Yokonuma–Hecke algebras

Yokonuma–Hecke algebras
Name	Yokonuma–Hecke algebras
Caption	Generators and relations visualization
Introduced	1960s–1980s
Related	Hecke algebra, Iwahori–Hecke algebra, Ariki–Koike algebra, Brauer algebra

Yokonuma–Hecke algebras are a family of associative algebras arising from the study of finite Chevalley groups, braid group centralizers, and framed braid groups. Introduced as a deformation of group algebras associated with Gyoji Yokonuma's work on finite groups of Lie type, the algebras have since appeared in connections with Vladimir Drinfeld's quantum groups, Vera Serganova's representation questions, and knot invariant constructions related to Joan Birman and Hitoshi Murakami.

Definition and Presentation

The algebra is generated by braid-like generators t_i and idempotent-like elements e_j together with parameters coming from a cyclic group; its presentation generalizes that of the Iwahori–Hecke algebra studied by Mark Haiman and Gordon James. In concrete form one fixes an integer r and a parameter q, and defines generators subject to braid relations modeled on Emil Artin's braid group, quadratic relations deforming Hecke algebra relations studied by Georges Lusztig, and commutation relations with the cyclic group elements reminiscent of constructions in Michio Jimbo's quantum algebras. The defining relations are often expressed to parallel presentations in work of Jun Murakami, Christian Blanchet, and Nicolas Reshetikhin in the theory of quantum invariants.

Algebraic Properties

Yokonuma–Hecke algebras are finite-dimensional over suitable base rings like those used by I. G. Macdonald and obey Poincaré–Birkhoff–Witt type bases analogous to results of Gerhard Hochschild and Bertram Kostant. They possess a cellular structure in the sense of Graham Lehrer and J. J. Graham and admit traces used by Tomotada Ohtsuki and Tatsuya Akutsu to define link invariants. The center exhibits connections to character theory as in work of Frobenius and Issai Schur, while semisimplicity criteria reflect parameter specializations studied by Claude Chevalley and Roger Carter. Deformation theoretic aspects link to Pierre Deligne's ideas on tensor categories and to Jean-Michel Bismut's index-theoretic perspectives.

Representations and Modules

Simple modules and decomposition matrices have been investigated using methods pioneered by Ariki Koike and Susumu Ariki, with branching rules echoing classical Schur–Weyl duality considered by Issai Schur and Hermann Weyl. Induction and restriction functors parallel constructions by J. Bernstein and André Henriques in representation categories; Specht-module analogues and cell modules were developed in works following Andrew Mathas and William Graham. Modular representation theory questions involve parameter regimes studied by Richard Brauer and Michel Broué, while connections to tilting theory evoke Henning Krause and Viktor Ostrik.

Connections to Knot Theory and Link Invariants

Using Markov traces in the spirit of Joan Birman and Vaughan Jones, Yokonuma–Hecke algebras produce link invariants related to those of Jones polynomial origin and to HOMFLY-PT-type invariants investigated by Przytycki and P. Traczyk. Framed braid interpretations tie to constructions by Louis Kauffman and Klaus Murasugi, and comparisons of resulting invariants have been made against Khovanov homology-inspired invariants studied by Mikhail Khovanov and Jacob Rasmussen. Applications to satellite knot operations evoke classical work by Ralph Fox and J. W. Alexander.

Relations to Hecke, Ariki–Koike, and Brauer Algebras

Yokonuma–Hecke algebras specialize to or contain subalgebras isomorphic to Iwahori–Hecke algebra and relate to the Ariki–Koike algebra through cyclotomic quotients studied by Jun Hu and Andrew Mathas. Brauer-type connections occur via Schur–Weyl dualities reminiscent of Richard Brauer's original duality and later developments by Anthony Temperley and Elliot Lieb in statistical mechanics. These relationships link to categorifications and to cellular algebra frameworks advanced by Hugh Morton and Vladimir Turaev.

Categorification and Homological Aspects

Categorification efforts connect Yokonuma–Hecke algebras to 2-representation theory as in work by Chuang Rouquier and relate to homological invariants à la Khovanov and derived equivalences studied by Raphaël Rouquier. Homological algebra tools from Alexandre Grothendieck's school and Hochschild cohomology techniques used by Gerhard Hochschild illuminate extension groups between cell modules, while link homology comparisons draw on ideas from Edward Witten and Maxim Kontsevich in topological quantum field theory. Derived and triangulated category approaches echo work by Bernhard Keller and Amnon Neeman.

Category:Algebraic structures