Rota–Baxter Operators on Cocommutative Weak Hopf Algebras

In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a new multiplication “∗” and an antipode SB from a Rota–Baxter operator B on H such that HB=(H,∗,η,Δ,ε,SB) becomes a new weak Hopf algebra. Finally, all Rota–Baxter operators on a weak Hopf algebra of a matrix algebra are given.

Automorphism groups of representation rings of the weak Sweedler Hopf algebras

<abstract><p>Let $ \mathfrak{w}^{s}_{2, 2}(s = 0, 1) $ be two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra, and $ r(\mathfrak{w}^{s}_{2, 2}) $ be the representation rings of $ \mathfrak{w}^{s}_{2, 2} $. In this paper, we investigate the automorphism groups $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $ of $ r(\mathfrak{w}^{s}_{2, 2}) $, and discuss some properties of $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $. We obtain that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2})) $ is isomorphic to $ K_4 $, where $ K_4 $ is the Klein four-group. It is shown that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is a non-commutative infinite solvable group, but it is not nilpotent. In addition, $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is isomorphic to $ (\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2} $, and its centre is isomorphic to $ \mathbb{Z}_{2} $.</p></abstract>

The category of affine algebraic regular monoids

<abstract><p>The main aim of this study is to characterize affine weak $ k $-algebra $ H $ whose affine $ k $-variety $ S = M_{k}(H, k) $ admits a regular monoid structure. As preparation, we determine some results of weak Hopf algebras morphisms, and prove that the anti-function from the category $ \mathcal{C} $ of weak Hopf algebras whose weak antipodes are anti-algebra morphisms is adjoint. Then, we prove the main result of this study: the anti-equivalence between the category of affine algebraic $ k $-regular monoids and the category of finitely generated commutative reduced weak $ k $-Hopf algebras.</p></abstract>

Weak Multiplier Hopf Algebras II: Source and Target Algebras

Let (A,Δ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Δ:A⟶M(A⊗A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps εs and εt are studied, and their symmetric pair of images, the source algebra and the target algebra εs(A) and εt(A), are also investigated. We show that the canonical idempotent E (which is eventually Δ(1)) belongs to the multiplier algebra M(B⊗C), where (B=εs(A), C=εt(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,Δ), it is possible to make C⊗B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the ’Hopf algebra part’ of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E∈M(B⊗C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known ’quantization’ of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).

Weak Multiplier Hopf Algebras II: The Source and Target Algebras

Let $(A,\Delta)$ be a {\it weak multiplier Hopf algebra} as introduced in [VD-W3] (see also [VD-W2]). It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. If the algebra has an identity and the coproduct is unital, then we have a Hopf algebra. If the algebra has no identity, but if the coproduct is non-degenerate (which is the equivalent of being unital if the algebra has an identity), then the pair would be a multiplier Hopf algebra. If the algebra has an identity, but the coproduct is not unital, we have a weak Hopf algebra. In the general case, we neither assume $A$ to have an identity nor do we assume $\Delta$ to be non-degenerate and so we work with a {\it genuine} weak multiplier Hopf algebra. It is called {\it regular} if its antipode is a bijective map from $A$ to itself. \snl In this paper, we {\it continue the study of weak multiplier Hopf algebras}. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the multiplier algebra $M(B\ot C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a {\it separability idempotent} (as studied in [VD4.v2]). If the weak multiplier Hopf algebra is regular, then also $E$ is a {\it regular} separability idempotent. \snl We also consider {\it special cases and examples} in this paper. In particular, we see how for any weak multiplier Hopf algebra $(A,\Delta)$, it is possible to make $C\ot B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\ot C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

Partial actions of weak Hopf algebras on coalgebras

In this work, the notions of a partial action of a weak Hopf algebra on a coalgebra and of a partial action of a groupoid on a coalgebra will be introduced, together with some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of a partial action on a coalgebra and of a partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.

Comparison of two notions of weak crossed product

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].