Cell structures for the Yokonuma–Hecke algebra and the algebra of braids and ties

Abstract

We construct a faithful tensor representation for the Yokonuma-Hecke algebra Y r,n , and use it to give a concrete isomorphism between Y r,n and Shoji's modified Ariki-Koike algebra. We give a cellular basis for Y r,n and show that the Jucys-Murphy elements for Y r,n are JM-elements in the abstract sense. Finally, we construct a cellular basis for the Aicardi-Juyumaya algebra of braids and ties.

Two Parameters bt-Algebra and Invariants for Links and Tied Links

“…. , E n−1 satisfying the relations (1)- (8), where σ i is replaced by T i and η i by E i , together with the relations…”

“…The V i 's still satisfy the defining relations (1) to (8), substituting σ i with V i , η i with E i , but equation (10) becomes…”

A braids and ties algebra of type B

“…For short we shall omit the subset of cardinality 1 (single blocks) in the partition. For example, the partition I = ({1, 2, 3}, {4, 6}, {5}, {7}) in P (7), will be simply written as I = ({1, 2, 3}, {4, 6}). Moreover, Supp(I) will be denote the union of non-single blocks of I.…”

“…For a positive integer n, the bt-algebra with parameter u is denoted by E n (u), and its definition is obtained by considering abstractly as the subalgebra of the Yokonuma-Hecke algebra Y d,n := Y d,n (u) generated by the braid generators and the family of idempotents that appear in the quadratic relations of these generators. Thus, there is a natural homomorphism from E n in Y d,n , which is injective for d ≥ n, see [7], cf. [2,Remark 3].…”

“…The bt-algebra have been studied by several researchers lately, which has helped to respond some important questions of its structure. In [7] J. Espinoza and Ryom-Hansen found a cellular basis for the bt-algebra, which is used to obtain an isomorphism theorem between E n and a sum of matrix algebras over certain wreath products. In her Ph.D. tesis [4] E. Banjo got an explicit isomorphism between the specialization E n (1) and the small ramified partition algebra [16], and using it, she determined the complex generic representation of E n .…”