Related papers: Matrix quantum groups as matrix product operator r…
We develop the Baxterization approach to (an extension of) the quantum group GL_q(2). We introduce two matrices which play the role of spectral parameter dependent L-matrices and observe that they are naturally related to two different…
We review the representation theory of the quantum group $U_\epsilon sl_2\mathbb{C}$ at a root of unity $\epsilon$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories (QHFT). Our…
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,…
The method of geometrical quantization of symplectic manifolds is applied to constructing infinite dimensional irreducible unitary representations of the algebra of functions on the compact quantum group $SU_q(2)$. A formulation of the…
The recently presented quantum antibrackets are generalized to quantum Sp(2)-antibrackets. For the class of commuting operators there are true quantum versions of the classical Sp(2)-antibrackets. For arbitrary operators we have a…
This expository article explains how planar diagrammatics naturally arise in the study of categorified quantum groups with a focus on the categorification of quantum sl2. We derive the definition of categorified quantum sl2 and highlight…
A systematic and compact treatment of arbitrary $su(2)$ invariant spin-$s$ quantum chains with nearest-neighbour interactions is presented. The ground-state is derived in terms of matrix product states (MPS). The fundamental MPS…
Lie bialgebra structures on $e(2)$ are classified. For two Lie bialgebra structures which are not coboundaries (i.e. which are not determined by a classical $r$-matrix) we solve the cocycle condition, find the Lie-Poisson brackets and…
We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions,…
Recently B. Hartwig and the second author found a presentation for the three-point $sl_2$ loop algebra via generators and relations. To obtain this presentation they defined an algebra $\boxtimes$ by generators and relations, and displayed…
There are two approaches to projective representation theory of symmetric and alternating groups, which are powerful enough to work for modular representations. One is based on Sergeev duality, which connects projective representation…
Using face algebras (i.e. algebras of L-operators of IRF models), we construct modular tensor categories with positive definite inner product, whose fusion rules and S-matrices are the same as (or slightly different from) those obtained by…
In the matrix product states approach to $n$ species diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process. The…
This article shows that one can consistently incorporate nonunitary representations of at least one group into the ``ordinary'' nonrelativistic quantum mechanics. This group turns out to be Lorentz group thus giving us an alternative…
Generalized symmetries have emerged as a powerful organizing principle for exotic quantum phases. However, their role in open quantum systems, especially for non-invertible cases, remains largely unexplored. We address this by applying a…
Quantum Teichmuller theory assigns invariants to three-manifolds via projective representations of mapping class groups derived from the representation of a noncommutative torus. Here, we focus on a representation of the simplest…
A new link between tetrahedra and the group SU(2) is pointed out: by associating to each face of a tetrahedron an irreducible unitary SU(2) representation and by imposing that the faces close, the concept of quantum tetrahedron is seen to…
Coherent state operators (CSO) are defined as operator valued functions on G=SL(n,C), homogeneous with respect to right multiplication by lower triangular matrices. They act on a model space containing all holomorphic finite dimensional…
A certain class of unitary representations of U_q(sl(2,R)) has the property of being simultanenously a representation of U_{tilde{q}}(sl(2,R)) for a particular choice of tilde{q}(q). Faddeev has proposed to unify the quantum groups…
We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same…