Marco Bonatto (Charles University, Prague)

Quandles and universal algebra.

Abstract:

Quandles are algebraic structures, defined by Joyce and Matveev indipendently in the '80s, in the context of knot theory. They also arise in other different areas as the study of braided vector spaces and the classification of pointed Hopf algebras. Most of the known results in quandle theory are obtained translating quandle-theoretical questions into equivalent group-theoretical problems. This approach turned out to be very useful, as one of the central notion of the theory is the so-called displacement group (i.e. a subgroup of the automorphism group of the quandle) and a lot of properties of the quandles correspond to suitable properties of its displacement group. I will briefly introduce the connection between quandles, knots, braided vector spaces and Hopf algebras and then I will present an universal algebraic approach to quandles. In the variety of quandles, some interesting properties of congruences have a group-theoretical counterpart in terms of property of the displacement group and its subgroups. As an application of this joint work with Prof. Stanovský I will provide a classification of latin quandle of size