Construction of Hopf algebroids

Cleft extensions of Hopf algebroids

The notions of a cleft extension and a cross product with a Hopf algebroid are introduced and studied. In particular it is shown that an extension (with a Hopf algebroid $H= (H_L,H_R)$) is cleft if and only if it is $H_R$-Galois and has a normal basis property relative to the base ring $L$ of $H_L$. Cleft extensions are identified as crossed products with invertible cocycles. The relationship between the equivalence classes of crossed products and gauge transformations is established. Strong connections in cleft extensions are classified and sufficient conditions are derived for the Chern-Galo…

单项式曲线上微分算子环是一个Hopf代数胚

The ring of differential operators on a monomial curve is a Hopf algebroid

本文证明数值半群代数对应的尖点曲线上，微分算子环构成余交换且余幂零的左Hopf代数胚，并在半群对称即曲线为Gorenstein时进一步成为具有对极的全Hopf代数胚。

The elliptic GL(n) dynamical quantum group as an h-Hopf algebroid

Using the language of h-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group, F_ell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra sl_n. We apply the generalized FRST construction and obtain an h-bialgebroid F_ell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing …

Multiplier Hopf algebroids arising from weak multiplier Hopf algebras

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This r…

Galois theory for Hopf algebroids

An extension B\subset A of algebras over a commutative ring k is an H-extension for an L-bialgebroid H if A is an H-comodule algebra and B is the subalgebra of its coinvariants. It is H-Galois if the canonical map A\otimes_B A\to A\otimes_L H is an isomorphism or, equivalently, if the canonical coring (A\otimes_L H:A) is a Galois coring. In the case of a Hopf algebroid H=(H_L,H_R,S) any H_R-extension is shown to be also an H_L-extension. If the antipode is bijective then also the notions of H_R-Galois extensions and of H_L-Galois extensions are proven to coincide. Results about bijective entwi…

Isotopy Quotients of Hopf Algebroids and the Fundamental Groupoid of Digraphs

We build on our construction of Hopf algebroids from noncommutative calculi under the further assumption of surjectivity for the calculus. We also introduce the notions of Hopf ideals and isotopy quotients for arbitrary Hopf algebroids. Using these ingredients, we prove a Riemann-Hilbert correspondence for digraphs, by showing that the groupoid algebra of the fundamental groupoid of a digraph is isomorphic to the isotopy quotient of the Hopf algberoid corresponding to flat connections over the digraph.

Tannaka-Krein duality for Hopf algebroids

We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The $\coend$ of such a functor turns out to be a Hopf algebroid over this ring. Using the result of a previous paper we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.

Hopf algebroid的对应定理及其在affine groupoids中的应用

Correspondence theorems for Hopf algebroids with applications to affine groupoids

本文建立了bialgebroid中单边coideal子环与单边理想双边coideal的对应关系，证明其在Hopf algebroids中满足附加条件时为双射，并应用于研究交换Hopf algebroids中的正规Hopf理想与正规affine subgroupoids的联系。

A Schneider type theorem for Hopf algebroids

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B\subseteq A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the non-commutative base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B\subseteq A and also to the equivalence of the category of relative Hopf modules to the category of B-modules. This extends a classical theorem by H.-J. Schneider on Galois extensions by a Hopf algebra. Our main tool is an observation that relative injectivity of…

关于Hopf algebroids的对偶代数

On Dual Algebras of Hopf Algebroids

本文研究了Hopf algebroids的对偶代数，并将余模转化为其对偶代数上的模结构。

通过 Hopf algebroid 研究 nil Hecke 代数

A study of nil Hecke algebras via Hopf algebroids

本文通过 Kostant 和 Kumar 定义的余乘法，利用 Demazure 算子与单反射间的混合二面体辫关系，将与 Coxeter 系统关联的仿射 nil Hecke 代数转化为无对极的 Hopf algebroid，并附带了相关领域的实例。

Hopf Algebroid Twists for Deformation Quantization of Linear Poisson Structures

In our earlier article [Lett. Math. Phys. 107 (2017), 475-503, arXiv:1409.8188], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every deformation quantization leads to a Drinfeld twist of the associative bialgebroid of h-adic series of differential operators on a fixed Poisson manifold. In the case of linear Poisson structures, the twisted bialgebroid essentially coincides with our construction. Using our explicit description of the Hopf algebroid, we compute the corresponding Drinfeld twist ex…

Partial Representations of Hopf Algebras

In this work, the notion of partial representation of a Hopf algebra is introduced and its relationship with partial actions of Hopf algebras is explored. Given a Hopf algebra $H$, one can associate it to a Hopf algebroid $H_{par}$ which has the universal property that each partial representation of $H$ can be factorized by an algebra morphism from $H_{par}$. We define also the category of partial modules over a Hopf algebra $H$, which is the category of modules over its associated Hopf algebroid $H_{par}$. The Hopf algebroid structure of $H_{par}$ enables us to enhance the category of partial…

标量扩张 Hopf algebroids

Scalar extension Hopf algebroids

本文补全了 Brzeziński 和 Militaru 关于 smash product 代数上 Hopf algebroid 的 antipode 反同态性质的完整证明，并提出不要求 $H$ 的 antipode 可逆的广义对称 Hopf algebroid 构造。

关于Hopf代数胚中理想余理想与余模子环之间Galois型对应的一个注记

A remark on the Galois-type correspondence between ideal coideals and comodule subrings of a Hopf algebroid

本文建立了Hopf代数胚中特定左理想余理想与右余理想子环之间的双射对应，推广了作者与合作者的近期结果，并在Hopf代数情形下扩展了已知对应关系。

Poincaré Duality for Hopf algebroids

We prove a twisted Poincaré duality for (full) Hopf algebroids with bijective antipode. As an application, we recover the Hochschild twisted Poincaré duality of Van Den Bergh [VDB]. We also get a Poisson twisted Poincaré duality, which was already stated for oriented Poisson manifolds in [CLYZ].

无对径的Hopf代数胚的裂扩张

Cleft Extensions for Hopf Algebroids without Antipodes

本文引入无对径Hopf代数胚的裂扩张，证明其与σ-扭曲交叉积、带正规基的Hopf-Galois扩张等价，研究相关Ehresmann Hopf代数胚的同构性，推广了相关理论。

A note on the restricted universal enveloping algebra of a restricted Lie-Rinehart Algebra

Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping algebra of a restricted Lie algebra admits a quotient Hopf algebra which is finite-dimensional if the Lie algebra is. Rumynin has shown that suitably defined restricted Lie algebroids allow to define restricted universal enveloping algebras that are finitely generated projective if the Lie algebroid is. T…

Hopf algebroids and Galois extensions

To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := \End_BA_B$ and $T := (A ø_B A)^B$ over the centralizer $R$ using the depth two theory in math.RA/0108067. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra \cite{KT,Doi} to depth two extensions using coring theory math.RA/0002105. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem 5.1 in math.QA/9505024 for smash products with special modules over the Drinfel'd double, the Miy…

Hopf代数胚的测量与循环（上）同调理论中的态射

Measurings of Hopf algebroids and morphisms in cyclic (co)homology theories

本文研究Hopf代数胚及SAYD模上的测量，证明其诱导循环同调与循环上同调态射，并给出SAYD模范畴及非Σ operad上循环单位comp模的充实结构。

Hopf代数胚的Hopf双Galois理论

Hopf BiGalois Theory for Hopf Algebroids

该研究发展了不要求对极映射的Hopf代数胚的Hopf双Galois理论，利用Chemla的结果解决了右Galois映射非良定义的障碍，构造了Ehresmann Hopf代数胚并将其应用于量子结构示例。

关于 Hopf algebroids 与 Lie algebroids 之间的微分与积分

Towards differentiation and integration between Hopf algebroids and Lie algebroids

本文在交换 Hopf algebroids 与 Lie-Rinehart algebras 背景下，构造了形式微分与积分的反变函子，揭示了二者的伴随关系，并利用微分函子分析了 Hopf algebroids 态射的几何可分性。

后霍普夫余代数、后李-里纳尔特代数与几何数值积分

Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration

本文提出后霍普夫余代数概念，构造作用后霍普夫余代数，并展示后李-里纳尔特代数的 universal enveloping 代数性质，应用于流形上的几何数值积分。

Representative functions on discrete groupoids and duality with Hopf algebroids

The aim of this paper is to establish a duality between the category of discrete groupoids and the category of geometrically transitive commutative Hopf algebroids in the sense of P. Deligne and A. Bruguières. In one direction we have the usual contravariant functor which assigns to each Hopf algebroid its characters groupoid (the fiber groupoid at the ground field). In the other direction we construct the contravariant functor which associated to each discrete groupoid its Hopf algebroid of representative functions. This duality extends the well known duality between discrete groups and commu…

Topological tensor product of bimodules, complete Hopf Algebroids and convolution algebras

Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the s…

Homotopy theory of comodules over a Hopf algebroid

Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Gamma). That is, we construct the derived category Stable(Gamma) of (A, Gamma) as the homotopy category of a Quillen model structure on the category of unbounded chain complexes of Gamma-comodules. This derived category is obtained by inverting the homotopy isomorphisms, NOT the homology isomorphisms. We establish the…

与Hopf代数胚相关的Brace $B_{\u221e}$代数

Brace $B_{\infty}$ algebras associated with Hopf algebroids

本研究将brace $B_{\u221e}$代数的operadic建模应用于Xu型Hopf代数胚，构造了I型与II型扭brace $B_{\u221e}$代数间的严格$B_{\u221e}$同构，并借此揭示了两类源自李代数对的brace $B_{\u221e}$代数间的隐藏关联。

Quantum Inverse Semigroups

In this work, the notion of a quantum inverse semigroup is introduced as a linearized generalization of inverse semigroups. Beyond the algebra of an inverse semigroup, which is the natural example of a quantum inverse semigroup, several other examples of this new structure are presented in different contexts, those are related to Hopf algebras, weak Hopf algebras, partial actions and Hopf categories. Finally, a generalized notion of local bisections are defined for commutative Hopf algebroids over a commutative base algebra giving rise to new examples of quantum inverse semigroups associated t…

The Hopf algebroid structure of differentially recursive sequences

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive sequences over a given field with a non-zero differential. We show that these sequences form a two-sided vector space that admits, in a canonical way, a structure of Hopf algebroid over the subfield of constant elements. We prove that it is the direct…

Morita theory for Hopf algebroids, principal bibundles, and weak equivalences

We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting them. This gives a positive answer to a conjecture due to Hovey and Strickland. We also prove that principal (left) bundles lead to a bicategory together with a 2-functor from flat Hopf algebroids to trivial principal bundles. This turns out to be the universal solution for 2-functors which send weak equivalences to invertible 1-cells. Our approach can be seen as an algebraic counterpart to Lie groupoid Morita theory.

Lie algebra type noncommutative phase spaces are Hopf algebroids

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a com…

Integral theory for left Hopf left bialgebroids

We study integral theory for left (or right) Hopf left bialgebroids. Contrary to Hopf algebroids, the latter ones don't necessary have an antipode $S$ but, for any element $u$, the elements $u_{(1)} \otimes S(u_{(2)})$ (or $u_{(2)}\otimes S^{-1}(u_{(1)})$ ) does exist. Our results extend those of G. Böhm who studied integral theory for Hopf algebroids. We make use of recent results about left Hopf left bialgebroids. We apply our results to the restricted enveloping algebra of a restricted Lie Rinehart algebra.

Pairing and duality of algebraic quantum groupoids

Algebraic quantum groupoids have been developed by two of the authors (AVD and SHW) of this note in a series of papers. Regular multiplier Hopf algebroids are obtained also by two authors (TT and AVD). Integral theory and duality for those have been studied by one author here (TT). Finally, again two authors of us (TT and AVD) have investigated the relation between weak multiplier Hopf algebras and multiplier Hopf algebroids. In the paper 'Weak multiplier Hopf algebras III. Integrals and duality' (by AVD and SHW), one of the main results is that the dual of an algebraic quantum groupoid, admit…

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 \cite{BB} with the notion of crossed product with a weak Hopf algebra introduced in~\cite{AG}

Finitary Galois extensions over noncommutative bases

We study Galois extensions Coinv(M)<M for M an H-comodule algebra and H a Frobenius Hopf algebroid. We obtain generalizations of various theorems in Hopf-Galois theory by Kreimer-Takeuchi, Doi-Takeuchi and Cohen-Fischman-Montgomery. An algebra extension is Galois precisely if it is balanced, depth 2, and Frobenius. Then we show that Yetter-Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by the Brzezinski-Militaru Theorem. Contravariant "fiber functors" are used to prove an analogue of Ulbrich's Theorem and t…

Graded comodule categories with enough projectives

It is well-known that the category of comodules over a flat Hopf algebroid is abelian but typically fails to have enough projectives, and more generally, the category of graded comodules over a graded flat Hopf algebroid is abelian but typically fails to have enough projectives. In this short paper we prove that the category of connective graded comodules over a connective, graded, flat, finite-type Hopf algebroid has enough projectives. Applications to algebraic topology are given: the Hopf algebroids of stable co-operations in complex bordism, Brown-Peterson homology, and classical mod $p$ h…

Dual Constructions for Partial Actions of Hopf Algebras

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.

Modified weak multiplier Hopf algebras

Let $(A,Δ)$ be a regular weak multiplier Hopf algebra. Denote by $E$ the canonical idempotent of $(A,Δ)$ and by $B$ the image of the source map. Recall that $B$ is a non-degenerate algebra, sitting nicely in the multiplier algebra $M(A)$ of $A$ so that also $M(B)$ can be viewed as a subalgebra of $M(A)$. Assume that $u,v$ are invertible elements in $M(B)$ so that $E(vu\otimes 1)E=E$. This last condition is obviously fulfilled if $u$ and $v$ are each other inverses, but there are also other cases. Now modify $Δ$ and define $Δ'(a)=(u\otimes 1)Δ(a)(v\otimes 1)$ for all $a \in A$. We show in this …

Heisenberg double H(B^*) as a braided commutative Yetter-Drinfeld module algebra over the Drinfeld double

We study the Yetter--Drinfeld D(B)-module algebra structure on the Heisenberg double H(B^*) endowed with a "heterotic" action of the Drinfeld double D(B). This action can be interpreted in the spirit of Lu's description of H(B^*) as a twist of D(B). In terms of the braiding of Yetter--Drinfeld modules, H(B^*) is braided commutative. By the Brzezinski--Militaru theorem, H(B^*)#D(B) is then a Hopf algebroid over H(B^*). For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_q(s…

Cohomology for partial actions of Hopf algebras

In this work, the cohomology theory for partial actions of co-commutative Hopf algebras over commutative algebras is formulated. This theory generalizes the cohomology theory for Hopf algebras introduced by Sweedler and the cohomology theory for partial group actions, introduced by Dokuchaev and Khrypchenko. Some nontrivial examples, not coming from groups are constructed. Given a partial action of a co-commutative Hopf algebra $H$ over a commutative algebra $A$, we prove that there exists a new Hopf algebra $\widetilde{A}$, over a commutative ring $E(A)$, upon which $H$ still acts partially a…

Skew Hopf algebras, irreducible extensions and the pi-method

To a depth two extension A | B, we associate the dual bialgebroids S := \End {}_BA_B and T := (A ø_B A)^B over the centralizer R=C_A(B). In the set-up where R is a subalgebra of B, which is quite common, two nondegenerate pairings of S and T will define an anti-automorphism τof the algebra S. Making use of a two-sided depth two structure, we prove that τis an antipode and S is a Hopf algebroid of a type we call skew Hopf algebra. A final section discusses how τand the nondegenerate pairings generalize to modules via the pi-method for depth two, and a certain derived mapping of cochain complexe…

Galois extensions over commutative and non-commutative base

This paper is a written form of a talk. It gives a review of various notions of Galois (and in particular cleft) extensions. Extensions by coalgebras,bialgebras and Hopf algebras (over a commutative base ring) and by corings,bialgebroids and Hopf algebroids (over a non-commutative base algebra) are systematically recalled and compared. In the first version of this paper, the journal version of [15, Theorem 2.6] was heavily used, in two respects. First, it was applied to establish an isomorphism between the comodule categories of two constituent bialgebroids in a Hopf algebroid. Second, it was …

Finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard-Vessiot theory

A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each finite-dimensional differential module over differential field in such a way that the category of differential modules it generates is equivalent, as a symmetric monoidal category, to the category of representations of the group. Its underlying set is then recognized as the group of differential automorphisms of the Picard-Vessiot field extension of the base field for this differential module. These results can be obtained by means of a Tannaka reconstruction process, applied to the abelian categor…

Semisimple Hopf algebras and their depth two Hopf subalgebras

We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field $k$ is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time creates no new simple constituents. This point of view was taken in the paper http://arxiv.org/abs/math/0409346 which established a normality result in case H and K are finite group algebras. On the other hand this means that K is normal in H when H | K is a Galois extension with respect to action o…

弱偏表示

Weak Partial Representations

本文引入了弱 Hopf 代数的偏表示概念，提出了泛代数 $H_{par}^w$ 并证明其具有 Hopf algebroid 与量子逆半群结构，揭示了模范畴代数对象与对称偏模代数的对应关系。

芳香和聚类多指标：代数结构与霍普夫嵌入

Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings

本文引入芳香和聚类多指标，通过代数结构简化体积保持方法的研究，同时保留原有结构，扩展霍普夫嵌入至芳香情境。

关于尖点Hopf代数的部分表示

On partial representations of pointed Hopf algebras

本文研究尖点Hopf代数的部分表示，证明当H的类群元素构成有限群G时，部分表示对应的Hopf algebroid H_par可分解为由G确定的groupoid分量索引的单位理想直和。

由弱霍普夫代数作用捕捉代数的对称性

Symmetries of algebras captured by actions of weak Hopf algebras

本文提出利用弱霍普夫代数作用捕捉代数对称性的方法，拓展了传统群和李代数作用的框架。

When Ext is a Batalin-Vilkovisky algebra

We show under what conditions the complex computing general Ext-groups carries the structure of a cyclic operad such that Ext becomes a Batalin-Vilkovisky algebra. This is achieved by transferring cyclic cohomology theories for the dual of a (left) Hopf algebroid to the complex in question, which asks for the notion of contramodules introduced along with comodules by Eilenberg-Moore half a century ago. Another crucial ingredient is an explicit formula for the inverse of the Hopf-Galois map on the dual, by which we illustrate recent categorical results and answer a long-standing open question. …

Regular and Biregular module algebras

Motivated by the study of von Neumann regular skew groups as carried out by Alfaro, Ara and del Rio in 1995 we investigate regular and biregular Hopf module algebras. If $A$ is an algebra with an action by an affine Hopf algebra $H$, then any $H$-stable left ideal of $A$ is a direct summand if and only if $A^H$ is regular and the invariance functor $(-)^H$ induces an equivalence of $A^H$-Mod to the Wisbauer category of $A$ as $A# H$-module. Analogously we show a similar statement for the biregularity of $A$ relative to $H$ where $A^H$ is replaced by $R=Z(A)\cap A^H$ using the module theory of …

Geometric structures encoded in the Lie structure of an Atiyah algebroid

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, we give two characterizations of the isomorphisms of the Lie algebras of sections for Atiyah algebroids associated to principle bundles with semisimple structure groups. For instance we prove that in the semisimple case the Lie algebras of sections are isomorphic if and only if th…

The Lie algebra of a Lie algebroid

Results describing Lie ideals and maximal finite-codimensional Lie subalgebras of the Lie algebras associated with Lie algebroids with non-singular anchor maps are presented. It is also proved that every isomorphism of such Lie algebras induces a diffeomorphism of base manifolds respecting the generalized foliations defined by the anchor maps.