An abstract approach to algebras of braids and ties

In the paper [1] Aicardi and Juyumaya studied an algebra En defined by generators and relations; the relations were devised by an abstracting procedure of a non-standard presentation for the Yokonuma–Hecke algebra [10], finalized to build a Markov trace on it [11]. They also provided a diagramatic interpretation of this algebra in terms of braid diagrams on n strands, by adding “ties” which can freely move along the strands. In subsequent papers, the algebra En , named the braids and ties algebra, has been extensively studied, e.g. in connection with link invariants [1, 2] or from a representation theoretic point of view [6, 16]. From its algebraic presentation the algebra En can be viewed as a “type A” object, and indeed (inequivalent) generalizations to type B have been introduced in [8, 13]. The paper [13] has been another source of inspiration for the present paper. Marin introduced an extension CW of the Iwahori–Hecke algebra of a Coxeter system (W , S) and built up a family of generically surjective morphisms of k[B W ] → CW (here B W is the Artin braid group attached to W ). When W is of type A n−1 , it turns out that CW ∼= En .