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

Lacabanne, Abel; Poulain d’Andecy, Loïc

doi:10.5802/art.20

Lacabanne, Abel ¹; Poulain d’Andecy, Loïc ²

¹ Laboratoire de Mathématiques Blaise Pascal (UMR 6620), Université Clermont Auvergne, Campus Universitaire des Cézeaux, 3 place Vasarely, 63178 Aubière Cedex, France
² Laboratoire de Mathématiques de Reims (UMR 9008), Université de Reims Champagne-Ardenne, Moulin de la Housse - BP 1039, 51100 Reims, France

Annals of Representation Theory, Volume 2 (2025) no. 1, pp. 1-35.

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.

Received: 2024-01-25
Revised: 2024-08-07
Accepted: 2024-10-21
Published online: 2025-01-16

DOI: 10.5802/art.20

Classification: 20C08, 17B37, 20F36
Mots-clés : Schur–Weyl duality, Yokonuma–Hecke algebras, framization, framed braid group, tied braid algebra

Author's affiliations:

Lacabanne, Abel ¹; Poulain d’Andecy, Loïc ²

¹ Laboratoire de Mathématiques Blaise Pascal (UMR 6620), Université Clermont Auvergne, Campus Universitaire des Cézeaux, 3 place Vasarely, 63178 Aubière Cedex, France
² Laboratoire de Mathématiques de Reims (UMR 9008), Université de Reims Champagne-Ardenne, Moulin de la Housse - BP 1039, 51100 Reims, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Lacabanne, Abel; Poulain d’Andecy, Loïc. Framization of Schur–Weyl duality and Yokonuma–Hecke type algebras. Annals of Representation Theory, Volume 2 (2025) no. 1, pp. 1-35. doi : 10.5802/art.20. https://art.centre-mersenne.org/articles/10.5802/art.20/

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