Tensor category equivariant KK-theory
Yuki Arano, Kan Kitamura, Yosuke Kubota
TL;DR
This work develops a tensor category–level generalization of Kasparov KK-theory, denoted $KK^\mathscr{C}$, where symmetries are encoded by a rigid C*-tensor category $\mathscr{C}$. It provides both a C*-algebraic and a module-category formulation of $\mathscr{C}$-actions, defines $\mathscr{C}$-Kasparov bimodules and the Kasparov product, and proves universality and triangulated-structure results for $\mathfrak{KK}^\mathscr{C}$. A key novelty is the weak Morita invariance: if two tensor categories are weakly Morita equivalent, their KK-categories are categorically equivalent, extending Takesaki–Takai duality to this setting. The framework is applied to 3-cocycle twists of discrete groups, yielding a Baum–Connes type theorem for certain twisted group actions and outlining untwisting criteria via cohomological obstructions. Together, these results provide a robust, categorical topological toolkit to study quantum group symmetries, subfactor–inclusion classifications, and BC-type phenomena in a broad, tensor-categorical context.
Abstract
In this paper, we introduce Kasparov's bivariant K-theory that is equivariant under symmetries of a C*-tensor category. It is motivated by some dualities in quantum group equivariant KK-theory, and the classification theory of inclusions of C*-algebras. The fundamental properties of the KK-theory, i.e., the existence of the Kasparov product, Cuntz's picture, universality, and triangulated category structure, hold true in this generalization as well. Moreover, we further prove a new property specific to this theory; the invariance of KK-theory under weak Morita equivalence of the tensor categories. As an example, we study the Baum-Connes type property for $3$-cocycle twists of discrete groups.