Markov Traces on Affine and Cyclotomic Yokonuma–Hecke Algebras

Abstract

In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can define Markov traces on them, which we in turn use to construct invariants for framed and classical knots in the solid torus. Finally, we study the Markov trace with zero parameters on the cyclotomic Yokonuma-Hecke algebras and determine the Schur elements with respect to that trace. We are grateful to Sofia Lambropoulou and Jesús Juyumaya for their useful suggestions and corrections to this paper. We would also like to thank Vincent Sécherre for our fruitful conversations and his interesting remarks concerning the affine case.

A braids and ties algebra of type B

“…For that, we use the method of relative traces, cf. [2,6], which consists in construct a family of linear maps ϑ n : E B n → E B n−1 , which gives step by step the desired Markov properties. Specifically, these properties are guaranteed by three key results, in our case these are Lemmas 8, Lemma 11 and (ii) Lemma 9 (ii), which are essential to prove that the trace defined by tr n = ϑ 1 • · · · • ϑ n is a Markov trace (Theorem 3).…”

Framization of a Temperley-Lieb algebra of type B

“…In brief, this method consists in constructing a certain family of linear maps tr n : Y B d,n −→ Y B d,n−1 , called relative traces, which builds step by step the desired Markov properties (see also cf. [1]). Finally, the Markov trace on Y B d,n is defined by Tr n := tr 1 • · · · • tr n .…”

“…However, there has been a growing interest also in the framization of algebras that are related to Coxeter systems of type B. Indeed, the affine and cyclotomic Yokonuma-Hecke algebra was introduced in [1], and recently the first author together Juyumaya and Lambopoulou [6] introduced a framization of the Hecke algebra of type B, Y B d,n (u, v), a construction analogous to the Yokonuma-Hecke algebra but in the context of Coxeter systems of the type B.…”

Extremal projectors of two-parameter Kashiwara algebras

“…In their subsequent paper [ChPA2], they studied the representation theory of the affine Yokonuma-Hecke algebra Y r,n (q) and the cyclotomic Yokonuma-Hecke algebra Y d r,n (q). In particular, they gave the classification of irreducible representations of Y d r,n (q) in the generic semisimple case.…”

“…In particular, they gave the classification of irreducible representations of Y d r,n (q) in the generic semisimple case. In [CW], we gave the classification of the simple Y r,n (q)-modules as well as the classification of the simple modules of the cyclotomic Yokonuma-Hecke algebras over an algebraically closed field K of characteristic p such that p does not divide r. In the past several years, the study of affine and cyclotomic Yokonuma-Hecke algebras has made substantial progress; see [ChPA1,ChPA2,ChS,C1,C2,CW,ER,JaPA,Lu,PA2,Ro].…”