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Framization of Schur--Weyl duality and Yokonuma--Hecke type algebras

Abel Lacabanne, Loïc Poulain d'Andecy

TL;DR

This work develops a general Schur--Weyl duality framework to systematically framize a broad class of algebras via framed braid groups. By representing framed braids in centralizers of tensor powers of quantum-group modules and exploiting $A_1\boxtimes\cdots\boxtimes A_d$-module structures, the Yokonuma--Hecke algebra and its framizations of Temperley--Lieb, Birman--Murakami--Wenzl, and related algebras arise naturally as End$_A(V^{\otimes n})$ for suitable $A$ and $V$. The paper provides a concrete procedure to construct framizations, describes one-boundary and affine extensions, and develops a robust theory for tied braids and fixed-point subalgebras, yielding explicit decompositions into matrix algebras over tensor products of centralizers. This unifies several knot-theory framings and leads to new framizations (e.g., a framized BMW) while clarifying when such algebras are isomorphic to centralizers, thus enabling a representation-theoretic and invariant-theoretic interpretation of framization. Overall, the results offer a versatile algebraic toolkit for generating and understanding framized algebras through quantum-group dualities and their centralizer structures, with potential applications to knot invariants and categorification.

Abstract

We study framizations of algebras through the idea of Schur--Weyl duality. We provide a general setting in which framizations of algebras such as the Yokonuma--Hecke algebra naturally appear and we obtain this way a Schur--Weyl duality for many examples of these algebras which were introduced in the study of knots and links. We thereby provide an interpretation of these algebras from the point of view of representations of quantum groups. In this approach the usual braid groups is replaced by the framed braid groups. This gives a natural procedure to construct framizations of algebras and we discuss in particular a new framized version of the Birman--Murakami--Wenzl algebra. The general setting is also extended to encompass the situation where the usual braid group is replaced by the so-called tied braids algebra, and this allows to collect in our approach even more examples of algebras introduced in the knots and links setting.

Framization of Schur--Weyl duality and Yokonuma--Hecke type algebras

TL;DR

This work develops a general Schur--Weyl duality framework to systematically framize a broad class of algebras via framed braid groups. By representing framed braids in centralizers of tensor powers of quantum-group modules and exploiting -module structures, the Yokonuma--Hecke algebra and its framizations of Temperley--Lieb, Birman--Murakami--Wenzl, and related algebras arise naturally as End for suitable and . The paper provides a concrete procedure to construct framizations, describes one-boundary and affine extensions, and develops a robust theory for tied braids and fixed-point subalgebras, yielding explicit decompositions into matrix algebras over tensor products of centralizers. This unifies several knot-theory framings and leads to new framizations (e.g., a framized BMW) while clarifying when such algebras are isomorphic to centralizers, thus enabling a representation-theoretic and invariant-theoretic interpretation of framization. Overall, the results offer a versatile algebraic toolkit for generating and understanding framized algebras through quantum-group dualities and their centralizer structures, with potential applications to knot invariants and categorification.

Abstract

We study framizations of algebras through the idea of Schur--Weyl duality. We provide a general setting in which framizations of algebras such as the Yokonuma--Hecke algebra naturally appear and we obtain this way a Schur--Weyl duality for many examples of these algebras which were introduced in the study of knots and links. We thereby provide an interpretation of these algebras from the point of view of representations of quantum groups. In this approach the usual braid groups is replaced by the framed braid groups. This gives a natural procedure to construct framizations of algebras and we discuss in particular a new framized version of the Birman--Murakami--Wenzl algebra. The general setting is also extended to encompass the situation where the usual braid group is replaced by the so-called tied braids algebra, and this allows to collect in our approach even more examples of algebras introduced in the knots and links setting.
Paper Structure (38 sections, 27 theorems, 149 equations)

This paper contains 38 sections, 27 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Algebraic preliminaries
  4. External tensor products of algebras.
  5. Consequences.
  6. One-boundary extension
  7. Centralizer and group action
  8. Main example
  9. Consequences
  10. Framed braid group and centralizers of tensor products
  11. Braid group and framed braid group
  12. The Artin--Tits braid group of type $A$
  13. The framed braid group
  14. Framed braid group and centralizers
  15. The idempotents $E_{i,j}$
  16. ...and 23 more sections

Key Result

Theorem 1

We have a morphism of algebras $\Phi\ :\ \Bbbk\mathcal{FB}_{d,n}\rightarrow\mathop{\mathrm{End}}\nolimits_A(V^{\otimes n})$.

Theorems & Definitions (64)

  • Theorem
  • Theorem
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 54 more