Integrability and Fusion Algebra for Quantum Mappings

TL;DR

This paper develops a fusion-based approach to quantum Yang-Baxter algebras, constructing higher-order invariants for integrable quantum mappings and illustrating the structure with Gel'fand-Dikii hierarchies.

Contribution

It introduces a general fusion procedure for quantum invariants in integrable quantum mappings, expanding the algebraic understanding of these systems.

Findings

Constructed higher-order quantum invariants for integrable systems.

Demonstrated the Yang-Baxter structure of Gel'fand-Dikii mappings.

Provided explicit commuting families of quantum invariants.

Abstract

We apply the fusion procedure to a quantum Yang-Baxter algebra associated with time-discrete integrable systems, notably integrable quantum mappings. We present a general construction of higher-order quantum invariants for these systems. As an important class of examples, we present the Yang-Baxter structure of the Gel'fand-Dikii mapping hierarchy, that we have introduced in previous papers, together with the corresponding explicit commuting family of quantum invariants.