Related papers: The half-quantum group U^>=0

We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson…

By means of the notions of cross product algebras of the theory of quantum groups, in the context of classical Hopf algebra structures, we deduce some known structures of Weyl algebras type (as the Drinfeld quantum double, the restricted…

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…

Given a Hopf algebra $H$ and a projection $H\to A$ to a Hopf subalgebra, we construct a Hopf algebra $r(H)$, called the partial dualization of $H$, with a projection to the Hopf algebra dual to $A$. This construction provides powerful…

We define an algebra $\mathcal{U}_0$ using a simplified set of generators for the quantum toroidal algebra $U_q(sl_{n+1}, tor)$ and show that there exists an epimorphism from $\mathcal{U}_0$ to $U_q(sl_{n+1}, tor)$. We derive a closed…

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 generalize various properties of Yetter-Drinfeld modules over Hopf algebras to quasi-Hopf algebras. The dual of a finite dimensional Yetter-Drinfeld module is again a Yetter-Drinfeld module. The algebra $H_0$ in the category of…

In this note we propose a construction of the Hopf algebra of a complex analog of devided powers of the Weyl generators of a semisimple simply-laced quantum group. Here we consider the generators as positive, self-adjoint operators. In…

In this letter we prove that the unrolled small quantum group, appearing in quantum topology, is a Hopf subalgebra of Lusztig's quantum group of divided powers. We do so by writing down non-obvious primitive elements with the correct…

The classification of finite-dimensional pointed Hopf algebras with group S_3 was finished in "The Nichols algebra of a semisimple Yetter-Drinfeld module", arXiv:0803.2430v1 [math.QA], by Andruskiewitsch, Heckenberger and Schneider: there…

We describe the structure of the Grothendieck ring of projective modules of basic Hopf algebras using a positive integer determined by the composition series of the principal indecomposable projective module.

Jensen, Su, and Yang described the projective indecomposable modules of the $0$-Schur algebra $\mathbf{S}_0(n,r)$ using its geometric realization. In this paper, the simple modules of $\mathbf{S}_0(n,r)$ are identified by computing the tops…

Let ${U}_q(sl_2)$ be the quantized enveloping algebra associated to the simple Lie algebra $sl_2$. In this paper, we study the quantum double $D_q$ of the Borel subalgebra ${U}_q((sl_2)^{\leq 0})$ of ${U}_q(sl_2)$. We construct an analogue…

Let $U_q'(\mathfrak{g})$ be an arbitrary quantum affine algebra of either untwisted or twisted type, and let $\mathscr{C}_{\mathfrak{g}}^0$ be its Hernandez-Leclerc category. We denote by $\mathsf{B}$ the braid group determined by the…

We use braided groups to introduce a theory of $*$-structures on general inhomogeneous quantum groups, which we formulate as {\em quasi-$*$} Hopf algebras. This allows the construction of the tensor product of unitary representations up to…

Let $H$ be a dual quasi-Hopf algebra. In this paper we will firstly introduce all possible categories of Yetter-Drinfeld modules over $H$, and give explicitly the monoidal and braided structure of them. Then we prove that the category…

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 introduce the notion of partial representation of a weak Hopf algebra. We present the universal algebra $H_{par}^w$, which factorizes these partial representations by algebra morphisms. Also, it is shown that $\Hp$ is isomorphic to a…

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…

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories…