Algebraic aspects of the quantum Yang-Baxter equation
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 \(_ H\)\textit{YB} of left quantum Yang-Baxter \(H\)-modules is studied. Many constructions in the module category \(_ H\)\textit{M} are generalised to the category \(_ H\)\textit{YB}. It is shown that the tensor product of objects of \(_ H\)\textit{YB} can be regarded as an object of \(_ H\)\textit{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.
scientific article; zbMATH DE number 130515