Sadek Alharbat (University of Leeds) – 45 years of Full-commutativity, 29 years of Fully commutative elements

Long before having taken their name, thus earned their distinguished place in an area in the intersection of  algebra, Knots Theory and combinatorics now a days, Fully Commutative Elements served in the shadows for years, being used as element of basis of the known Temperley-Lieb algebras. It turns out that "they" have their own story. I shall enter by the pure algebra door by giving the general definition, which is simple "by definition". It is a fact that 55 minutes would not be enough to tell the importance of such a class of elements, yet I hope having enough time to talk about the present rather than the past and and so to present some interesting open problems. Please enjoy.