Baxterization of GL_q(2) and its application to the Liouville model and some other models on a lattice

TL;DR

This paper extends the Baxterization method to the quantum group GL_q(2) and related algebras, deriving spectral parameter dependent L-matrices, R-operators, and Hamiltonians, with applications to various integrable lattice models including Liouville and Toda models.

Contribution

It introduces a novel Baxterization framework for GL_q(2) and related algebras, connecting algebraic structures to integrable lattice models with explicit Hamiltonians.

Findings

Derived spectral parameter dependent L-matrices and R-operators.

Constructed quantum local Hamiltonians in terms of logarithms of positive operators.

Applied the framework to lattice Liouville, q-DST, Volterra, and Toda models.

Abstract

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 comultiplications. Using these comultiplication structures, we find the related fundamental R-operators in terms of powers of coproducts and also give their equivalent forms in terms of quantum dilogarithms. The corresponding quantum local Hamiltonians are given in terms of logarithms of positive operators. An analogous construction is developed for the q-oscillator and Weyl algebras using that their algebraic and coalgebraic structures can be obtained as reductions of those for the quantum group. As an application, the lattice Liouville model, the q-DST model, the Volterra model, a lattice regularization of the free field, and the relativistic Toda model are considered.