# Meeting Details

Title: The modular class of quantum permutations GAP Seminar Tomasz Maszczyk, Polish Academy of Sciences We introduce a new notion of a Radon-Nikodym differentiable structure on an algebra. This allows us to speak about the quantum fundamental class. Many algebras of geometric origin admit a unique fundamental class. For the algebra of smooth functions on a compact smooth manifold it coincides with its classical counterpart. For any Hopf algebra $H$ with bijective antipode coacting on an algebra $A$ with a fundamental class one could ask if the coaction preserves the fundamental class, as it is so for classical group actions in the classical geometry. If it is so, there is a canonical cohomology class in ${\rm H}^{1}(H, A^{\times})$ which is the obstruction to the existence of an invariant nondegenerate trace supported on the fundamental class. We call it the \emph{modular class}. We show that the universal Hopf algebra with bijective antipode coaction on the algebra of functions on a finite set preserves the fundamental class and for a finite set of cardinality bigger than one the modular class is nontrivial.