Affine cellularity of affine Yokonuma-Hecke algebras

Cui, Weideng

doi:10.48550/arxiv.1510.02647

Cited by 6 publications

(7 citation statements)

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“…We start with an isomorphism between the algebra Y d,n and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras. As done in [4], the isomorphism can be proved repeating the same arguments as for Y d,n (see [5] where the proof for Y d,n is presented, as a particular case of a more general result by G. Lusztig [8, §34]). Here we sketch a short different proof for Y d,n using the known result for Y d,n .…”

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confidence: 97%

Invariants for Links from Classical and Affine Yokonuma–Hecke Algebras

d'Andecy

2017

Springer Proceedings in Mathematics &Amp; Statistics

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We present a construction of invariants for links using an isomorphism theorem for affine Yokonuma-Hecke algebras. The isomorphism relates affine Yokonuma-Hecke algebras with usual affine Hecke algebras. We use it to construct a large class of Markov traces on affine Yokonuma-Hecke algebras, and in turn, to produce invariants for links in the solid torus. By restriction, this construction contains the construction of invariants for classical links from classical Yokonuma-Hecke algebras. In general, the obtained invariants form an infinite family of 3-variables polynomials. As a consequence of the construction via the isomorphism, we reduce the number of invariants to study, given the number of connected components of a link. In particular, if the link is a classical link with N components, we show that N invariants generate the whole family. L. Poulain d'Andecycan deform the standard surjective morphism from the framed braid group algebra to its quotient Y d,n into a family of morphisms (depending on γ) respecting the braid relations and the Markov conditions. Another way of interpreting the parameter γ is that it modifies the quadratic relation satisfied by the generators of Y d,n . Its existence explains (or is reflected in) the fact that different presentations for Y d,n were used before. Juyumaya-Lambropoulou invariants correspond to certain specialisations of this parameter γ, depending on the chosen presentation. So the parameter γ unifies every possible choices and yields more general invariants. It is indicated in [3, Remark 8.5] that changing the presentation seems to give a non-equivalent topological invariant.2. In this paper, we consider the affine Yokonuma-Hecke algebras (of type GL), denoted Y d,n . They were introduced in [1] in connections with the representation theory and the Jucys-Murphy elements of the classical Yokonuma-Hecke algebras. Our main goal here is to generalise for Y d,n the whole approach to link invariants via the isomorphism theorem. The invariants are in general for links in the solid torus. The classical links are naturally contained in the solid torus links and, restricted to them, the obtained invariants correspond to the invariants obtained in [5] from Y d,n (naturally seen as a subalgebra of Y d,n ). Specialising the parameter γ, we identify the Juyumaya-Lambropoulou invariants among them. For those invariants, we emphasize that we recover some known results [3] by a different method and furthermore obtain some new results already in this particular case.We start with an isomorphism between the algebra Y d,n and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras. As done in [4], the isomorphism can be proved repeating the same arguments as for Y d,n (see [5] where the proof for Y d,n is presented, as a particular case of a more general result by G. Lusztig [8, §34]). Here we sketch a short different proof for Y d,n using the known result for Y d,n . We also prove the analogous theorem for the cyclotomic quotients of Y d,n (with A...

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confidence: 97%

Invariants for Links from Classical and Affine Yokonuma–Hecke Algebras

d'Andecy

2017

Springer Proceedings in Mathematics &Amp; Statistics

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“…Finally, we can lift trivially the length function on W n to W d,n . Indeed, we deduce from relations (6), that every x ∈ W d,n can be written in the form x = wt a 1 1 • • • t an n , with w ∈ W n . Thus, we define the length of x by (w).…”

Section: Framizations Of Type Bmentioning

confidence: 99%

“…Before giving the organization of the article we note that, by taking into account the various articles generated recently from the algebra of Yokonuma-Hecke of type A (view for example [8,5,6,3] among others), the framization proposed here indicates that the algebra Y B d,n (u, v) should be interesting in itself.…”

Section: Introductionmentioning

confidence: 99%

A framization of the Hecke algebra of type B

Flores

Juyumaya

Lambropoulou

2018

Journal of Pure and Applied Algebra

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“…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].…”

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confidence: 99%