Self-Inverse Yang–Baxter Operators from (Co)Algebra Structures

Abstract

Some attempts to find Yang–Baxter operators in a systematic way have led to the theory of quantum groups (see [4]). Many papers in the literature are devoted to the construction of them (see [5, p. 198]). We adopt the terminology of [5] in giving some relevant examples: the twist, operators on modules over braided bialgebras, and operators on comodules over cobraided bialgebras. The FRT construction showed that any Yang–Baxter operator on a finite dimensional space V can be obtained from a cobraided bialgebra coacting on V . (Other authors might use the term “quasi triangular bialgebras” instead of “braided bialgebras”; see, for example, [4, 7].) All these constructions have proved of great utility in low-dimensional topology. We present here a new method to construct self-inverse Yang–Baxter operators. Thus, to every (co)algebra structure on a space we associate an operator. We describe the algebra (respective coalgebra) structures producing the same operator. Yang–Baxter operators associated to algebras are different from those associated to coalgebras except for a special case. We give characterizations for operators which are associated to (co)algebra structures, and construct the structures which produce them. We give an