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Register a new userHopf algebras arising from partial (co)actions
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In this paper, extending the idea presented by M. Takeuchi in [13], we introduce the notion of partial matched pair $(H,L)$ involving the concepts of partial action and partial coaction between two Hopf algebras $H$ and $L$. Furthermore, we present necessary conditions for the corresponding bismash product $L# H$ to generate a new Hopf algebra and, as illustration, a family of examples is provided.
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In this work the notions of partial action of a weak Hopf algebra on a coalgebra and partial action of a groupoid on a coalgebra will be introduced, just as some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of partial action on a coalgebra and partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.
In this work we study the deformations of a Hopf algebra $H$ by partial actions of $H$ on its base field $Bbbk$, via partial smash product algebras. We introduce the concept of a $lambda$-Hopf algebra as a Hopf algebra obtained as a partial smash product algebra, and show that every Hopf algebra is a $lambda$-Hopf algebra. Moreover, a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras.
In this paper we determine all partial actions and partial coactions of Taft and Nichols Hopf algebras on their base fields. Furthermore, we prove that all such partial (co)actions are symmetric.
We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.
Let $U_q(mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $mathscr{C}_{mathfrak{g}}$ be the category of finite-dimensional integrable $U_q(mathfrak{g})$-modules. We write $mathscr{C}_{mathfrak{g}}^0$ for the monoidal subcategory of $mathscr{C}_{mathfrak{g}}$ introduced by Hernandez-Leclerc. In this paper, we associate a simply-laced finite type root system to each quantum affine algebra $U_q(mathfrak{g})$ in a natural way, and show that the block decompositions of $mathscr{C}_{mathfrak{g}}$ and $mathscr{C}_{mathfrak{g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $mathcal{W}$ (resp. $mathcal{W}_0$) arising from simple modules of $ mathscr{C}_{mathfrak{g}}$ (resp. $mathscr{C}_{mathfrak{g}}^0$) by using the invariant $Lambda^infty$ introduced in the previous work by the authors. The groups $mathcal{W}$ and $mathcal{W}_0$ have the subsets $Delta$ and $Delta_0$ determined by the fundamental representations in $ mathscr{C}_{mathfrak{g}}$ and $mathscr{C}_{mathfrak{g}}^0$ respectively. We prove that the pair $( mathbb{R} otimes_mathbb{Z} mathcal{W}_0, Delta_0)$ is an irreducible simply-laced root system of finite type and the pair $( mathbb{R} otimes_mathbb{Z} mathcal{W}, Delta) $ is isomorphic to the direct sum of infinite copies of $( mathbb{R} otimes_mathbb{Z} mathcal{W}_0, Delta_0)$ as a root system.
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