Graziela Fonseca, Eneilson Fontes, Grasiela Martini
In this work we define partial (co)actions on multiplier Hopf algebras, we also present examples and properties. From a partial comodule coalgebra we construct a partial smash coproduct generalizing the constructions made by the L. Delvaux, E. Batista and J. Vercruysse.
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We describe the Hopf algebra quotients and Hopf images of the smash coproduct of a group algebra by the algebra of functions on a finite group.
We give some general results on the ring structure of Hochschild cohomology of smash products of algebras with Hopf algebras. We compute this ring structure explicitly for a large class of finite dimensional Hopf algebras of rank one.
Let $H$ be a Hopf algebra. Ju and Cai introduced the notion of twisting of an $H$-module coalgebra. In this note, we study the relationship between twistings, crossed coproducts and Hopf-Galois coextensions. In particular, we show that a twisting of an $H$-Galois coextension remains $H$-Galois if the twisting is invertible.
Partial Actions of a Hopf algebra on its base field and the corresponding partial smash product algebra
本研究引入λ-Hopf代数概念并证明所有Hopf代数均为λ-Hopf代数，提出Hopf代数在基域上部分作用的计算方法，还给出若干Hopf代数族的全部此类部分作用。
We introduce the Hom-analogue of the L-R-smash product and use it to define the Hom-analogue of the diagonal crossed product. When H is a finite dimensional Hom-Hopf algebra with bijective antipode and bijective structure map, we define the Drinfeld double of H; its algebra structure is a Hom-diagonal crossed product and it has all expected properties, namely it is quasitriangular and modules over it coincide with left-right Yetter-Drinfeld modules over H.
We give some applications of a Hopf algebra constructed from a group acting on another Hopf algebra A as Hopf automorphisms, namely Molnar's smash coproduct Hopf algebra. We find connections between the exponent and Frobenius-Schur indicators of a smash coproduct and the twisted exponents and twisted Frobenius-Schur indicators of the original Hopf algebra A. We study the category of modules of the smash coproduct.
The theory of R-smash products for Hopf quasigroups is developed.
Partial representations of connected and smash product Hopf algebras
本文证明连通Hopf代数的部分表示皆为全局表示，研究了Smash积Hopf代数的部分表示，并在特定条件下将$H_{par}$描述为源自Hopf范畴的弱Hopf代数。
For a semisimple quasi-triangular Hopf algebra $\left( H,R\right) $ over a field $k$ of characteristic zero, and a strongly separable quantum commutative $H$-module algebra $A$ over which the Drinfeld element of $H$ acts trivially, we show that $A\#H$ is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra $\operatorname{End}A^{\ast}\otimes H$. With these structure, $_{A\#H}\operatorname{Mod}$ is the monoidal category introduced by Cohen and Westreich, and $_{\operatorname{End}A^{\ast}\otimes H}\mathcal{M}$ is tensor equivalent to $_{H}\mathcal{M}$. If $A$ is in the M{ü}ger cen…
Let H be a Hopf algebra and let B be a Koszul H-module algebra. We provide necessary and sufficient conditions for a filtered algebra to be a Poincare-Birkhoff-Witt (PBW) deformation of the smash product algebra B#H. Many examples of these deformations are given.
We analyze some relations between quasi-Hopf smash products and certain twisted tensor products of quasialgebras. Along the way we obtain also some results of independent interest, such as a duality theorem for finite dimensional quasi-Hopf algebras and a universal property for generalized diagonal crossed products.
We introduce a more general version of the so-called L-R-smash product and study its relations with other kinds of crossed products (two-sided smash and crossed product and diagonal crossed product). We also give an interpretation of the L-R-smash product in terms of an L-R-twisting datum.
It is an open question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central $H$-invariant elements of the Martindale ring of quotients of a module algebra form a von Neumann regular and self-injective ring whenever $A$ is semiprime. For a semiprime Goldie PI $H$-module algebra $A$ with central invariants we show that $\AH$ is semiprime if and only if the $H$-action c…
In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may be applied to H^*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the "generating matrix" formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.
In this paper we introduce the notion of partial action of a weak Hopf algebra on algebras, unifying the notions of partial group action [11], partial Hopf action ([2],[3],[9]) and partial groupoid action [4]. We construct the fundamental tools to develop this new subject, namely, the partial smash product and the globalization of a partial action, as well as, we establish a connection between partial and global smash products via the construction of a surjective Morita context. In particular, in the case that the globalization is unital, these smash products are Morita equivalent. We show tha…
In this work we investigate partial actions of a Hopf algebra H on nonunital algebras and the associated partial smash products. We show that our partial actions correspond to nonunital algebras in the category of partial representations of H. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and w…
Motivated by partial group actions on unital algebras, in this article we extend many results obtained by Exel and Dokuchaev to the context of partial actions of Hopf algebras, according to Caenepeel and Jansen. First, we generalize the theorem about the existence of an enveloping action, also known as the globalization theorem. Second, we construct a Morita context between the partial smash product and the smash product related to the enveloping action. Third, we dualize the globalization theorem to partial coactions and finally, we define partial representations of Hopf algebras and show som…
PBW-deformations of smash products involving Hopf algebra of Kac-Paljutkin type
本文研究了Kac-Paljutkin型Hopf代数的PBW变形，探讨了smash积和辫状张量积代数的变形性质及其条件。
In the theory of dynamical Yang-Baxter equation, with any Hopf algebra $H$ and a certain $H$-module and $H$-comodule algebra $L$ (base algebra) one associates a monoidal category. Given an algebra $A$ in that category, one can construct an associative algebra $A\rtimes L$, which is a generalization of the ordinary smash product when $A$ is an ordinary $H$-algebra. We study this "dynamical smash product" and its modules induced from one-dimensional representation of the subalgebra $L$. In particular, we construct an analog of the Galois map $A\otimes_{A^H} A\to A\otimes H^*$.
The characterization of $H$-prime radical is given in many ways. Meantime, the relations between the radical of smash product $R # H$ and the $H$-radical of Hopf module algebra $R$ are obtained.
Definitions of actions of Hopf quasigroups are discussed in the context of Long dimodules and smash products. In particular, Long dimodules are defined for Hopf quasigroups and coquasigroups, and solutions to Militaru's D-equation are constructed. A necessary compatibility condition between action and multiplication of a Hopf quasigroup acting on its quasimodule Hopf quasigroup for a smash product construction is derived.
Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto-Reutenauer, convolution, etc.) have been defined in the following objects : species, representations of the symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, etc. With the purpose of simplifying and unifying this diversity we introduce yet, another -non graded- product the Heisenberg product, that for the highest and lowest degrees produces the cl…
Reflections on the Larson-Sweedler theorem for (weak) multiplier Hopf algebras
本文证明乘子Hopf代数的Larson-Sweedler定理可在不要求余积满性的条件下成立，并指出Hopf代数情形中余单位的存在性可由其他条件推出，同时讨论了弱乘子Hopf代数的相关问题。
Let $G$ be a group which admits a generating set consisting of finite order elements. We prove that any Hopf algebra which factorizes through the Taft algebra and the group Hopf algebra $K[G]$ (equivalently, any bicrossed product between the aforementioned Hopf algebras) is isomorphic to a smash product between the same two Hopf algebras. The classification of these smash products is shown to be strongly linked to the problem of describing the group automorphisms of $G$. As an application, we completely describe by generators and relations and classify all bicrossed products between the Taft a…
Prime and semiprime quantum linear space smash products
本文研究量子线性空间玻色子化与结合代数的 smash 积，给出其为（半）素的条件，并应用于量子仿射空间，推广了 Bergen 的结果。
Characterization Results on Generalized Smash Biproduct Hopf Algebras over the Partial Dual Construction
本文在辫子幺半范畴中，当$B$有左对偶$B^\ast$时构造$D\times B^\ast$，证明$B\times D$为其左smash积，并建立Yetter-Drinfeld模范畴的同构，涵盖代数、双代数、Hopf代数三个层次。
Generalized Connes-Kreimer Hopf algebras on decorated rooted forests by weighted cocycles
本文提出了一种广义Connes-Kreimer霍普夫代数，通过加权余圈条件扩展了根树结构，并证明其作为自由对象的性质。
A Hopf algebra on nonplanar binary forests
本文研究非平面二叉树生成的多项式代数，通过定义禁止同时切断任意顶点两个子节点的余乘积，构建霍普夫代数结构，并揭示其与布伦德霍普夫代数及预李运算的联系。
We show that the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual Hochschild cohomology ring of a smash product and the (derived) invariants of its braided Hochschild cohomology ring. We apply our results to identify the associative formal deformation theory of a smash product with its formal deformation theory as a module algebra over the given Hopf algebra (when the Hopf algebra is sufficiently semisimple). As a second …
The duality between partial actions (partial $H$-module algebras) and co-actions (partial $H$-comodule algebras) of a Hopf algebra $H$ is fully explored in this work. A connection between partial (co)actions and Hopf algebroids is established under certain commutativity conditions. Moreover, we continue this duality study, introducing also partial $H$-module coalgebras and their associated $C$-rings, partial $H$-comodule coalgebras and their associated cosmash coproducts, as well as the mutual interrelations between these structures.
We prove that both, the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras, have right adjoints; in other words: every bialgebra has a Hopf coreflection and on every algebra there exists a cofree Hopf algebra. In this way we give an affirmative answer to a forty years old problem posed by Sweedler. On the route the coequalizers and the coproducts in the category of Hopf algebras are explicitly described.
A weakly equivariant Hopf algebra is a Hopf algebra A with an action of a finite group G up to inner automorphisms. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra B with a strict action of G and with a coalgebra structure that leads to a tensor equivalent representation category. However, the coproduct of this strictification cannot, in general, be chosen to be unital, so that a strictification of the G-action can only be found on a weak Hopf algebra B.
The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra $H$ is studied. Such a Hopf algebra $H$ is an extension of an affine commutative Hopf algebra $A$ by a finite dimensional Hopf algebra $F$. The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $F^{\ast}$ by the finite dual of $A$. This decomposition is then further analysed using the Cartier- Gabriel-Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$. The detailed consequences for a number of f…
One-dimensional partial actions of 8-dimensional Hopf algebras and right coideal subalgebras
本文完成8维Hopf代数一维部分作用的分类，证明其对称性，并证明有限维余半单Hopf代数的右余理想子代数均可实现为部分smash积。
Quantum group coproducts and universality under scalar extensions
本文刻画了域上双代数、Hopf代数及余代数族乘积有限维的条件，并纠正了标量扩张保持乘积的文献错误，明确了有限与代数扩张的普适性特征。
In this paper, we obtain a canonical factorization of basis elements in free Rota-Baxter algebras built on bracketed words. This canonical factorization is applied to give a coalgebra structure on the free Rota-Baxter algebras. Together with the Rota-Baxter algebra multiplication, this coproduct gives a bialgebra structure on the free Rota-Baxter algebra of rooted forests. When the weight of the Rota-Baxter algebra is zero, we further obtain a Hopf algebra structure.
We construct a Hopf algebra cocycle in the Yangian double $DY(SL_{2})$, conjugating Drinfeld's coproduct to the usual one. To do that, we factorize the twist between two ``opposite'' versions of Drinfeld's coproduct, introduced in earlier work by V. Rubtsov and the first author, using the decomposition of the algebra in its negative and non-negative modes subalgebras.
We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative variety. We relate the coproduct with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees - a planar decorated version of the Hopf algebra defined by Connes and Kreimer. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. We give a criteria for a motivic iterated integral…
The effectiveness of the aplication of constructions in $G$-graded $k$-categories to the computation of the fundamental group of a finite dimensional $k$-algebra, alongside with open problems still left untouched by those methods and new problems arisen from the introduction of the concept of fundamental group of a $k$-linear category, motivated the investigation of $H$-module categories, i.e., actions of a Hopf algebra $H$ on a $k$-linear category. The $G$-graded case corresponds then to actions of the Hopf algebra $k^G$ on a $k$-linear category. In this work we take a step further and introd…
Factorization of quasitriangular structures of smash biproduct bialgebras
本文研究了smash双积双代数拟三角结构的分解与重构，在σ右余正规条件下给出其拟三角性的等价刻画及显式结构，推广了Radford双积、双交叉积Hopf代数等已有结果。
Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.
We introduce Yetter-Drinfeld modules over a weak Hopf algebra $H$, and show that the category of Yetter-Drinfeld modules is isomorphic to the center of the category of $H$-modules. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If $H$ is finitely generated and projective, then we introduce the Drinfeld double using duality results between entwining structures and smash product structures, and show …
In this paper we prove a ``Leray theorem'' for preLie algebras. We define a notion of ''Hopf'' preLie algebra: it is a preLie algebra together with a non associative permutative coproduct D and a compatibility relation between the preLie product and the coproduct D. A nonassociative permutative algebra is a vector space together with a product satisfying the relation (ab)c=(ac)b. A non associative permutative coalgebra is the dual notion. We prove that any such graded connected "Hopf" preLie algebra is a free preLie algebra. It uses the description of preLie algebras in term of rooted trees de…
Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions. This allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we explore some properties of the fixed point subalgebra with relations to a partial action of a Hopf algebra. We also construct, for partial actions of finite dimensional Hopf algebras a Morita context relating the fixed point subalgebra and the…
In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the renormalization procedure. Both structures are related to some semi-direct coproduct of Hopf algebras.
We show how to get a non-commutative product for functions on space-time starting from the deformation of the coproduct of the Poincare' group using the Drinfel'd twist. Thus it is easy to see that the commutative algebra of functions on space-time (R^4) can be identified as the set of functions on the Poincare' group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincare' group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and invol…
In this paper, we generalize Majid's bicrossproduct construction. We start with a pair (A,B) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A # B. The left coaction of B on A gives a possible coproduct on A # B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for this to be a proper coproduct on A # B. The result is again a regula…
Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf
本文构造显式链映射，在温和假设下证明Hopf-Galois扩张的Hochschild上同调与H-不变子代数同构，推广群作用经典结果，并计算量子(-1)-平面与Kac-Paljutkin Hopf代数smash积的上同调及杯积。
If H is a finite dimensional quasi-Hopf algebra and A is a left H-module algebra, we prove that there is a Morita context connecting the smash product A#H and the subalgebra of invariants A^{H}. We define also Galois extensions and prove the connection with this Morita context, as in the Hopf case.
Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)} which is a left-left Yetter-Drinfeld module over H, having extra properties that allow to make a smash product B^{co(H)}# H which is an H-bicomodule algebra, isomorphic to B.
Let $H$ be a finite dimensional semisimple Hopf algebra and $R$ a braided Hopf algebra in the category of Yetter-Drinfeld modules over $H$. When $R$ is a Calabi-Yau algebra, a necessary and sufficient condition for $R#H$ to be a Calabi-Yau Hopf algebra is given. Conversely, when $H$ is the group algebra of a finite group and the smash product $R#H$ is a Calabi-Yau algebra, we give a necessary and sufficient condition for the algebra $R$ to be a Calabi-Yau algebra.
For an action $α$ of a group $G$ on an algebra $R$ (over $\Bbb C$), the crossed product $R\times_αG$ is the vector space of $R$-valued functions with finite support in $G$, together with the twisted convolution product given by $$(ξη)(p) = \sum_{q \in G} ξ(q) α_q (η(q^{-1}p))$$ where $p\in G$. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebra $A$ on an algebra $R$, it is possible to make the tensor product $R\ot A$ into an algebra by using a twisted product, involving the action. In this case, the algebra is called the smash product and denot…
We construct the analogue of Takeuchi's free Hopf algebra in the setting of Poisson Hopf algebras. More precisely, we prove that there exists a free Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful functor from the category of Poisson Hopf algebras to the category of coalgebras has a left adjoint. In particular, we also prove that the category of Poisson Hopf algebras is a reflective subcategory of the category of Poisson bialgebras. Along the way, we describe coproducts and coequalizers in the category of Poisson Hopf algebras, therefore showing that the latter catego…
In this work we deal with partial (co)action of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra $R^{\underline{coA}}$ with a certain subalgebra of the smash product $R\#\widehat{A}$. Besides this we present the notion of partial Galois coaction, which is closely related to this Morita context.
An internal coproduct is described, which is compatible with Hoffman's quasi-shuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with J. Ecalle's mould calculus, i.e. mould composition and contracting arborification, is precised.
Let $A$ and $B$ be algebras and coalgebras in a braided monoidal category $\Cc$, and suppose that we have a cross product algebra and a cross coproduct coalgebra structure on $A\ot B$. We present necessary and sufficient conditions for $A\ot B$ to be a bialgebra, and sufficient conditions for $A\ot B$ to be a Hopf algebra. We discuss when such a cross product Hopf algebra is a double cross (co)product, a biproduct, or, more generally, a smash (co)product Hopf algebra. In each of these cases, we provide an explicit description of the associated Hopf algebra projection.
For any strong smash product algebra $A\#_{_R}B$ of two algebras $A$ and $B$ with a bijective morphism $R$ mapping from $B\ot A$ to $A\ot B$, we construct a cylindrical module $A\natural B$ whose diagonal cyclic module $Δ_{\bullet}(A\natural B)$ is graphically proven to be isomorphic to $C_{\bullet}(A\#_{_R}B)$ the cyclic module of the algebra. A spectral sequence is established to converge to the cyclic homology of $A\#_{_R}B$. Examples are provided to show how our results work. Particularly, the cyclic homology of the Pareigis' Hopf algebra is obtained in the way.
Let $R$ be an associative ring and $e,f$ idempotent elements of $R$. In this paper we introduce the notion of $(e,f)$-invertibility for an element of $R$ and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra $H$ and an algebra $A$, we present sufficient conditions for $A$ to admit an inner action of $H$. We also prove that if $A$ is a left $H$-module algebra then $H$ acts innerly on the smash product $A\#H$ if and only if $H$ is a quantum commutative weak Hopf algebra.
A vector species is a functor from the category of finite sets with bijections to vector spaces; informally, one can view this as a sequence of $S_n$-modules. A Hopf monoid (in the category of vector species) consists of a vector species with unit, counit, product, and coproduct morphisms satisfying several compatibility conditions, analogous to a graded Hopf algebra. We say that a Hopf monoid is strongly linearized if it has a "basis" preserved by its product and coproduct in a certain sense. We prove several equivalent characterizations of this property, and show that any strongly linearized…
We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted Hopf pairing. We state a Stone--von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction.
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