Related papers: Antipodes of monoidal decomposition spaces
The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite simple graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new…
We introduce topological invariants of semi-decompositions (e.g. filtrations, semi-group actions, multi-valued dynamical systems, combinatorial dynamical systems) on a topological space to analyze semi-decompositions from a dynamical…
We give a new formula for the antipode of the algebra of rooted trees, directly in terms of the bialgebra structure. The equivalence, proved in this paper, among the three available formulae for the antipode, reflects the equivalence among…
A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the `base algebras') are shown to carry coseparable co-Frobenius coalgebra…
The permutation pattern Hopf algebra is a commutative filtered and connected Hopf algebra. Its product structure stems from counting patterns of a permutation, interpreting the coefficients as permutation quasi-shuffles. The Hopf algebra…
We give explicit formulas for the coproduct and the antipode in the Connes-Moscovici Hopf algebra $\mathcal{H}_{\tmop{CM}}$. To do so, we first restrict ourselves to a sub-Hopf algebra $\mathcal{H}^1_{\tmop{CM}}$ containing the nontrivial…
We initiate the study of a large class of species monoids and comonoids which come equipped with a poset structure that is compatible with the multiplication and comultiplication maps. We show that if a monoid and a comonoid are related…
We prove that there exist Hopf algebras with surjective, non-bijective antipode which admit no non-trivial morphisms from Hopf algebras with bijective antipode; in particular, they are not quotients of such. This answers a question left…
Let $k$ be a field and let $H$ denote a pointed Hopf $k$-algebra with antipode $S$. We are interested in determining the order of $S$. Building on the work done by Taft and Wilson $[7]$, we define an invariant for $H$, denoted $m_{H}$, and…
Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B\subseteq A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the non-commutative base algebra of H,…
In this paper we study the separately continuous actions of semitopological monoids on pseudocompact spaces. The main aim of this paper is to generalize Lawson's results to some class of pseudocompact spaces. Also, we introduce a concept of…
Certain sufficient homological and ring-theoretical conditions are given for a Hopf algebra to have bijective antipode with applications to noetherian Hopf algebras regarding their homological behaviors.
We extend to the co-Frobenius case a result of Drinfeld and Radford related to the fourth power of the antipode of a finite dimensional (co)quasitriangular Hopf algebra.
We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with…
We consider the property of unique parallel decomposition modulo branching and weak bisimilarity. First, we show that infinite behaviours may fail to have parallel decompositions at all. Then, we prove that totally normed behaviours always…
We establish a Crapo complementation formula for the M\"obius function $\mu^X$ in a general decomposition space $X$ in terms of a convex subspace $K$ and its complement: $\mu^X \simeq \mu^{X\setminus K} + \mu^X*\zeta^K*\mu^X$. We work at…
We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration and…
Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided…
We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual…
In arXiv:1709.07504 Ardila and Aguiar give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on…