Strong forms of self-duality for Hopf monoids in species (Q2790701)

By definition, a vector species is a functor from the category of finite sets with bijections to the category of vector spaces. All vector species \(\mathcal{C}\) forms a symmetric tensor category naturally, which has recently attracted much attention. The tensor product is given by so called Cauchy product. So we can talk about monoids, comonoids, bimonoids and Hopf monoids in this category. The structures of Hopf monoids in this category can be very complicated, but we still can consider some special cases, say, the finite dimensional commutative and cocommutative ones. The paper under reviewed goes on along this line.NEWLINENEWLINETo describe the author's results, we need some definitions. A vector species has a basis if it is a linearization of a set species. A vector species \(\mathbf{p}\) is called connected if \(\mathbf{p}[\varnothing]\) is isomorphic to the base filed \(k\). We say that a Hopf monoid is freely self-dual (FSD, for short) if it is connected, finite-dimensional and has a basis in which the structure constants of its product and coproduct coincide. At first, the author proved that a FSD is always commutative and cocommutative. Then, the author gave some classification results about some special kinds of FSDs. At last, the author gave an equivalent condition to determine when a finite dimensional connected commutative and cocommutative Hopf monoid has a basis in which its product and coproduct are both linearized.