Quantum Chains with $GL_q(2)$ Symmetry

TL;DR

This paper introduces a method to construct quantum chains with $C_q(G)$ symmetry, exemplified by a $GL_q(2)$ symmetric chain that interpolates between classical Ising models and satisfies a generalized braid group algebra.

Contribution

It presents a novel approach to build quantum chains with $C_q(G)$ symmetry, expanding beyond the usual $U_q(g)$ framework, and demonstrates a specific $GL_q(2)$ chain with unique properties.

Findings

Constructed a $GL_q(2)$ symmetric quantum chain interpolating between Ising chains.

The Hamiltonian satisfies a generalized braid group algebra.

Provides a new method for quantum chain construction with quantum group symmetries.

Abstract

Usually quantum chains with quantum group symmetry are associated with representations of quantized universal algebras $U_q(g) $ . Here we propose a method for constructing quantum chains with $C_q(G)$ global symmetry , where $C_q(G)$ is the algebra of functions on the quantum group. In particular we will construct a quantum chain with $GL_q(2)$ symmetry which interpolates between two classical Ising chains.It is shown that the Hamiltonian of this chain satisfies in the generalised braid group algebra.