单子理论及其在Hom量子群中的应用

关于单子理论和Hom量子群的讨论是近年来的学术热点，其对研究张量范畴及表示、量子离散型形变具有重要意义。本项目将主要研究双单子理论及其在Hom-Hopf代数上的应用。具体地，拟从如下几个方面着手研究：首先，建立双(余)单子的混合双模范畴中的辫子结构，并将我们的理论应用到Hom量子群上，得到一组QYBE的解系；其次，讨论双(余)单子的smash(余)积体系，作为应用，将引入Hom缠绕结构，并给出Hom化的中心结构定理；最后，将构造新的2-张量范畴，讨论Hom-双代数的范畴化构造，并刻画Hom量子群的规格化变换。

Currently the theory of monads and Hom-quantum groups have been one of the most powerful tools to study the theory of q-deformations, and monoidal categories and their representations. Based on the applicant's work, this proposal will focus on the bimonad theory and the application on Hom-Hopf algebras. Indeed, we will first compute the braided structure in the category of mixed bimodules over a bimonad and a bicomonad, and specially discuss the applications on Hom-Hopf algebras and the solutions of the QYBE. In order to investigate the Hom-entwinings structures and the Hom-type center construction, we will also discuss the smash (co)product system of bi-(co)monads. Finally, we will construct the new 2-monoidal cateogries and study the categorification of Hom-bialgebras, and compute the gauge transformation for Hom-quantum groups.