On the automorphisms of crossed modules

Abstract

The concept of crossed module was introduced by J. H. C. Whitehead on his combinatorial homotopy theory work. Crossed modules and its applications have very important roles in category theory, homology, cohomology of groups, homotopy theory, algebra, k-theory and etc. At the same time Yang-Baxter equation has fundamental role in various areas such as apparently distant fields as statistical mechanics, particle physics, quantum field theory and quantum groups. Most suitable applications of Yang-Baxter equation are crossed modules; where actor crossed module of algebroids was detailed by Alp; Nilpotent, Solvable, n− Complete and Representations of crossed modules were studied by Dehghanizadeh and Davvaz. In this paper, these two important tools are combined to solve Yang-Baxter equation and also some concepts of center, central automorphisms, n− central automorphism groups for crossed modules are described and some results are obtained.