The notion of monoidal equivalence for compact quantum groups was recently introduced by Bichon, De Rijdt and Vaes. In this paper we prove that there is a natural bijective correspondence between actions of monoidally equivalent quantum groups on unital $C^{*}$-algebras or on von Neumann algebras. This correspondence turns out to be very useful to obtain the behavior of Poisson and Martin boundaries under monoidal equivalence of quantum groups. Finally, we apply these results to identify the Poisson boundary...

Let $U$ be the two-parameter quantized enveloping algebra $U_{r,s}\left({\mathrm{𝔰𝔩}}_2\right)$ and $F\left(U\right)$ the locally finite subalgebra of $U$ under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of $F\left(U\right)$ in the case when $r{s}^{-1}$ is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of $U$ by generators.

We construct bar-invariant $\mathbb{Z}\left[q^{\pm 1/2}\right]$-bases of the quantum cluster algebra of the valued quiver $A_2^{\left(2\right)}$, one of which coincides with the quantum analogue of the basis of the corresponding cluster algebra discussed in P. Sherman, A. Zelevinsky: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., 4, 2004, 947–974.

Let $H_{m,n}$ be the $m{n}^2$-dimensional Radford Hopf algebra over an algebraically closed field of characteristic zero. We give the classification of all ideals of $8$-dimensional Radford Hopf algebra $H_{2,2}$ by generators.

We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra $U_q\left(s{l}_2\left(ℂ\right)\right)$. We discuss two cases, according to whether the parameter $q$ is a root of unity. We show that the universal enveloping algebra of $s{l}_2\left(ℂ\right)$ embeds in a non-principal ultraproduct of $U_q\left(s{l}_2\left(ℂ\right)\right)$, where $q$ varies over the primitive roots of unity.

The half-liberated orthogonal group $O_n^{*}$ appears as intermediate quantum group between the orthogonal group $O_n$, and its free version $O_n^{+}$. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between $O_n^{*}$ and $U_n$, a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that...

We analyse perturbative expansions of the invariants defined from unitary representations of the Quantum Lorentz Group in two different ways, namely using the Kontsevich Integral and weight systems, and the R-matrix in the Quantum Lorentz Group defined by Buffenoir and Roche. The two formulations are proved to be equivalent; and they both yield ℂ[[h]]h-valued knot invariants related with the Melvin-Morton expansion of the Coloured Jones Polynomial.

We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_q\left(2\right)$, which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_j:=I_j\otimes \alpha \otimes I_{n-2-j}$ on ${\left(ℂ³\right)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra ${\mathscr{H}}_{q,n}\left(2\right)$ associated with the quantum group $U_q\left(2\right)$. The purpose of this note is to present the construction.

We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups.

Let $\Lambda$ be a preprojective algebra of type $A,D,E$, and let $G$ be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories $\mathrm{Sub}Q$ for $Q$ an injective $\Lambda$-module, and we introduce a mutation operation between complete rigid modules in $\mathrm{Sub}Q$. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to $G$.

We describe how the constructions of quantum homogeneous spaces using infinitesimal invariance and quantum coisotropic subgroups are related. As an example we recover the quantum 4-sphere of [2] through infinitesimal invariance with respect to ${}_q\left(SU\left(2\right)\right)$.

We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group $S{U}_q\left(2\right)$ of Woronowicz. As an illustration of this result we determine the K-groups of quantum automorphism groups of simple matrix algebras.

We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if ${}_{\Gamma }$ is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space $k{\oplus }_{\Gamma }$ is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra $U{(}_{\Gamma })$ is a bialgebra in the category of A-comodules.

A list of known quantum spheres of dimension one, two and three is presented.

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if G is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra A, then the algebra of invariants $A^G$ is finitely generated. We also prove that in characteristic 0 a quantum group G is geometrically reductive if and only if every rational G-module is semisimple, and that in...