弱Hopf代数上的Militaru-Stefan提升定理

本文研究了弱Hopf-Galois扩张的扩张模.利用忠实平坦的弱Hopf-Galois扩张理论,研究了弱Hopf代数上的Militaru-Stefan提升定理,推广了文献[10]中的相应结果.进一步地,通过诱导模的自同态环的cleft扩张刻画了弱稳定模.     

Galois theory for bialgebroids, depth two and normal Hopf subalgebras

We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a leftT-Galois extension for some right finite projective left bialgebroid over some algebraR if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

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Sunto Riduciamo alcune prove di [16,11,12] a quasibasi di profondità due da un lato solo, un approccio minimalistico che conduce ad una caratterizzazione di estensioni di Galois per bialgebroidi proietivi finiti senza la proprietà di estensione di Frobenius. Dimostriamo che un'algebra che sia un'estensione propria è un'estensioneT-Galois sinistra per qualche bialgebroide finito proiettivo a sinistra su qualche algebraR se, e solo se, è un'estensione di profondità due a sinistra e bilanciata a sinistra. Scambiando destra e sinistra nell'enunciato, otteniamo una caratterizzazione di estensioni di Galois destre per bialgebroidi finiti proiettivi a destra. Guardando ad esempi di profondità due, otteniamo che una sottoalgebra di Hopf è normale se, e solo se, è un'estensione Hopf-Galois. Caratterizziamo le estensioni Hopf-Galois deboli finite usando un'applicazione canonica di Galois alternativa ottenendo parecchi corollari: queste sono di profondità due e la suriettività dell'applicazione di Galois implica la sua biiettività.

     

Coloured peak algebras and Hopf algebras

For G a finite abelian group, we study the properties of general equivalence relations on G _n = G ⁿ ⋊ _n , the wreath product of G with the symmetric group _n , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of G _n as well as graded connected Hopf subalgebras of ⨁ _{n≥ o} G _n . In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects. 2000 Mathematics Subject Classification Primary: 16S99; Secondary: 05E05, 05E10, 16S34, 16W30, 20B30, 20E22Bergeron is partially supported by NSERC and CRC, CanadaHohlweg is partially supported by CRC     

Two-Parameter Quantum Groups and Drinfel'd Doubles

We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras and . We show that these quantum groups can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix and a quantum Casimir element. We discuss isomorphisms among these quantum groups and connections with multiparameter quantum groups.     

Extensions of the endomorphism algebra of weak comodule algebras

Let H be a weak Hopf algebra, and let A/B be a weak right H-Galois extension. In this paper, we mainly discuss the extension of the endomorphism algebra of a module over A. A necessary and sufficient condition for such an extension of the endomorphism algebra to be weak H-Galois is obtained by using Hopf-Galois theory and Morita theory.     

Frobenius extensions of subalgebras of Hopf algebras

We consider when extensions of subalgebras of a Hopf algebra are -Frobenius, that is Frobenius of the second kind. Given a Hopf algebra , we show that when are Hopf algebras in the Yetter-Drinfeld category for , the extension is -Frobenius provided is finite over and the extension of biproducts is cleft.

More generally we give conditions for an extension to be -Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants.

We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.

     

Subgroup Depth and Twisted Coefficients

Danz computes the depth of certain twisted group algebra extensions in [10 Danz, S. (2011). The depth of some twisted group extensions. Comm. Alg. 39:1–15. [Google Scholar]], which are less than the values of the depths of the corresponding untwisted group algebra extensions in [8 Burciu, S., Kadison, L., Külshammer, B. (2011). On subgroup depth (with an appendix by B. Külshammer and S. Danz). I. E. J. A. 9:133–166. [Google Scholar]]. In this article, we show that the closely related h-depth of any group crossed product algebra extension is less than or equal to the h-depth of the corresponding (finite rank) group algebra extension. A convenient theoretical underpinning to do so is provided by the entwining structure of a right H-comodule algebra A and a right H-module coalgebra C for a Hopf algebra H. Then A ? C is an A-coring, where corings have a notion of depth extending h-depth. This coring is Galois in certain cases where C is the quotient module Q of a coideal subalgebra R ? H. We note that this applies for the group crossed product algebra extension, so that the depth of this Galois coring is less than the h-depth of H in G. Along the way, we show that the subgroup depth behaves exactly like the combinatorial depth with respect to the core of a subgroup, and extend results in [22 Kadison, L. (2014). Hopf subalgebras and tensor powers of generalized permutation modules. J. Pure Appl. Alg. 218:367–380.[Crossref], [Web of Science ®] , [Google Scholar]] to coideal subalgebras of finite dimension.     

Hopf subalgebras of pointed Hopf algebras and applications

In this paper we construct certain Hopf subalgebras of a pointed Hopf algebra over a field of characteristic 0. Some applications are given in the case of Hopf algebras of dimension 6, and , where and are different prime numbers.

     

Bialgebra Cyclic Homology with Coefficients

We show that one can extend the definition of Hopf cyclic homology with coefficients such that one can use bialgebras and a larger class of coefficient module/co-modules. With the help of this extension, we calculate the bialgebra cyclic homology of U_q() the quantum deformation of an arbitrary semi-simple Lie algebra and (N) the Hopf algebra of foliations of codimension N, with several coefficient modules.     

Pointed Hopf Algebras with Triangular Decomposition

In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) ? BC) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to \(\mathfrak {sl}_{2}\).     

BRST Operator for Quantum Lie Algebras and Differential Calculus on Quantum Groups

For a Hopf algebra , we define the structures of differential complexes on two dual exterior Hopf algebras: (1) an exterior extension of and (2) an exterior extension of the dual algebra ^*. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on . The first differential complex is an analogue of the de Rham complex. When ^* is a universal enveloping algebra of a Lie (super)algebra, the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST operator Q. We give a recursive relation that uniquely defines the operator Q. We construct the BRST and anti-BRST operators explicitly and formulate the Hodge decomposition theorem for the case of the quantum Lie algebra U _q(gl(N)).     

Normal Hopf subalgebras of semisimple Drinfeld doubles

In this paper we study normal Hopf subalgebras of a semisimple Drinfeld double. This is realized by considering an analogue of Goursat’s lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension.     

THE NICHOLS ZOELLER THEOREM FOR HOPF ALGEBRAS IN THE CATEGORY OF YETTER DRINFELD MODULES

In 1989 Nichols and Zoeller [NZ] Nichols, W. D. and Zoeller, M. B. 1989. A Hopf algebra freeness theorem. Amer. J. Math., 111: 381–385. [Crossref], [Web of Science ®] , [Google Scholar] showed that finite dimensional k-Hopf algebras are free over Hopf subalgebras. An analog result for Yetter Drinfeld Hopf algebras was not known. In this paper the existence of such a basis will be proved. Moreover the existence of a basis in a certain categorial sense cannot be expected.     

On 2-Homogeneity of Monounary Algebras

Fraïssé introduced the notion of a k-set-homogeneous relational structure. In the present paper the following classes of monounary algebras are described: --the class of all algebras which are 2-set-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively, and --the class of all algebras which are 2-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively.     

External Homogenization for Hopf Algebras: Applications to Maschke's Theorem

We apply to Hopf algebras a construction from graded rings, called the group ring of a graded ring. By using this construction we study the transfer of properties between certain categories of relative Hopf modules. As another application, we obtain a Maschke-type theorem for a Galois extension over a semisimple Hopf algebra.     

Relativizations of relation algebras by the diversity

The elements of a relation algebra A that are below a fixed elementa form a relative subalgebrab Aa of A. It was shown by H. Andréka that the class of all relative subalgebras of relation algebras is not a variety, but it follows immediately from results of R. L. Kramer that the closure of this class under subalgebras is a finitely based variety. We show that the relative subalgebras A0 with 0 the diversity element of A, form a finitely based variety. We also show that A is determined by A0, up to a direct factor that is Boolean (the relative product coincides with the meet).Presented by R. McKenzie.