Solutions of the braid equation attached to power series

Abstract

Let K be an algebraically closed characteristic 0 field and let V be the K-vector space with basis { x 0 , x 1 , x 2 , … } . In this paper we use the theory developed in [17] to associate with each power series f ∈ K [ [ X ] ] satisfying f ( 0 ) = 1 , an involutive solution s ( f ) : V ⊗ V → V ⊗ V , of the braid equation, and we begin the study of this correspondence. For n ≥ 2 , let Vn be the subspace of V generated by { x 0 , … , x n − 1 } . Each one of the solution s(f) induces by restriction a solution s n ( f ) on V n ⊗ V n . We also begin the study of these solutions.