Groups of order 4p, twisted wreath products and hopfgalois. The results, not previously published in book form, show that hopf algebras play a natural role in local galois module theory. Then hopfgalois descent theory for linear hopf categories, the hopf algebra version of a linear category, is developed. After giving a number of examples, we discuss galois extensions with additional properties, such as having a normal basis. Among the topics covered are results toward the classification of finitedimensional hopf algebras semisimple and nonsemisimple, as well as what is known about the extension theory of hopf algebras. Artins classic treatment of galois theory we know that l l. Here is a very brief indication of the origins of those connections. Pdf a galoistype correspondence theory for actions of.
Click download or read online button to get hopf algebras book now. All the maps in the diagrams are algebra homomorphisms given our hy. Twisted kummer and kummerartinschreier theories suwa, noriyuki, tohoku mathematical journal, 2008. This leads to the notion of hopf galois category extension. Journal of algebra 121, 488516 1989 hopfgalois extensions of algebras, the miyashitaulbrich action, and azumaya algebras yukio doi department of mathematics, fukui university, fukui 910, japan and mitsuhiro takeuchi institute of mathematics, university of tsukuba, ibaraki 305, japan communicated by nathan jacobson received march 20, 1987 to the memory of akira hattori 19271986, who. An extension b\subset a of algebras over a commutative ring k is an hextension for an lbialgebroid h if a is an hcomodule algebra and b is the subalgebra of. We investigate the category amh b of relative hopf bimodules, and the morita equivalences between a and b induced by them. A hopf algebra h over the commutative unital ring k gives an example of a hopf algebroid. Galois theory for multiplier hopf algebras with integrals. Hopf module duality applied to xouter galois theory akira masuokaa and tadashi yanaib a institute of mathematics, university of tsukuba, ibaraki 3058571, japan b niihama national college of technology, ehime 7928580, japan. Let lkbe a eld extension, ha cocommutative k hopf algebra. The talk is based on the paper 3, so no proofs are given, but we. Hopf galois theory and elliptic curves jean gillibert abstract. Our methods are based on the recently introduced galois theory for corings.
Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Chase and sweedler 8 generalized the concept of galois extension by replacing the group ring of the galois group by a hopf algebra, as follows. The structure of hopf algebras acting on galois extensions. Childs, on hopf galois extensions of elds and number rings, in perspectives in ring theory, f. Hopf algebras and galois theory lecture notes in mathematics. However, formatting rules can vary widely between applications and fields of interest or study. Forms of hopf algebras and galois theory citeseerx. Some tools 39 references 45 1 introduction hopfgalois extensions were introduced by chase and sweedler 8 in the commutative case and kreimer and takeuchi 25 in the case of.
Hopf module duality applied to xouter galois theory. Arithmetic structure of galois groups of pextensions of local elds with nilpotent class and m is a natural number. The aim of this note is to develop galois theory for weak hopf algebras. It includes many illustrating exam ples some of which. In this dissertation, we introduce the notions of hopf algebra and hopf galois extension, and discuss some results of hopf galois theory in the case of separable.
Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Hopf algebras arrived to the galois theory of rings as early as the 1960s independently of, but in fact similarly to, the way in which algebraic group schemes were introduced to the theory of etale. We intend the outline below to summarize how we achieve these goals. Hopfgalois theory and elliptic curves jean gillibert abstract. An extension b\subset a of algebras over a commutative ring k is an hextension for an lbialgebroid h if a is an hcomodule algebra and b is the. If a is generated as an algebra by a set b, to prove that a is a bialgebra we only need to check the diagrams in 1. Let lkbe a galois extension of elds with galois group gan. Descent and galois theory for hopf categories journal of.
Hopf galois theory expands the classical galois theory by considering the galois property in terms of the action of the group algebra kg on kk and then. Finally insection 7we consider a different approach to the galois correspondence via actions. Hopf galois objects were introduced by chase and sweedler 7, and was. F orms of hopf algebras and galois theor y bodo p areigis the theory of hopf algebras is closely connected with v arious applications, in particular to algebraic and formal groups. Childs, azumaya algebras which are not smash products. We will discuss an application to the computation of galois objects over tensor products, and to the problem of reducing the hopf algebra in a hopf galois object to a quotient hopf algebra here, however, larises because of a lack of commutativity rather than cocommutativity.
This was the starting point for the development of a galois theory for hopf categories in 10. This is the text of a presentation at the workshop on hopf algebras and galois module theory held at the university of nebraska at omaha, may 23, 2016. Later, hopfgalois theory for separable extensions of elds was studied by greither and pareigis gp87. This leads to the notion of hopfgalois category extension. A survey susan montgomery we consider a hopf galois extension b. This site is like a library, use search box in the widget to get ebook that you want. Numerous and frequentlyupdated resource results are available from this search. In 1969 chase and sweedler extended these ideas to coactions of hopf algebras acting on a commutative kalgebra, for k a commutative ring 5. Hopf galois theory and galois module structure university of exeter, 2326 june 2015 abstracts of talks.
A primer of hopf algebras pierre cartier institut math. The concept of hopfgalois extensions for arbitrary eld extensions is due to chase and sweedler cs69. Pdf forms of hopf algebras and galois theory researchgate. The study of hopf algebras spans many fields in mathematics including topology, algebraic geometry, algebraic number theory, galois module theory, cohomology of groups, and formal groups and has wideranging connections to fields from theoretical physics to computer science. August 1998, ihesm9860 abstract we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations.
Finally in section 7 we consider a different approach to the galois correspondence via actions. Galois theory, hopf algebras, and semiabelian categories. Although the rst o ccurence of hopf algebras w as in algebraic top ology, they are no w found in areas as remote as com binatorics and analysis. We consider a hopf algebra h and two h galois extensions a and b. Then hopf galois descent theory for linear hopf categories, the hopf algebra version of a linear category, is developed. We develop hopf galois theory for weak hopf algebras, and recover analogs of classical results for hopf algebras. An important wellknown result of rota describes the relationship between the mo. This book studies hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. A hopf galois structure on a finite extension of fields kk is. A purely algebraic approach can be found in 2 and 3. Hopf algebras and graded ring theory, held in brussels, september 1923, 2006. Given a linear category as an extension of a diagonal category, we introduce descent data, and the category of descent data is isomorphic to the c. Hopf algebras and galois module theory keele university.
We are interested in a systematic treatment of hopf algebras with the techniques of forms and descent. Galois and hopfgalois theory for associative salgebras. Hopf galois theory and galois module structure university of exeter, 2326 june 2015 abstracts of talks victor abrashkin. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. We focus on the situatation where the weak hopf algebra is a groupoid algebra or its dual. Dec 23, 2016 brauer groups, hopf algebras and galois theory k. Hopf galois structures on symmetric and alternating extensions. This volume is based on talks given at the workshop on categorical structures for descent and galois theory, hopf algebras, and semiabelian categories held at the fields institute for research in mathematical sciences toronto, on, canada. A galois type correspondence theory for actions of finitedimensional pointed hopf algebras on prime algebras. Buy hopf algebras and galois theory lecture notes in mathematics on free shipping on qualified orders. Depth two hopf subalgebra, hochschild complex and noncommutative galois theory. On the galois correspondence theorem in separable hopf galois theory crespo, teresa, rio, anna, and vela, montserrat, publicacions matematiques, 2016. Hopf galois theory expands the classical galois theory by considering the galois property in terms of the action of the group algebra k g on kk and then replacing it by the action of a hopf algebra.
Groups of order 4p, twisted wreath products and hopf. Unlimited viewing of the articlechapter pdf and any associated supplements and figures. We have a dual theory, where actions by dual linear hopf categories on linear categories are considered. Galois extensions with galois group a frobenius group g h. They showed how to recast the problem of classifying all hopfgalois structures on lkas a problem in group theory. Sweedler, hopf algebras and galois theory, lecture notes in math. Commutative hopf galois module structure of tame extensions. Pdf we consider a hopf galois extension b a, for a a comodule algebra over the hopf algebra h with coinvariant algebra b. Pdf the theory of hopf algebras is closely connected with various applications, in particular to algebraic and formal groups. Examples of nonisomorphic hopf galois structures with isomorphic underlying hopf algebras have previously be given in and all of them for galois extensions k k and the hopf algebra h 1 that gives the standard nonclassical hopf galois structure, obtained by taking n. Nonisomorphic hopf galois structures with isomorphic. Hopf categories as multiobjected versions of hopf algebras. A hopfgalois correspondence for free algebras request pdf. Many results on hopf algebras can be generalized to hopf categories, for example the fundamental theorem on hopf algebras, see 3, sec.
In mathematics, a hopf algebra, named after heinz hopf, is a structure that is simultaneously an unital associative algebra and a counital coassociative coalgebra, with these structures compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. In 1969 chase and sweedler extended these ideas to coactions of hopf algebras acting on a commutative k algebra. Hopf algebras, renormalization and noncommutative geometry. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.
The main aim of this paper is to develop hopf galois theory for hopf categories. Some papers consider hopf versions of classical topics, such as the brauer group, while others are closer to work in quantum groups. A, for a a comodule algebra over the hopf algebra h with coinvariant algebra b. Commutative nilpotent algebras and hopf galois structures abstract. Hopf algebras can be generalized to hopf categories, see 3 for some examples. A hopf algebra also has representations which carry additional structure, namely they are algebras. Hopfgalois extensions of algebras, the miyashitaulbrich. We focus on the situatation where the weak hopf algebra is a groupoid algebra or. Hopf algebroids, hopf categories, and their galois theories arxiv. Hopf algebra, hopf galois theory, galois correspondence. Rosenberg, galois theory and galois cohomology of commutative rings, mem. Pdf file 1551 kb djvu file 293 kb article info and citation. A possible strategy could be to try to adapt the methods from classical hopf. Compatibility of hopf galois structures for separable.
Pdf galois theory for multiplier hopf algebras with. The rst three paragraphs of this paper give a survey of the present state of the theory of forms of hopf algebras and of hopf galois theory especially for separable extensions. New york journal of mathematics biskew braces and hopf. Hopf galois theory expands the classical galois theory by considering the galois property in terms of the action of the group algebra kg on kk and then replacing it by the action of a hopf algebra.
Hopf algebras, renormalization and noncommutative geometry alain connes ihes dirk kreimer mainz univ. Hopf algebras and local galois module theory, 2000. One of the results in 3 is the fundamental theorem for hopf modules, opening the way to hopf galois theory. Hopf algebras download ebook pdf, epub, tuebl, mobi. Galois theories, hopf algebras, and semiabelian categories.