Integrable models and star structures

TL;DR

This paper explores how star structures on Hopf algebras, arising from physical models like S-matrix models and quantum spin chains, relate to hermiticity and their compatibility with algebraic structures, revealing differences between models.

Contribution

It introduces the concept of twisted star structures on Hopf algebras in statistical models and analyzes their properties and implications for physical representations.

Findings

Stars are compatible with Hopf structures in FSMs and QSCs.

Statistical models exhibit twisted star structures that are not Hopf-compatible.

Real representations of twisted star Hopf algebras do not close under usual tensor products.

Abstract

We consider the representations of Hopf algebras involved in some physical models, namely, factorizable S-matrix models (FSM's), one-dimensional quantum spin chains (QSC's) and statistical vertex models (SVM's). These physical representations have definite hermiticity assignments and lead to star structures on the corresponding Hopf algebras. It turns out that for FSM's and the quantum mechanical time-evolution of QSC's the corresponding stars are compatible with the Hopf structures. However, in the case of statistical models the resulting star structure is not a Hopf one but what we call a twisted star. Real representations of a twisted star Hopf algebra do not close under the usual tensor product of representations. We briefly comment on the relation of these results with the Wick rotation.