## Evento

Dove: Palazzo Campana - TORINO

Relatore: Gabriella Bohm

Afferenza: Wigner Research Centre for Physics, Budapest, Ungheria.

Locandina:

## Descrizione:

**Abstract :**Bihom bialgebras viewed as bimonoids

In recent years, lead by diverse motivations, several different generalizations of Hopf algebra have been proposed. Their similar features naturally raise the question whether they are instances of the same, more general notion. In [2,3] several examples, such as groupoids, Hopf monoids in braided monoidal categories, Hopf algebroids over central base algebras, weak Hopf algebras, Turaev’s Hopf group algebras, Batista-Caenepeel-Vercruysse’s Hopf categories and Bruguiéres’ Hopf polyads were unified as Hopf monoids in suitable duoidal (called 2-monoidal in [1]) categories.

BiHom-bimonoids in [5] however, do not seem to fit this picture. In order to find a framework describing them, we introduce monoidal categories whose monoidal products of any posiitive number of factors are lax coherent and whose nullary products are oplax coherent. We call them Lax^{+}Oplax^{0}-monoidal. Dually, we consider Lax_{0}Oplax_{+}-monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define Lax^{+ }_{0 }Oplax^{0}_{+} -duoidal categories with compatible Lax^{+}Oplax^{0}- and Lax_{0}Oplax_{+}-monoidal structures. We introduce comonoids in Lax^{+}Oplax^{0}-monoidal categories, monoids in Lax_{0}Oplax_{+}-monoidal categories and bimonoids in Lax^{+} _{0}Oplax^{0} _{+} -duoidal categories.

The unital BiHom-monoids, counital BiHom-comonoids, and unital and counital BiHom-bimonoids of [5] are identified with the monoids, comonoids and bimonoids in a suitable Lax^{+} _{0}Oplax^{0}_{+} -duoidal category, constructed by a method generalized from [4].

This is a joint work with Joost Vercruysse at ULB, Brussels.

[1] Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, volume 29 of CRM Monograph Series. American Mathematical Society, Providence, RI, 2010. 1, 2

[2] Gabriella Böhm, Hopf polyads, Hopf categories and Hopf group monoids viewed as Hopf monads, Theor. and Appl. of Categories 32 no. 37 (2017) 1229–1257. 1

[3] Gabriella Böhm and Stephen Lack, Hopf comonads on naturally Frobenius map-monoidales, J. Pure Appl. Algebra 220 no. 6 (2016) 2177–2213. 1

[4] Stefaan Caenepeel and Isar Goyvaerts, Monoidal Hom-Hopf algebras, Comm. Algebra 39 (2011) 2216–2240.

[5] Giacomo Grazianu, Abdenacer Makhlouf, Claudia Menini and Florin Panaite, BiHom- Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras, SIGMA 11 (2015), 086, 34 pages.