Algebra Seminar

Let G⁺ be a monoid and Delta\in G⁺. We say that Delta is balanced if

{g\in G⁺ | g\prec \Delta} = { g\in G⁺ | \Delta \succ g}.

In that case the set of divisors of Delta is denoted M_\Delta the pair G⁺,\Delta) is a Garside monoid if G⁺ is Noetherian and cancellative, Delta is balanced and M_\Delta is a finite generating set of G⁺ such (M_\Delta,\prec) and (M_\Delta,\succ) are lattices. A Garside group is the group of fractions of a Garside monoid. For instance, braid groups,or more generally Artin groups of spherical type, are Garside groups. One of the fundamental properties in the study of Artin groups is the existence of natural subgroups, the so-called standard parabolic subgroups. Therefore it is natural to ask what the definition of a standard parabolic subgroup should be in the context of a Garside group. We will define a family of good candidates for such subgroups; more generally we will defined what we call Garside subgroups, which are motivated by several examples and will investigate some of their properties.