Abdenacer Makhlouf

In this talk we deal with Hom-type and BiHom-type algebras, bialgebras and Hopf algebras. 

The first instances comes from quantum deformations of algebras of vector fields. 

The relevant examples are obtained for Witt and Virasoro algebra, where usual derivation is replaced by a Jackson derivation, the Leibniz identity being twisted by an algebra map. 

The description of the new structure gave rise to a structure generalizing Lie algebras, called Hom-Lie algebras or quasi-Lie algebras introduced by Larsson and Silvestrov. 

A pending Hom-associative algebras and some other Hom-type generalizations of non-associative algebras were introduced by Makhlouf and Silvestrov. 

A study from the point of view of monoidal categories were considered by Caenepeel and Goyvaerts for Hom-algebras, then generalized by Graziani, Makhlouf, Menini and Panaite to introduce BiHom-type algebras.

The purpose of my talk is to give an overview of recent developments and provide some key constructions.