Related papers: Braided quantum $\mathrm{SU}(2)$ group - a case st…
Consider the compact quantum group $U_q(2)$, where $q$ is a non-zero complex deformation parameter such that $|q|\neq 1$. Let $C(U_q(2))$ denote the underlying $C^*$-algebra of the compact quantum group $U_q(2)$. We prove that if $q$ is a…
The representations of the pointed Hopf algebras $U$ and $\su$ are described, where $U$ and $\su$ can be regarded as deformations of the usual quantized enveloping algebras $U_q(\mathfrak{sl}(3))$ and the small quantum groups respectively.…
A ${\mathbb Z}_2$-graded qubit represents an even (bosonic) "vacuum state" and an odd, excited, Majorana fermion state. The multiparticle sectors of $N$, braided, indistinguishable Majorana fermions are constructed via first quantization.…
Let ${\mathfrak g}$ be a finite dimensional complex semisimple Lie algebra. The finite dimensional representations of the quantized enveloping algebra $U_q({\mathfrak g})$ form a braided monoidal category $O_{int}$. We show that the…
In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a…
We describe a method for quantization of Poisson Hopf algebras in $\mathbb Q$-linear symmetric monoidal categories. It is compatible with tensor products and can also be used to produce braided Hopf algebras. The main idea comes from the…
Motivated by our attempts to construct an analogue of the Dirac operator in the setting of $U_q(\mathfrak{sl}_n)$, we write down explicitly the braided coproduct, antipode, and adjoint action for quantum algebra $U_q(\mathfrak{sl}_2)$. The…
In the present paper we prove decomposition formulae for the braided symmetric powers of simple modules over the quantized enveloping algebra $U_q(sl_2)$; natural quantum analogues of the classical symmetric powers of a module over a…
We study braided Hochschild and cyclic homology of ribbon algebras in braided monoidal categories, as introduced by Baez and by Akrami and Majid. We compute this invariant for several examples coming from quantum groups and braided groups.
Fully braided analog of Faddeev-Reshetikhin-Takhtajan construction of quasitriangular bialgebra $A(X,R)$ is proposed. For given pairing $C$ factor-algebra $A(X,R;C)$ is a dual quantum braided group. Corresponding inhomogeneous quantum group…
We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras,…
Global gauge symmetry becomes more intricate in low dimensional QFT. We survey the mathematical concepts leading to the relevant analogues of the (D=4) Doplicher-Haag-Roberts theory of superselection sectors and internal symmetry. We also…
We study the TQFT mapping class group representations for surfaces with boundary associated with the $SU(2)$ gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all…
We introduce a large class of bicovariant differential calculi on any quantum group $A$, associated to $Ad$-invariant elements. For example, the deformed trace element on $SL_q(2)$ recovers Woronowicz' $4D_\pm$ calculus. More generally, we…
We generalise the quantum double construction of Drinfel'd to the case of the (Hopf) algebra of suitable functions on a compact or locally compact group. We will concentrate on the *-algebra structure of the quantum double. If the conjugacy…
We introduce and study symmetric and exterior algebras in braided monoidal categories such as the category O for quantum groups. We relate our braided symmetric algebras and braided exterior algebas with their classical counterparts.
We show that every Lie algebra or superLie algebra has a canonical braiding on it, and that in terms of this its enveloping algebra appears as a flat space with braided-commuting coordinate functions. This also gives a new point of view…
In this article, when G is a locally compact quantum group, we associate to a braided-commutative G-Yetter-Drinfel'd algebra $(N,a,\hat{a})$ equipped with a normal faithful semi-finite weight verifying some appropriate condition, a…
The braided approach to q-deformation (due to the author and collaborators) gives natural algebras $R_{21}u_1Ru_2=u_2R_{21}u_1R$ and $R_{21}x_1x_2=x_2x_1R$ for q-Minkowski and q-Euclidean spaces respectively. These algebras are covariant…
A two parametric deformation of the enveloping Heisenberg algebra ${\cal H}(4)$ which appear as a combination of the standard and a nonstandard quantization given by Ballesteros and Herranz is defined and proved to be Ribbon Hopf algebra.…