Related papers: Introduction to quantum groups
Starting from a Hopf algebra endowed with an action of a group G by Hopf automorphisms, we construct (by a twisted double method) a quasitriangular Hopf G-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular…
The notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a…
We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear…
We introduce the framework of Hopf algebra field theory (HAFT) which generalizes the notion of group field theory to the quantum group (Hopf algebra) case. We focus in particular on the 3d case and show how the HAFT we considered is…
We introduce a notion of partial algebraic quantum group. This is an important special case of a weak multiplier Hopf algebra with integrals, as introduced in the work of Van Daele and Wang. At the same time, it generalizes the notion of…
Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set…
We describe a topological ribbon Hopf algebra whose elements are sequences of matrices. The algebra is a quantum version of U(sl_2).
Discrete quantum groups were introduced as duals of compact quantum groups by Podle\'s and Woronowicz in 1990. Shortly after, they were defined and studied intrinsically by Effros and Ruan, and by this author. In 1998, with the introduction…
This tutorial is intended to give an accessible introduction to Hopf algebras. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics,…
We provide a classification of compact quantum groups, which can be obtained by the Woronowicz construction, when the arrays used in the twisted determinant condition are extensions of functions on permutations. General properties of such…
We investigate the theory of induction in the setting of doubles of coideal $*$-subalgebras of compact quantum group Hopf $*$-algebras. We then exemplify parts of this theory in the particular case of quantum $SL(2,\mathbb{R})$, and compute…
Given a Hecke symmetry $R$, one can define a matrix bialgebra $E_R$ and a matrix Hopf algebra $H_R$, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to $R$. We show that for an even…
We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now well-known construction for Hopf algebras, first studied by G. I.…
We introduce the notion of characters of comodules over coribbon Hopf algebras. The case of quantum groups of type $A_n$ is studied. We establish a characteristic equation for the quantum matrix and a q-analogue of Harish-Chandra-…
In this Letter, we introduce the Hopf algebra structure of the quantum quaternionic group GL(1,H$_q)$ and discuss the isomorphism between the quantum symplectic group SP$_q(1)$ and the quantum unitary group SU$_q(2)$.
Coquasitriangular universal ${\cal R}$ matrices on quantum Lorentz and quantum Poincar\'e groups are classified. The results extend (under certain assumptions) to inhomogeneous quantum groups of [10]. Enveloping algebras on those objects…
The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier…
By using twist construction, we obtain a quantum groupoid $\cald\ot_{q}\uqg$ for any simple Lie algebra $\frakg$. The underlying Hopf algebroid structure encodes all the information of the corresponding elliptic quantum group-the quasi-Hopf…
We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups $G$ such that every element of $G$ is contained in a compact open normal subgroup of $G$. For…
Intrinsic Hopf algebra structure of the Woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups -- coordinate functions, differential…