Gran, Université Catholique de Louvain Title:
Groupoids, commutators and cocommutative Hopf algebras.
Abstract: Internal structures are useful to
understand some fundamental constructions in commutator theory.
In this talk we shall first explain the relationship
between groupoids and commutators in the so-called Mal’tsev
varieties , that are the varieties in the sense of
universal algebra whose algebraic theory has a ternary
operation p(x,y,z) satisfying the identities p(x,y,y)= x
and p(x,x,y)=y. Typical examples of Mal’tsev varieties are those
of groups, where such an operation is given by p(x,y,z)= x - y +
z, quasi-groups, rings, Lie algebras, Boolean algebras and
shall then explain how some of these results can be naturally
extended to a categorical context [2,3], that also includes the
categories of compact groups and of cocommutative Hopf
 J. D. Smith, Mal’cev varieties, Springer
Lect. Notes in Math. 554 (1976).
M.C. Pedicchio, A categorical approach to commutator
theory, J. Algebra 177 (1995) 647-657.
A. Duvieusart and M. Gran, Higher commutator conditions
for extensions in Mal’tsev categories, J. Algebra 515
M. Gran, F. Sterck, J. Vercruysse, A semi-abelian
extension of a theorem by Takeuchi, J.
Pure Appl. Algebra (2019), pub.online.