Marino Gran, Université Catholique de Louvain Title: Groupoids, commutators and cocommutative Hopf algebras.

: 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 [1], 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 crossed modules.

We 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 algebras [4].


[1] J. D. Smith, Mal’cev varieties, Springer Lect. Notes in Math. 554 (1976).
[2] M.C. Pedicchio, A categorical approach to commutator theory, J. Algebra 177 (1995) 647-657.
[3] A. Duvieusart and M. Gran, Higher commutator conditions for extensions in Mal’tsev categories, J. Algebra 515 (2018) 298-327.
[4] M. Gran, F. Sterck, J. Vercruysse, A semi-abelian extension of a theorem by Takeuchi, J. Pure Appl. Algebra (2019),