Related papers: Serre-Lusztig relations for $\imath$quantum groups

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathbf{U}_q(\mathfrak{g})$ denote the associated Drinfel'd Jimbo quantized enveloping algebra. In this paper we study spherical functions of $\mathbf{U}_q(\mathfrak{g})$…

Quantum field theory allows more general symmetries than groups and Lie algebras. For instance quantum groups, that is Hopf algebras, have been familiar to theoretical physicists for a while now. Nowdays many examples of symmetries of…

We prove a general theorem for constructing integral quantum cluster algebras over ${\mathbb{Z}}[q^{\pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster…

Dynamical quantum field theories (QFTs), such as those in which spacetimes are equipped with a metric and/or a field in the form of a smooth map to a target manifold, can be formulated axiomatically using the language of…

Let $A$ be a symmetrizable generalized Cartan matrix, which is not of finite or affine type. Let $\mathfrak{g}$ be the corresponding Kac-Moody algebra over a commutative ring $R$ with $1$. We construct an infinite-dimensional group $G_V(R)$…

We obtain a presentation of quantum Schur algebras (over the field Q(v)) by generators and relations. This presentation is compatible with the usual presentation of the quantized universal enveloping algebra of the Lie algebra gl(2). We…

We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual…

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…

Let ${\mathbf U}^-_q$ be the negative half of the quantum group associated to a Kac-Moody algebra ${\mathfrak g}$, and $\underline{\mathbf U}^-_q$ the quantum group obtained by a folding of ${\mathfrak g}$. Let ${\mathbf A} = {\mathbf…

In this text, we study derived versions of the fusion category associated to Lusztig's quantum group $\textbf{U}_q$. The categories that so arise are non-semisimple but recovers the usual fusion ring when passing to complexified…

In this paper, we shall study the structure of the Grothendieck group of the category consisting of Lusztig's perverse sheaves and give a decomposition theorem of it. By using this decomposition theorem and the geometric realizations of…

We introduce the $\alpha,\beta$-symmetric difference derivative and the $\alpha,\beta$-symmetric N\"orlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generalization of the forward and backward…

We establish PBW type bases for $\imath$quantum groups of arbitrary finite type, using the relative braid group symmetries. Explicit formulas for root vectors are provided for $\imath$quantum groups of each rank 1 type. We show that our PBW…

Let $\mathbf U^{\imath}\equiv\mathbf U^{\imath} (\mathfrak{sl}_2)$ be Letzter's coideal subalgebra of quantum $\mathfrak{sl}_2$ corresponding to the symmetric pair $(\mathfrak{sl}_2(\mathbb C),\mathbb C)$. As a subalgebra of quantum…

The purpose of this paper is to consider some basic constructions in the category of compact quantum groups --for example de case of extensions, of Drinfeld twists, of matched pairs, of extensions, of linked pairs and of cocycle Singer…

We prove the existence of a universal braided compact quantum group acting on a graph $\mathrm{C}^*$-algebra in the category of $\mathbb{T}$-$\mathrm{C}^*$-algebras with a twisted monoidal structure, in the spirit of the seminal work of S.…

Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced.It is shown that, given a groupoid $G\rightrightarrows \Omega$ associated with a…

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…

For the quantum symmetric pair $(\textbf{U}, \textbf{U}^{\imath})$ of type AIII/AIV, we show various positivity properties of the $\imath$-canonical bases on finite-dimensional simple $\textbf{U}$-modules, as well as their tensor product.

We develop invariant theory for the quantum group ${\rm U}_q$ of $G_2$ at generic $q$ in the setting of braided symmetric algebras. Let ${\mathcal A}_m$ be the braided symmetric algebra over $m$-copies of the $7$-dimensional simple ${\rm…