Solutions of Yang-Baxter equation in an endomorphism semigroup and quasi-(Co)braided almost bialgebras

The aim of this paper is to study the solutions of the Yang-Baxter equation in the endomorphism semigroup of the tensor product of a vector space. As preparation, we introduce the concepts of quasi-braided almost bialgebra (see also [10]) and quasi-cobraided almost bialgebra, and discuss some of their properties. In particular, it is shown that the quasi-R-matrix R of every quasi-braided almost weak Hopf algebra is regular under von Neumann's meaning. The solutions of the Yang-Baxter equation in the endomorphism semigroups are constructed respectively from every quasi-braided almost bialgebra and every quasi-cobraided almost bialgebra. As examples, we explain how to build solutions of the Yang-Baxter equation from some weak Hopf algebras and all Clifford monoids. Finally, the FRT construction is given so-as to build every solution of the Yang-Baxter equation from a quasi-cobraided bialgebra.