Lambe, Larry A.; Radford, David E. Algebraic aspects of the quantum Yang-Baxter equation. (English) Zbl 0778.17009 J. Algebra 154, No. 1, 228-288 (1993). For a finite-dimensional vector space \(M\) over a field \(k\) it was shown that a solution of the quantum Yang-Baxter equation \(R: M\otimes M\to M\otimes M\) can be derived from a left \(H\)-module structure and a right \(H\)-comodule structure on \(M\) for some bialgebra \(H\) over \(k\). The module and comodule structure satisfy a natural compatibility condition. \(M\) together with this structure is called a left quantum Yang-Baxter \(H\)- module.In this paper the category \(_ H\)YB of left quantum Yang-Baxter \(H\)-modules is studied. Many constructions in the module category \(_ H\)M are generalised to the category \(_ H\)YB. It is shown that the tensor product of objects of \(_ H\)YB can be regarded as an object of \(_ H\)YB in two different ways. A duality between finite-dimensional left and right quantum Yang-Baxter modules is established. The definition of graded Yang-Baxter equation and natural examples arising from homological algebra are given. The \(A(R)\) construction of Faddeev, Reshetikhin and Takhtadzhan is considered in connection with quantum Yang-Baxter modules. Module algebra and comodule algebra structures are used in the calculus for \({\mathcal U}_ q(\text{sl}_ 2)\). Two families of Yang-Baxter solutions for the quantum plane \(k[x,y]_ q\) are constructed. Reviewer: A.S.Dzhumadil’daev (Alma-Ata) Cited in 34 Documents MSC: 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W30 Hopf algebras (associative rings and algebras) (MSC2000) Keywords:quantum deformations; bialgebras; Hopf algebras; left quantum Yang-Baxter \(H\)-modules; graded Yang-Baxter equation; comodule algebra; quantum plane PDFBibTeX XMLCite \textit{L. A. Lambe} and \textit{D. E. Radford}, J. Algebra 154, No. 1, 228--288 (1993; Zbl 0778.17009) Full Text: DOI Link