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In this paper, we introduce and investigate emph{bisemialgebras}andemph{ Hopf semialgebras} over commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebras over rings including the main reconstruction theorems and the emph{Fundamental Theorem of Hopf Algebras}. We also provide a notion of emph{quantum monoids} as Hopf semialgebras which are neither commutative nor cocommutative; this extends the Hopf algebraic notion of a quantum group. The generalization to the semialgebraic context is neither trivial nor straightforward due to the non-additive nature of the base category of Abelian monoids which is also neither Puppe-exact nor homological and does not necessarily have enough injectives.
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We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.
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.
The Calabi-Yau property of cocommutative Hopf algebras is discussed by using the homological integral, a recently introduced tool for studying infinite dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group algebra of a universal enveloping algebra of a finite dimensional Lie algebra $g$ with a finite subgroup $G$ of automorphisms of $g$ is Calabi-Yau if and only if the universal enveloping algebra itself is Calabi-Yau and $G$ is a subgroup of the special linear group $SL(g)$. The Noetherian cocommutative Calabi-Yau Hopf algebras of dimension not larger than 3 are described. The Calabi-Yau property of Sridharan enveloping algebras of finite dimensional Lie algebras is also discussed. We obtain some equivalent conditions for a Sridharan enveloping algebra to be Calabi-Yau, and then partly answer a question proposed by Berger. We list all the nonisomorphic 3-dimensional Calabi-Yau Sridharan enveloping algebras.
We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors, where the statement was established for semisimple H. The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on A modulo prime powers rather than primes. We also show that the result holds, more generally, for algebras of differential operators. This gives an affirmative answer to a question posed by the last two authors.
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.
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