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In this paper we study weighted versions of Fourier algebras of compact quantum groups. We focus on the spectral aspects of these Banach algebras in two different ways. We first investigate their Gelfand spectrum, which shows a connection…

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group $G(d,1,n)$. The construction of the category follows the decomposition of the Fourier matrix…

We introduce the first provably efficient algorithm to check if a finitely generated subgroup of an almost simple semi-simple group over the rationals is Zariski-dense. We reduce this question to one of computing Galois groups, and to this…

We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if $L^\infty(\mathbb{G})$ contains non-zero compact operators on $L^2(\mathbb{G})$. As a corollary we classify all discrete quantum groups among…

Let $X$ be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that $X$ is equipped with several non-degenerate fixed point free $SL_2$-actions satisfying some mild additional assumption. Then…

Let $G$ be a non-discrete LCA group with the dual group $\Gamma$. We define generalized group algebra, ${\mathcal L}^1(G)$, and generalized measure algebra, ${\mathcal M}(G),$ on $G$ as generalizations of the group algebra $L^1(G)$ and the…

We prove that amenability of a unitary co-representation $U$ of a locally compact quantum group passes to unitary co-representations that weakly contain $U$. This generalizes a result of Bekka, and answers affirmatively a question of…

This is a survey of recent results on classification of compact quantum groups of Lie type, by which we mean quantum groups with the same fusion rules and dimensions of representations as for a compact connected Lie group $G$. The…

In this paper, we prove that for any semisimple simply connected algebraic group $G$, for any regular dominant character $\lambda$ of a maximal torus $T$ of $G$ and for any element $\tau$ in the Weyl group $W$, the character $e^{\rho}\cdot…

In this paper we compute the K-theory (algebraic and topological) and entire periodic cyclic homology for compact quantum groups, define Chern characters between them and show that the Chern characters in both topological and algebraic…

This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative…

A locally compact groupoid is said to have the weak containment property if its full $C^*$-algebra coincides with its reduced one. This property is strictly weaker than amenability and is known to be equivalent to amenability for…

Let $G$ be a locally compact group with the group algebra $L^1(G)$ and $N$ be a closed normal subgroup of $G$. Suppose that $\xi:N\to\mathbb{T}$ is a continuous character and $L_\xi^1(G,N)$ is the $L^1$-space of all covariant functions of…

The Banach algebra LUC(G)* associated to a topological group G has been of interest in abstract harmonic analysis. A number of authors have studied the topological centre of LUC(G)*, which is defined as the set of elements in LUC(G)* for…

A locally compact group $ G $ is discrete if and only if the Fourier algebra $ A(G) $ has a non-zero (weakly) compact multiplier. We partially extend this result to the setting of ultraspherical hypergroups. Let $H$ be an ultraspherical…

We expose a K-theoretic approach to study group C*-algebras and C*-algebraic compact quantum groups: 1. The conception of multidimensional geometric quantization and the index of group C*-algebras; 2. the entire homology of noncommutative…

The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on $\mathbb{G}$ and $\widehat{\mathbb{G}}$ act on the level of direct…

Let $G$ be a locally compact group, and let $1\le p < \infty$. Consider the weighted $L^p$-space $L^p(G,\omega)=\{f:\int|f\omega|^p<\infty\}$, where $\omega:G\to \mathbb R$ is a positive measurable function. Under appropriate conditions on…

Quantum groups of semisimple Lie algebras at roots of unity admit several different forms. Among them is the De Concini-Kac form, which is the easiest to define but, perhaps, hardest to study. In this paper, we propose a suitable…

We show that (central) Cowling-Haagerup constant of discrete quantum groups is multiplicative, which extends the result of Freslon to general (not necesarilly unimodular) discrete quantum groups. The crucial feature of our approach is…