We study the free complexification operation for compact quantum groups, $G\to G^c$. We prove that, with suitable definitions, this induces a one-to-one correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying $G=G^c$.

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.

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k {0, and z, t ∈ X is satisfied...

Motivated by our attempts to construct an analogue of the Dirac operator in the setting of $U_q\left({\mathrm{𝔰𝔩}}_n\right)$, we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra $U_q\left({\mathrm{𝔰𝔩}}_2\right)$. The braided adjoint action is seen to coincide with the ordinary quantum adjoint action, which also follows from the general results of S. Majid.

We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products $H_4\bowtie H_4$. There are three steps in our approach. First, we explicitly describe the set of all matched pairs $(H_4,H_4,▹,◃)$ by proving that, with the exception of the trivial pair, this set is parameterized by the ground field $k$. Then, for...

We consider circulant graphs having $p$ vertices, with $p$ prime. To any such graph we associate a certain number $k$, that we call type of the graph. We prove that for $p\gg k$ the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

We determine the Hochschild homology and cohomology of the generalized Weyl algebras of rank one which are of ‘quantum’ type in all but a few exceptional cases.

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 define algebraic families of (all) morphisms which are purely algebraic analogs of quantum families of (all) maps introduced by P. M. Sołtan. Also, algebraic families of (all) isomorphisms are introduced. By using these notions we construct two classes of Hopf-algebras which may be interpreted as the quantum group of all maps from a finite space to a quantum group, and the quantum group of all automorphisms of a finite noncommutative (NC) space. As special cases three classes of NC objects are...

We propose a definition of a quantised ${\mathrm{𝔰𝔩}}_2$-differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of ${\mathrm{𝔰𝔩}}_2$ are natural examples of such algebras.

The goal of this expository paper is to give a quick introduction to $q$-deformations of semisimple Lie groups. We discuss principally the rank one examples of ${𝒰}_q\left({\mathrm{𝔰𝔩}}_2\right)$, $𝒪\left({\mathrm{SU}}_q\left(2\right)\right)$, $𝒟\left({\mathrm{SL}}_q(2,ℂ)\right)$ and related algebras. We treat quantized enveloping algebras, representations of ${𝒰}_q\left({\mathrm{𝔰𝔩}}_2\right)$, generalities on Hopf algebras and quantum groups, $*$-structures, quantized algebras of functions on $q$-deformed compact semisimple groups, the Peter-Weyl theorem, $*$-Hopf algebras associated to complex semisimple Lie groups and the Drinfeld double, representations...

We give a complete classification of right coideal subalgebras that contain all grouplike elements for the quantum group $U_q^{+}\left({\mathrm{𝔰𝔬}}_{2n+1}\right)$ provided that $q$ is not a root of 1. If $q$ has a finite multiplicative order $t>4$; this classification remains valid for homogeneous right coideal subalgebras of the Frobenius–Lusztig kernel $u_q^{+}\left({\mathrm{𝔰𝔬}}_{2n+1}\right)$. In particular, the total number of right coideal subalgebras that contain the coradical equals $\left(2n\right)!!$; the order of the Weyl group defined by the root system of type $B_n$.