Ultralocal solutions for quantum integrable non-ultralocal models

Frank Willem Nijhoff, Hans W. Capel

Anton Vladimirovich Zabrodin

Anjan Kundu

This chapter discusses a new type of vertex model will be constructed from the ancestor Lax operator using non-fundamental representations. It examines how these Lax operators can be generated in a systematic way confirming their integrability. The chapter focuses on the ultralocal systems due to their relative simplicity and examines a unifying scheme for generating quantum and statistical integrable models. It describes a new direction of study would be to investigate whether there could be any meaningful statistical model corresponding to the integrable non-ultralocal models. To describe an integrable system with such an involved structure, naturally one can no longer start from the Hamiltonian of the model as is customary in physics, since the Hamiltonian is merely one among many commuting conserved charges. Non-ultralocal models are mostly non-fundamental systems with the infinite dimensional representations defined in some Hilbert space.

Anjan Kundu

The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures.We restrict to quantum and statistical models belonging to trigonometric and rational classes with 2 × 2 Lax operators.The ultralocal models can be classified successfully through their associated quantum algebra and are governed by the Yang-Baxter equation, while the nonultralocal models, the theory of which is still in the developmental stage, allow systematization through the braided Yang-Baxter equation.Along with the known integrable models some possible directions for investigation in this field and generation of such new models are suggested.

LADISLAV HLAVATÝ, ANJAN KUNDU

A scheme suitable for describing quantum non-ultralocal models, including supersymmetric ones, is proposed. Braided algebras are generalized to be used through Baxterization for constructing braided quantum Yang—Baxter equations. Supersymmetric and some known non-ultralocal models are derived in the framework of this approach. As further applications of the scheme, the construction of new quantum integrable non-ultralocal models, like mKdV and anyonic supersymmetric models including deformed anyonic superalgebra, is outlined.

Günter Mahler A