Related papers: Representations of the small quasi-quantum group $…

A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for…

Let g be a complex semisimple Lie algebra, tau a point in the upper half-plane, and h a complex deformation parameter such that the image of h in the elliptic curve E_tau is of infinite order. In this paper, we give an intrinsic definition…

In this paper, we study the quantum virtual Grothendieck ring, denoted by $\frakK_q(\g)$, which was introduced in [39], and further investigated in [26, 25]. Our approach involves examining this ring from two perspectives: first, by…

We show that the representation category of the quantum group of a non-degenerate bilinear form is monoidally equivalent to the representation category of the quantum group SL_q(2), for a well chosen non-zero parameter q. The main…

We consider the quantum algebra $U_q(\mathfrak{sl}_2)$ with $q$ not a root of unity. We describe the finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules from the point of view of the equitable presentation.

Let ($\mathfrak{g},\mathsf{g})$ be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with $\mathsf{g}$ being of simply-laced type. We construct a collection of ring isomorphisms…

The structure of irreducible representations of (restricted) U_q(sl(3)) at roots of unity is understood within the Gelfand--Zetlin basis. The latter needs a weakened definition for non integrable representations, where the quadratic Casimir…

We show that the reduced $\mathrm{SL}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of…

We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation we develop a general method of constructing…

The coadjoint representation of a connected algebraic group $Q$ with Lie algebra $\mathfrak q$ is a thrilling and fascinating object. Symmetric invariants of $\mathfrak q$ (= $\mathfrak q$-invariants in the symmetric algebra $S(\mathfrak…

Every simple Hermitian Lie group has a unique family of spherical representations induced from a maximal parabolic subgroup whose unipotent radical is a Heisenberg group. For most Hermitian groups, this family contains a complementary…

We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups…

We study irreducible spherical unitary representations of the Drinfeld double of a $q$-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In…

Starting from a Hecke $R-$matrix, Jing and Zhang constructed a new deformation $U_{q}(sl_{2})$ of $U(sl_{2})$, and studied its finite dimensional representations in \cite{JZ}. Especically, this algebra is proved to be just a bialgebra, and…

Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view…

This paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is…

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an…

We study Lusztig's theory of cells for quantum affine $\mathfrak{sl}_n$. Using the geometric construction of the quantum group due to Lusztig and Ginzburg--Vasserot, we describe explicitly the two-sided cells, the number of left cells in a…

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…

We show that the quantum invariants arising from typical representations of the quantum group $U_h\mathfrak{sl}(2|1)$ are q-holonomic. In particular, this implies the existence of an underlying field theory for which this family of…