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We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang-Baxter homology with a combinatorial formulation for the boundary map and specializes to N-reduced rack homology when the birack is a rack. We introduce augmented birack 2-cocycle invariants of classical and virtual knots and links and provide examples. [ABSTRACT FROM AUTHOR]

We show that for any compact connected group G the second cohomology group defined by unitary invariant two-cocycles on Ĝ is canonically isomorphic to $H^2(\widehat{Z(G)};{\mathbb T})$. This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to $H^2(\widehat{Z(G)};{\mathbb T})\rtimes {\rm Out}(G)$. We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof-Gelaki and Izumi-Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. We give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analog of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups. [ABSTRACT FROM AUTHOR]

We consider a minimal, free action, φ, of the group ℤd on the Cantor set X, for d ≥ 1. We introduce the notion of small positive cocycles for such an action. We show that the existence of such cocycles allows the construction of finite Kakutani–Rohlin approximations to the action. In the case, d = 1, small positive cocycles always exist and the approximations provide the basis for the Bratteli–Vershik model for a minimal homeomorphism of X. Finally, we consider two classes of examples when d = 2 and show that such cocycles exist in both. [ABSTRACT FROM AUTHOR]