The Heisenberg–Lorentz quantum group

The Hyperoctahedral Quantum Group

We present a definition for free quantum groups. The idea is that these must satisfy S n ⊂ G ⊂ U + n , along with a technical representation theory condition. We work out in detail the case of quantum analogues of the hyperoctahedral group Hn. We first consider the hypercube in R , and show that its quantum symmetry group is in fact a q-deformation of On at q = −1. Then we consider the space fo...

Quantum Group and Quantum Symmetry

This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum group is primarily introduced as a systematic method for solving the Yang-Baxter equation. Quantum group theory is presented within the framework of quantum...

ON QUANTUM GROUP OF UNITARY OPERATORS QUANTUM ‘az + b’ GROUP

The concept of a quantum group of unitary operators is relevant for the theory of non-compact locally compact quantum groups. It plays similar role as the concept of a quantum matrix group in the compact case. To show the usefulness of this notion we present an approach to a construction of quantum ‘az + b’ group based on this idea. A brief survey of the present status of quantum group theory i...

QUANTUM GROUP Categorical Models of Quantum Circuits

Quantum Spin Chains with Quantum Group Symmetry

We consider actions of quantum groups on lattice spin systems. We show that if an action of a quantum group respects the local structure of a lattice system, it has to be an ordinary group. Even allowing weakly delocalized (quasi-local) tails of the action, we find that there are no actions of a properly quantum group commuting with lattice translations. The non-locality arises from the orderin...