Braid Tensor Category And Its Construction

In this article, we focus on the construction of braided tensor category. From the classical theory, we know that the center of monoidal category is braided tensor category. The aim of this article is to extend the existing theory to monoidal Hom-algebra and monoidal Hom-coalgebra and then proceed to get some conclusion respect to Hom.This article starts off with Sweedler Hopf algebra, we first prove that it is self-dual, then calculate it’s R- matrix using the method of undetermined coefficients.Next, this article studies the necessary and sufficient condition of the bimodule over monoidal Hom-algebra and bicomodule over monoidal Hom-coalgebra as braided tensor category. We get the following conclusions: when A is a monoidal Hom-algebra over commutative ring k, then we can conclude that the R-matrix in A(?) A(?) A which satisfies specific conditions have a bijection with the braid of A-bimodule category, and we can prove that the braid is symmetric. Furthermore, the R- matrix gives the solution of the quantum Yang Baxter equation and the braid equation. Duality theory, when C is a monoidal Hom-coalgebra over commutative ring K, then we can conclude that the R-matrix a in C (?) C (?) C which satisfies specific conditions has a bijection with the braid of C-bicomodule category.Finally, we prove the following isomorphisms:MA(?)A≌ yDAe,Z(AMA)≌ YDAe AMA is typical monoidal category, through the center structure we can know that Z(AMA) is braided tensor category. At last, we get that (MA(?)A, -(?)A-,A)is a braided tensor category and the braid is symmetric.