Related papers: Categorically Morita equivalent compact quantum gr…

For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined…

We take advantage of the correspondence between pseudogroups and inverse quantal frames, and of the recent description of Morita equivalence for inverse quantal frames in terms of biprincipal bisheaves, to define Morita equivalence for…

We define an extension of the affine Brauer algebra, the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group and it naturally acts on $END_K(X \otimes V^{\otimes k})$ for Orthogonal and Symplectic groups.…

We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group $G$ is strongly-graded-equivalent to the skew group algebra by a product partial action of $G$. As to a…

We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal…

We compare actions on C*-algebras of two constructions of locally compact quantum groups, the bicrossed product and the double crossed product. We give a duality between them as a generalization of Baaj-Skandalis duality. In the case of…

In this paper, we introduce notions called inverse set and inverse correspondence over inverse semigroups. These are analogies of Hilbert $C^*$-modules and \Ccorrs in the $C^*$-algebra theory. We show that inverse semigroups and inverse…

We suggest a new generalization of Pontryagin duality from the category of Abelian locally compact groups to a category which includes all Moore groups, i.e. groups whose irreducible representations are finite-dimensional. Objects in this…

We compute the homotopy groups at each unital abelian C*-algebra $C(T)$ in the Morita $3$-category of abelian C*-algebras, C*-algebras with central maps, C*-correspondences, and adjointable bimodule maps. We describe these groups in terms…

We explicitly show that symmetric Frobenius structures on a finite-dimensional, semi-simple algebra stand in bijection to homotopy fixed points of the trivial SO(2)-action on the bicategory of finite-dimensional, semi-simple algebras,…

In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier…

In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting…

In this paper we show that to a unital associative algebra object (resp. co-unital co-associative co-algebra object) of any abelian monoidal category $\mathcal{C}$ endowed with a symmetric $2$-trace, one can attach a cyclic (resp. cocyclic)…

In this paper, we introduce actions of fusion algebras on unital $C^*$-algebras, and define amenability for fusion algebraic actions. Motivated by S.\ Neshveyev et al.'s work, considering the co-representation ring of a compact quantum…

The Krein--Tannaka duality for compact groups was a generalization the Pontryagin--Van Kampen duality for locally compact abelian groups and a remote predecessor of the theory of tensor categories. It is less known that it found…

The notion of qausi-product actions of a compact group on a C$^*$-algebra was introduced by Bratteli et al. in their attempt to seek an equivariant analogue of Glimm's characterization of non-type I C$^*$-algebras. We show that a faithful…

Given a locally compact group $G$ and a unitary representation $\rho:G\to U({\mathcal H})$ on a Hilbert space ${\mathcal H}$, we construct a $C^*$-correspondence ${\mathcal E}(\rho)={\mathcal H}\otimes_{\mathbb C} C^*(G)$ over $C^*(G)$ and…

Let $\mathcal{A}= \{A_t \}_{t \in G}$ and $\mathcal{B}= \{B_t \}_{t\in G}$ be $C^*$-algebraic bundles over a finite group $G$. Let $C=\oplus_{t \in G}A_t$ and $D=\oplus_{t\in G}B_t$. Also, let $A=A_e$ and $B=B_e$, where $e$ is the unit…

We show that for any compact connected group G the second cohomology group defined by unitary invariant 2-cocycles on \hat G is canonically isomorphic to H^2(\hat{Z(G)};T). This implies that the group of autoequivalences of the C*-tensor…

For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion…