Products of Linear Maps on Bialgebras,with Applications to Left Hopf Algebras and Hopf Algebroids

The Doi–Koppinen generalized smash product can be applied to linear maps on bialgebras, and we define a new associative product for linear maps on bialgebras. We comment on the applications of these products to left Hopf algebras and 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–Galois characters to be independent on the choice of strong connections. The results concerning cleft extensions and crossed product are then extended to the case of weak cleft extensions of Hopf algebroids hereby defined. Dedicated to Stef Caenepeel on the occasion of his 50th birthday.     

Integral Theory for Hopf Algebroids

The theory of integrals is used to analyze the structure of Hopf algebroids. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it is a semi-simple extension and if and only if the Hopf algebroid possesses a normalized integral. It is a Frobenius extension if and only if the Hopf algebroid possesses a nondegenerate integral. We give also a sufficient and necessary condition in terms of integrals, under which it is a quasi-Frobenius extension, and illustrate by an example that this condition does not hold true in general. Our results are generalizations of classical results on Hopf algebras.     

Corrigendum to Cleft Extensions of Hopf Algebroids

Theorem 2.2 stated a monoidal isomorphism between the comodule categories of two bialgebroids in a Hopf algebroid. The proof of Theorem 2.2 was based on the journal version of Brzeziński (Ann Univ Ferrara Sez VII (NS) 51:15–27, 2005, Theorem 2.6), whose proof turned out to contain an unjustified step. Here we show that all other results in our paper remain valid if we drop unverified Theorem 2.2, and return to an earlier definition of a comodule of a Hopf algebroid that distinguishes between comodules of the two constituent bialgebroids.     

Duality for Hopf orders

In this paper we use duality to construct new classes of Hopf orders in the group algebra , where is a finite extension of and denotes the cyclic group of order . Included in this collection is a subcollection of Hopf orders which are realizable as Galois groups.

     

相关Hopf模的对偶

本文的目的就是给出相关Hopf模的对偶性质.在第一部分,证明了相关Hopf模的对偶模仍是相关Hopf模.特别地,Hopf模的对偶仍是Hopf模.在第二、第三两部分,分别给出相关Hopf模的对偶相关Hopf模的基本结构定理及Maschke定理.     

Duality for Finite Hopf Algebras Explained by Corings

We give a coring version for the duality theorem for actions and coactions of a finitely generated projective Hopf algebra. We also provide a coring analogue for a theorem of H.-J. Schneider, which generalizes and unifies the duality theorem for finite Hopf algebras and its refinements. This paper was written while the first author visited the Mathematics Departments of Syracuse University and California State University Dominguez Hills. He would like to thank both departments for their hospitality.     

有限Hopf C~*-代数的表示与对偶

设A是有限维Hopf C-代数,H是Hilbert空间.如果存在A在L(H)上的作用γ,在此作用下,L(H)成为具有共轭性质的模代数且H上内积是A-不变的,则A存在惟一的C-表示(θ,H),L(H)的A-不变子空间恰好是θ(A)的换位子.     

弱Hopf代数的对偶定理及Smash积的模结构

本文主要包括两方面内容:首先将Hopf代数理论中的对偶定理部分地推广到弱Hopf代数的情况;然后讨论弱Hopf代数上的Smash积的模及模同态,并部分地推广了Maschke-type定理．     

The Duality Theorem for Weak Hopf Algebra (Co) Actions

In this article, we mainly study a new notion of a generalized smash product for weak Hopf comodule algebras and provide a new version of the duality theorem for weak smash products as an application.     

Left counital Hopf algebras on free Nijenhuis algebras

Factorization in algebra is an important problem. In this paper, we first obtain a unique factorization in free Nijenhuis algebras. By using of this unique factorization, we then define a coproduct and a left counital bialgebraic structure on a free Nijenhuis algebra. Finally, we prove that this left counital bialgebra is connected and hence obtain a left counital Hopf algebra on a free Nijenhuis algebra.     

On Deformations of Lie Algebroids

For any Lie algebroid A, its 1-jet bundle ${\mathfrak{J} A}$ is a Lie algebroid naturally and there is a representation ${\pi:\mathfrak{J} A\longrightarrow\mathfrak{D} A}$ . Denote by ${{\rm d}_{\mathfrak{J}}}$ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A,A)_{{\mathfrak{D}} A}),{\rm d}_{\mathfrak{J}})}$ of the cochain complex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A, A)),{\rm d}_\mathfrak{J})}$ .     

Foliated Lie and Courant Algebroids

If A is a Lie algebroid over a foliated manifold (M, F){(M, {\mathcal {F}})}, a foliation of A is a Lie subalgebroid B with anchor image TF{T{\mathcal {F}}} and such that A/B is locally equivalent with Lie algebroids over the slice manifolds of F{\mathcal F}. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's supervector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last section, we define a corresponding notion of a foliation of a Courant algebroid A as a bracket–closed, isotropic subbundle B with anchor image TF{T{\mathcal {F}}} and such that B ^{^} /B{B^{ \bot } /B} is locally equivalent with Courant algebroids over the slice manifolds of F{\mathcal F}. Examples that motivate the definition are given.     

Distinguished Connections on Finsler Algebroids

Considering the prolongation of a Lie algebroid,the authors introduce Finsler algebroids and present important geometric objects on these spaces.Important endomorphisms like conservative and Barthel,Cartan tensor and some distinguished connections like Berwald,Cartan,Chern-Rund and Hashiguchi are introduced and studied.     

On Almost Complex Lie Algebroids

The almost complex Lie algebroids over smooth manifolds are considered in the paper. In the first part, we give some examples and we extend some basic results from almost complex manifolds to almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied.     

Vanishing Theorems on Holomorphic Lie Algebroids

The paper describes a Bochner-type study for holomorphic horizontal vector fields defined on a holomorphic Finsler algebroid E. We obtain in this setting a vanishing theorem for horizontal fields with compact support on E.     

Vertex Poisson Algebras Associated with Courant Algebroids

We study relationships between vertex Poisson algebras and Courant algebroids. For any ?-graded vertex Poisson algebra A = ? _n∈? A _(n), we show that A ₍₁₎ is a Courant A ₍₀₎-algebroid. On the other hand, for any Courant 𝒜-algebroid ?, we construct an ?-graded vertex Poisson algebra A = ? _n∈? A _(n) such that A ₍₀₎ is 𝒜 and the Courant 𝒜-algebroid A ₍₁₎ is isomorphic to ? as a Courant 𝒜-algebroid.     

Lie Algebroids, Holonomy and Characteristic Classes

We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and prove a Stability Theorem. We also introduce secondary or exotic characteristic classes, thus providing invariants which generalize the modular class of a Lie algebroid.