On the partial categorification of some Hopf algebras using the representation theory of towers of J -trivial monoids and semilattices
نویسنده
چکیده
This paper considers the representation theory of towers of algebras of J -trivial monoids. Using a very general lemma on induction, we derive a combinatorial description of the algebra and coalgebra structure on the Grothendieck rings G0 and K0. We then apply our theory to some examples. We first retrieve the classical Krob-Thibon’s categorification of the pair of Hopf algebras QSym/NCSF as representation theory of the tower of 0-Hecke algebras. Considering the towers of semilattices given by the permutohedron, associahedron, and Boolean lattices, we categorify the algebra and the coalgebra structure of the Hopf algebras FQSym, PBT, and NCSF respectively. Lastly we completely describe the representation theory of the tower of the monoids of Non Decreasing Parking Functions. Résumé. Cet article traite de la théorie des représentations des tours d’algèbres de monöıdes J -triviaux. Nous introduisons un lemme général d’induction, duquel nous déduisons une description combinatoire des algèbres et cogèbres des groupes de Grothendieck G0 et K0. Nous appliquons ensuite notre théorie pour retrouver le théorème de Krob-Thibon qui catégorifie la paire QSym/NCSF comme les algèbres de Hopfs duales K0 et G0 de la tour des algèbres 0-Hecke. En considérant les tours de semi-treillis du permutohedron, associahedron et booléen, nous catégorifions les structures d’algèbre et de cogèbre des algèbres de Hopf FQSym, PBT et NCSF. Enfin, nous décrivons complètement la théorie des représentations de la tour des monöıdes des fonctions de parking croissantes.
منابع مشابه
Adjunctions between Hom and Tensor as endofunctors of (bi-) module category of comodule algebras over a quasi-Hopf algebra.
For a Hopf algebra H over a commutative ring k and a left H-module V, the tensor endofunctors V k - and - kV are left adjoint to some kinds of Hom-endofunctors of _HM. The units and counits of these adjunctions are formally trivial as in the classical case.The category of (bi-) modules over a quasi-Hopf algebra is monoidal and some generalized versions of Hom-tensor relations have been st...
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