scholarly journals Multiplier Hopf algebras: Globalization for partial actions

2019 ◽  
Vol 30 (03) ◽  
pp. 539-565
Author(s):  
Graziela Fonseca ◽  
Eneilson Fontes ◽  
Grasiela Martini
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Action Theory ◽  
Natural Extension ◽  
Partial Action ◽  
Pertinent Question ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra

In partial action theory, a pertinent question is whenever given a partial action of a Hopf algebra [Formula: see text] on an algebra [Formula: see text], it is possible to construct an enveloping action. The authors Alves and Batista, in [M. Alves and E. Batista, Globalization theorems for partial Hopf (co)actions and some of their applications, groups, algebra and applications, Contemp. Math. 537 (2011) 13–30], have shown that this is always possible if [Formula: see text] is unital. We are interested in investigating the situation, where both algebras [Formula: see text] and [Formula: see text] are not necessarily unitary. A nonunitary natural extension for the concept of Hopf algebras was proposed by Van Daele, in [A. Van Daele, Multiplier Hopf algebras, Trans. Am. Math. Soc. 342 (1994) 917–932], which is called multiplier Hopf algebra. Therefore, we will consider partial actions of multipliers Hopf algebras on algebras with a nondegenerate product and we will present a globalization theorem for this structure. Moreover, Dockuchaev et al. in [Globalizations of partial actions on nonunital rings, Proc. Am. Math. Soc. 135 (2007) 343–352], have shown when group partial actions on nonunitary algebras are globalizable. Based on this paper, we will establish a bijection between globalizable group partial actions and partial actions of a multiplier Hopf algebra.

scholarly journals A CLASS OF QUASITRIANGULAR GROUP-COGRADED MULTIPLIER HOPF ALGEBRAS

2018 ◽  
Vol 62 (1) ◽  
pp. 43-57
Author(s):  
TAO YANG ◽  
XUAN ZHOU ◽  
HAIXING ZHU
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Finite Dimensional ◽  
Special Cases ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Classical Construction

AbstractFor a multiplier Hopf algebra pairing 〈A,B〉, we construct a class of group-cograded multiplier Hopf algebras D(A,B), generalizing the classical construction of finite dimensional Hopf algebras introduced by Panaite and Staic Mihai [Isr. J. Math. 158 (2007), 349–365]. Furthermore, if the multiplier Hopf algebra pairing admits a canonical multiplier in M(B⊗A) we show the existence of quasitriangular structure on D(A,B). As an application, some special cases and examples are provided.

scholarly journals A duality theorem for weak multiplier Hopf algebra actions

2017 ◽  
Vol 28 (05) ◽  
pp. 1750032 ◽  
Author(s):  
Nan Zhou ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Duality Theorem ◽  
Smash Product ◽  
Module Algebra ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Weak Hopf Algebras

The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by Van Daele and Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, multiplier Hopf algebras and groupoids.

A Note on the Antipode for Algebraic Quantum Groups

2012 ◽  
Vol 55 (2) ◽  
pp. 260-270
Author(s):  
L. Delvaux ◽  
A. Van Daele ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Fourth Power ◽  
General Situation ◽  
Locally Compact Quantum Groups ◽  
Algebraic Quantum ◽  
Multiplier Hopf Algebras ◽  
Compact Quantum Groups ◽  
Multiplier Hopf Algebra

AbstractRecently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals.We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

COREPRESENTATION THEORY OF MULTIPLIER HOPF ALGEBRAS II

2000 ◽  
Vol 11 (02) ◽  
pp. 233-278 ◽  
Author(s):  
HIDEKI KUROSE ◽  
ALFONS VAN DAELE ◽  
YINHUO ZHANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Covariant Theory ◽  
Crossed Products ◽  
Quantum Double ◽  
Multiplier Algebra ◽  
Algebraic Quantum ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Invertible Elements

We continue our development of the corepresentation theory of multiplier Hopf algebras. In this paper, we consider the corepresentations of a multiplier Hopf algebra A in a nondegenerate algebra B rather than on a vector space (cf. [25]). We concentrate ourself on those corepresentations of A in B which are invertible elements of the multiplier algebra M(B⊗A). They are called the unitary corepresentations of A. In particular, the generalized R-matrices or quasi-triangular structures of a regular multiplier Hopf algebra are unitary (bi)corepresentations. As an application the quantum double of an algebraic quantum group can be constructed by means of the universal unitary corepresentation. Moreover, a unitary corepresentation of A in B can implement an inner coaction of A on B which allows us to study the covariant theory and crossed products.

MULTIPLIER BI- AND HOPF ALGEBRAS

2010 ◽  
Vol 09 (02) ◽  
pp. 275-303 ◽  
Author(s):  
K. JANSSEN ◽  
J. VERCRUYSSE
Keyword(s):  
Hopf Algebra ◽  
Commutative Ring ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Left And Right ◽  
Projective Algebra

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

scholarly journals Weak Multiplier Hopf Algebras II: Source and Target Algebras

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1975
Author(s):  
Alfons Van Daele ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Hopf Algebras ◽  
Symmetric Pair ◽  
Multiplier Algebra ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Weak Hopf Algebras ◽  
Algebra Part

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).

scholarly journals Weak Multiplier Hopf Algebras II: The Source and Target Algebras

2020 ◽  
Author(s):  
Shuanhong Wang ◽  
Alfons Van Daele
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Weak Hopf Algebra ◽  
Multiplier Algebra ◽  
Special Cases ◽  
Multiplier Hopf Algebras ◽  
Group A ◽  
Multiplier Hopf Algebra ◽  
Weak Hopf Algebras ◽  
Algebra Part

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).

scholarly journals Algebraic quantum hypergroups of discrete type

2011 ◽  
Vol 108 (2) ◽  
pp. 198 ◽  
Author(s):  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Hopf Algebras ◽  
Algebraic Quantum ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Equivalent Conditions ◽  
Discrete Type

In this paper we will study some structures of algebraic quantum hypergroups. First, we construct more examples of algebraic quantum hypergroups of discrete type. Next, we introduce the notion of a generalized quasi-Frobenius multiplier Hopf algebra and then show that generalized quasi-Frobenius multiplier Hopf algebras are a class of algebraic quantum hypergroups of discrete type. We also give some equivalent conditions for an algebraic quantum group to be of discrete type. Finally, we study sub-algebraic quantum hypergroups of discrete type and quotients of algebraic quantum hypergroups of discrete type.

The shuffle Hopf algebra and noncommutative full completeness

1998 ◽  
Vol 63 (4) ◽  
pp. 1413-1436 ◽  
Author(s):  
R. F. Blute ◽  
P. J. Scott
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Natural Extension ◽  
Additive Group ◽  
Completeness Theorem ◽  
Vector Spaces ◽  
Invariance Condition ◽  
Shuffle Algebra ◽  
Topological Vector Spaces

AbstractWe present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces.This paper is a natural extension of the authors' previous work, “Linear Läuchli Semantics”, where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals.

scholarly journals Hopf algebras arising from partial (co)actions

2020 ◽  
pp. 2140006
Author(s):  
Danielle Azevedo ◽  
Grasiela Martini ◽  
Antonio Paques ◽  
Leonardo Silva
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Product Formula ◽  
Matched Pair ◽  
Matched Pairs ◽  
Partial Action

In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981) 841–882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130(1) (1990) 17–64.], we introduce the notion of partial matched pair [Formula: see text] involving the concepts of partial action and partial coaction between two bialgebras [Formula: see text] and [Formula: see text]. Furthermore, we present sufficient conditions for the corresponding bismash product [Formula: see text] to generate a new Hopf algebra and, as illustration, a family of examples is provided. Moreover, under some hypotheses such sufficient conditions are also necessary conditions.

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