Research
My main area of research is weak bialgebras and weak Hopf algebras, and incidentally fusion categories (look at the File page for some technical notes about WBAs). These objects were introduced by Böhm, Nill and Szlachányi in 1999 in their paper [BNS99] and much research has been done in this area since then. In particular it has been proven by Hayashi that any fusion category is equivalent to a category of modules of a weak Hopf algebra (look at [Hay99], [Thm. 4, Ost03] and [Cor. 2.22, ENO05]).
My work focuses on two constructions in weak bialgebras, namely the localization in weak bialgebras and the weak Hopf envelope.
Concerning the former, let H be a coquasi-triangular weak bialgebra and let G be a monoid of group-like elements. Then how can we invert these elements, i.e. how can we construct the coquasi-triangular weak bialgebra H[G^{-1}] in which all elements of G are invertible and homomorphism i : H --> H[G^{-1}] having the following universal property : for any weak bialgebra homomorphism f : H --> H’ with f(g) invertible for any g in G, there exists a unique weak bialgebra homomorphism f’ : H[G^{-1}] --> H’ such that f = i ◦ f’.
For the weak Hopf envelope, it is a generalization of the Hopf envelope construction due to Takeuchi [Tak71] and Manin [Man88]. The goal here is to generalize this construction to the weak case. Namely, starting with a weak bialgebra B, we want to construct the weak Hopf algebra W(B) and homomorphism i : B --> W(B) such that for any weak Hopf algebra H and weak bialgebra homomorphism f : B --> H, there exists a unique weak Hopf algebra homomorphism f’ : W(B) --> H such that f = i ◦ f’. The long-term goal is the use the technique developed by Pfeiffer in [Pfe11] to produce new examples of fusion categories and better understand the known ones.
For more details on this, you can look the Weak Bialgebra of Fractions paper or at my thesis (see below).
My work focuses on two constructions in weak bialgebras, namely the localization in weak bialgebras and the weak Hopf envelope.
Concerning the former, let H be a coquasi-triangular weak bialgebra and let G be a monoid of group-like elements. Then how can we invert these elements, i.e. how can we construct the coquasi-triangular weak bialgebra H[G^{-1}] in which all elements of G are invertible and homomorphism i : H --> H[G^{-1}] having the following universal property : for any weak bialgebra homomorphism f : H --> H’ with f(g) invertible for any g in G, there exists a unique weak bialgebra homomorphism f’ : H[G^{-1}] --> H’ such that f = i ◦ f’.
For the weak Hopf envelope, it is a generalization of the Hopf envelope construction due to Takeuchi [Tak71] and Manin [Man88]. The goal here is to generalize this construction to the weak case. Namely, starting with a weak bialgebra B, we want to construct the weak Hopf algebra W(B) and homomorphism i : B --> W(B) such that for any weak Hopf algebra H and weak bialgebra homomorphism f : B --> H, there exists a unique weak Hopf algebra homomorphism f’ : W(B) --> H such that f = i ◦ f’. The long-term goal is the use the technique developed by Pfeiffer in [Pfe11] to produce new examples of fusion categories and better understand the known ones.
For more details on this, you can look the Weak Bialgebra of Fractions paper or at my thesis (see below).
Publications
• Steve Bennoun and Hendryk Peiffer, Weak Bialgebras of Fractions, J. Algebra 385 (2013) 145-163, arXiv:1212.5775 [math.QA].
• Steve Bennoun, Localization in Weak Bialgebras and Hopf Envelopes, PhD thesis, available here.
• Steve Bennoun, Tannaka Duality for Comodules over a Hopf Algebra, Master thesis.
• Steve Bennoun, Localization in Weak Bialgebras and Hopf Envelopes, PhD thesis, available here.
• Steve Bennoun, Tannaka Duality for Comodules over a Hopf Algebra, Master thesis.