Narrow Results

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group ${Ū}_q({𝔰𝔩}_2)$. This is a diagrammatic description of $En{d}_{{Ū}_q({𝔰𝔩}_2)}(X^{\otimes n})$, where $X:={{𝒳}}_2^{+}$ is a two-dimensional ${Ū}_q({𝔰𝔩}_2)$ module.

We show that the left regular representation π_l of a discrete quantum group (A, Δ) has the absorbing property and forms a monoid in the representation category Rep(A, Δ).

Next we show that an absorbing monoid in an abstract tensor *-category gives rise to an embedding functor (or fiber functor) , and we identify conditions on the monoid, satisfied by , implying that E is *-preserving.

As is well-known, from an embedding functor the generalized Tannaka theorem produces a discrete quantum group (A, Δ) such that . Thus, for a C*-tensor category with conjugates and irreducible unit the following are equivalent: (1) is equivalent to the representation category of a discrete quantum group (A, Δ), (2) admits an absorbing monoid, (3) there exists a *-preserving embedding functor .

We present explicit examples of finite tensor categories that are C₂-graded extensions of the corepresentation category of certain finite-dimensional non-semisimple Hopf algebras.

We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that ${\bigcup }_{k\ge 1}{M}_k(D)$ admits a factor trace if and only if $D\in \{2cos(\pi /n)+1|n\ge 3\}\cup [3,\infty )$ and the higher commutants of these factors depend on $D$. Then a family of irreducible bimodules over these factors is constructed. A tensor category with $A_n$ fusion rule is obtained from these bimodules.

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to two-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

It is known that there are exactly five natural products, which are universal products fulfilling two normalization conditions simultaneously. We classify universal products without these extra conditions. We find a two-parameter deformation of the Boolean product, which we call (r, s)-products. Our main result states that, besides degnerate cases, these are the only new universal products. Furthermore, we introduce a GNS-construction for not necessarily positive linear functionals on algebras and study the GNS-construction for (r, s)-product functionals.

Given a discrete quantum group $H$ with a finite normal quantum subgroup $G$, we show that any positive, possibly unbounded, harmonic function on $H$ with respect to an irreducible invariant random walk is $G$-invariant. This implies that, under suitable assumptions, the Poisson and Martin boundaries of $H$ coincide with those of $H/G$. A similar result is also proved in the setting of exact sequences of C${}^{\ast }$-tensor categories. As an immediate application, we conclude that the boundaries of the duals of the group-theoretical easy quantum groups are classical.

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group $O_n^{+}$, the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.

We give a complete classification of pointed fusion categories over $ℂ$ of global dimension $p^3$ for $p$ any odd prime. We proceed to classify the equivalence classes of pointed fusion categories of dimension $p^3$ and we determine which of these equivalence classes have equivalent categories of modules.

For a finite tensor category $\mathcal{C}$, the action functor $\rho :\mathcal{C}\to \mathrm{Re}\text{x}\left(\mathcal{C}\right)$ is defined by ρ(X) = X ⨂ (−) for $X\in \mathcal{C}$, where $\mathrm{Re}\text{x}\left(\mathcal{C}\right)$ is the category of linear right exact endofunctors on $\mathcal{C}$. We show that ρ has a left and a right adjoint and demonstrate that adjoints of ρ are useful for dealing with certain (co)ends in $\mathcal{C}$. As an application, we give relations between some ring-theoretic notions and particular (co)ends.

We report, from an algebraic point of view, on some methods and results on the classification problem of fusion categories over an algebraically closed field of characteristic zero.

This article is a summary of our talk in QBIC2010. We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group in a Hilbert space with equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group in a finite von Neumann algebra , we show that the set of all generators of strongly continuous one-parameter subgroups of G forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra of all densely defined closed operators affiliated with from the viewpoint of a tensor category.