \]
by virtue of Vaes’ quantum version of Green’s imprimitivity theorem [Th. 2.6. in S. Vaes, J. Funct. Anal. 229, No. 2, 317–374 (2005; Zbl 1087.22005). The authors say that not only the results, but also the techniques developed in this paper rely at several points on this paper]. It is also shown
\[ \text{ind}^G_H(C^r_0(H)\otimes B)\cong \text{ind}^G_E \text{res}^H_E(B)= C^r_0(G)\otimes B, \]
if \(E\) is the trivial group. By these facts, the functor \(\text{ind}^G_H: H\text{-Alg}\to G\)-Alg is obtained.
§3 deals with YD-\(C^*\)-algebras and braided tensor product. It is remarked that a YD module in the algebraic setting [S. Majid, Foundations of quantum group theory. Cambridge: Cambridge Univ. Press (1995; Zbl 0857.17009)] works with the opposite comultiplication on the dual. If \(G\) is an ordinary locally compact group, then every \(G\)-\(C^*\)-algebra becomes a \(G\)-YD-algebra. By using the Drinfeld double \(D(G)\); \(C^r_0(D(G))= C^r_0(G)\otimes C^*_r(G)\), a \(G\)-YD-\(C^*\)-algebra is the same thing as a \(D(g)\)-\(C^*\)-algebra if \(G\) is a loclly compact quantum group (Prop. 3.2). Let \(G\) be a locally compact quantum group, \(A\), \(B\) are its \(G\)-YD-algebra and \(G\)-\(C^*\)-algebra, respectively. Then, \(*\)-homomorphisms \(\iota_A=\lambda_{12}: A\to \mathbb{L}(\mathbb{H}_G\otimes A\otimes B)\), \(\iota_B= \beta_{13}: B\to \mathbb{L}(\mathbb{H}_G\otimes A\otimes B)\) are defined. The braided tensor product \(A\boxtimes_G B\) is the \(C^*\)-subalgebra of \(\mathbb{L}(\mathbb{H}_G\otimes A\otimes B)\) generated by \(\{\iota_A(a)\iota_B(b)\mid a\in A,b\in B\}\) (Def. 3.3). The braided tensor product is compatible with restriction and induction:
\[ \text{ind}^G_H(A\boxtimes_H \text{res}^G_H(B))\cong \text{ind}^G_H(A)\boxtimes_G B \]
(Th. 3.6). Definition and stablity properties of braided tensor products of Hilbert modules (Prop. 3.7) are also explained in this section.
In §4, the \(S\)-equivariant Kasparov group \(KK^S(A,B)\), \(S\) a Hopf \(C^a\)st-algebra and \(A\), \(B\) are graded \(S\)-\(C^*\)-algebras, is defined according to S. Baaj and G. Skandalis [K-Theory 2, No. 6, 683–721 (1989; Zbl 0683.46048) (Def. 4.1)]. If \(G\) is a regular locally compact quantum group, then there is an alternative definition of \(KK^G(A,B)\), which is given as Th. 4.3 [R. Meyer,, K-Theory 21, No. 3, 201–228 (2000; Zbl 0982.19004)]. By using this definition, the category \(KK^G\) is shown to be a triangulated category [Prop. 4.5. in R. Meyer and R. Nest, Topology 45, No. 2, 209–259 (2006; Zbl 1092.19004)]. Then, applying the braided tensor product, the exterior product in equivariant Kasparove theory is introduced (Prop. 4.8, 4.9). This exterior Kasparov products has analogous properties to the ordinary exterior Kasparov product [Th. 4.10. cf. B. Blackadar, K-Theory for operator algebras, Cambridge: Cambridge Univ. Press (1998; Zbl 0913.46054)].
The statements in §2–§4 are sufficient to give the definition of the equivariant Poincaré duality. But before providing the definition of the Poincaré duality, defintions and constructions related to the compact quantum group \(SU_q(2)\), which are used to show that the Podleś sphere is equivariantly Poincaré dual to itself, is given in §5. Then, in §6, after giving the definition of equivariant Poincaré duality, the existence of a natural isomorphism
\[ KK^{DG_q}_*(C(G_q/T)\boxtimes_{G_q}, A,B)\cong KK^{DG_q}_*(A, C(G_q/T)\boxtimes_{G_q},B), \]
where \(C(G_q/T)\) is the Podleś sphere, is shown via the study of the Dirac operator of \((G_q/T)\). This shows that the Podleś sphere is \(G_q\)-equivariant Poincaré dual to itself (Th. 6.5).