Related papers: Infinite-dimensional Compact Quantum Semigroup

In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and…

In this expository article, we give a self-contained introduction to the wonderfully well-behaved class of pseudocompact algebras, focusing on the foundational classes of semisimple and separable algebras. We give characterizations of such…

We analyze Weyl algebra of quantum angular momentum system and construct qubit subalgebra out of it. We show that the commutant of this qubit subalgebra is isomorphic to the original algebra and prove the tensor product structure between…

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv…

We show that any compact quantum group having the same fusion rules as the ones of $SO(3)$ is the quantum automorphism group of a pair $(A, \varphi)$, where $A$ is a finite dimensional $C^*$-algebra endowed with a homogeneous faithful…

An algorithm for the explicit computation of a complete set of primitive central idempotents, Wedderburn decomposition and the automorphism group of the semisimple group algebra of a finite metabelian group is developed. The algorithm is…

For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra L, let T(L) be the vector space of tensors over L equipped with the Ito Hopf algebra structure derived from the associative multiplication in L.…

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H_1, the second term in the…

We define the double quantum affinization $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ of type $\mathfrak{a}_1$ as a topological Hopf algebra. We prove that it admits a subalgebra $\ddot{\mathrm{U}}_q'(\mathfrak a_1)$ whose completion is…

This thesis is devoted to the study of Lie bialgebra and Hopf algebra structures related to certain versions of non-commutative geometry constructed on infinite-dimensional Lie algebras that arise in the context of asymptotic symmetries of…

Let R be a finite Blaschke product of degree at least two with R(0)=0. Then there exists a relation between the associated composition operator C_R on the Hardy space and the C*-algebra associated with the complex dynamical system on the…

In this article we use a parametrized version of the FRT construction to construct two new coquasitriangular Hopf algebras. The first one, $\widehat{SL_q(2)}$, is a quantization of the coordinate ring on affine $SL(2)$. We show that there…

We replace the group of group-like elements of the quantized enveloping algebra $U_q({\frak{g}})$ of a finite dimensional semisimple Lie algebra ${\frak g}$ by some regular monoid and get the weak Hopf algebra ${\frak{w}}_q^{\sf d}({\frak…

We study topological properties of the symmetric inverse topological semigroup of finite transformations $\mathscr{I}_\lambda^n$ of the rank $\leqslant n$. We show that the topological inverse semigroup $\mathscr{I}_\lambda^n$ is…

Let $U_\hbar\mathfrak{g}$ denote the Drinfeld-Jimbo quantum group associated to a complex semisimple Lie algebra $\mathfrak{g}$. We apply a modification of the $R$-matrix construction for quantum groups to the evaluation of the universal…

We consider finite-dimensional Hopf algebras $u$ which admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers,…

Partial actions of groups on C*-algebras and the closely related actions and coactions of Hopf algebras received much attention over the last decades. They arise naturally as restrictions of their global counterparts to non-invariant…

We provide an analog of the Drinfeld quantum double construction in the context of crossed Hopf group coalgebras introduced by Turaev. We prove that, provided the base group is finite, the double of a semisimple crossed Hopf group coalgebra…

We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups.

We show that provided $n\ne 3$, the involutive Hopf *-algebra $A_u(n)$ coacting universally on an $n$-dimensional Hilbert space has enough finite-dimensional representations in the sense that every non-zero element acts non-trivially in…