Semientwining Structures and Their Applications

Semientwining structures are proposed as concepts simpler than entwining structures, yet they are shown to have interesting applications in constructing intertwining operators and braided algebras, lifting functors, finding solutions for Yang-Baxter systems, and so forth. While for entwining structures one can associate corings, for semientwining structures one can associate comodule algebra structures where the algebra involved is a bialgebra satisfying certain properties. 1. Introduction and Preliminaries Quantum groups appeared as symmetries of integrable systems in quantum and statistical mechanics in the works of Drinfeld and Jimbo. They led to intensive studies of Hopf algebras from a purely algebraic point of view and to the development of more general categories of Hopf-type modules (see [1] for a recent review). These serve as representations of Hopf algebras and related structures, such as those described by the solutions to the Yang-Baxter equations. Entwining structures were introduced in [2] as generalized symmetries of noncommutative principal bundles and provide a unifying framework for various Hopf-type modules. They are related to the so-called mixed distributive laws introduced in [3]. The Yang-Baxter systems emerged as spectral-parameter independent generalization of the quantum Yang-Baxter equation related to nonultra-local integrable systems [4, 5]. Interesting links between the entwining structures and Yang-Baxter systems have been established in [6, 7]. Both topics have been a focus of recent research (see, e.g., [8–13]). In this paper, we propose the concepts of semientwining structures and cosemientwining structures within a generic framework incorporating results of other authors alongside ours. The semientwining structures are some kind of entwining structures between an algebra and a module which obey only one-half of their axioms, while cosemientwining structures are kind of entwining structures between a coalgebra and a module obeying the other half of their axioms. The main motivations for this terminology are the new constructions which require only the axioms selected by us (constructions of intertwining operators and Yang-Baxter systems of type II or liftings of functors), our new examples of semientwining structures, simplification of the work with certain structures (Tambara bialgebras, lifting of functors, braided algebras, and Yang-Baxter systems of type I), the connections of the category of semientwining structures with other categories, and so forth. Let us observe that while for entwining structures one can