On the representation theory of an algebra of braids and ties

Abstract

Ryom-Hansen, S (reprint author), Univ Talca, Inst Matemat & Fis, Talca, Chile.We consider the algebra epsilon(n)(u) introduced by Aicardi and Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for epsilon(n)(u) and show that this is faithful. We use it to give a basis of epsilon(n)(u) and to classify its irreducible representations

A braids and ties algebra of type B

“…Also, we give some relations that hold on it (which are, mostly, direct consequence of results from [8] and [18]), and we propose a diagrammatical interpretation for E n , in the sense of [2,Section 6]. In Section 4 we construct two linear bases for E B n readjusting the ideas given in [18]. Similarly as in [8], one of these bases has a technical role, and the other is used for define a Markov trace in the last section.…”

“…Proof. For a) see [18,Corollary 1], and for b) the result follows by applying the defining relations (30) and (28).…”

“…In Section 3, making an analogy with the classical case, we introduce the algebra E B n , which contains the bt-algebra. Also, we give some relations that hold on it (which are, mostly, direct consequence of results from [8] and [18]), and we propose a diagrammatical interpretation for E n , in the sense of [2,Section 6]. In Section 4 we construct two linear bases for E B n readjusting the ideas given in [18].…”

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

“…Definition 2 (See [1,14,3]). The two-parameter bt-algebra, denoted by E n (u, v), is defined by E 1 (u, v) := K and, for n > 1, as the unital associative K-algebra, with unity 1, presented by the braid generators R 1 , .…”

On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

“…65. Let E n (u), u = 0, 1, be the Yokonuma-Hecke algebra, see, e.g., [124] and the literature quoted therein. It is known that the dimension of the Yokonuma-Hecke algebra E n (u) is equal to n!B n , where B n denotes as before the n-th Bell number.…”