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. Many results on hopf algebras can be generalized to hopf categories, for example the fundamental theorem on hopf algebras, see 3, sec. They showed how to recast the problem of classifying all hopfgalois structures on lkas a problem in group theory. Rosenberg, galois theory and galois cohomology of commutative rings, mem.
An important wellknown result of rota describes the relationship between the mo. This was the starting point for the development of a galois theory for hopf categories in 10. Forms of hopf algebras and galois theory citeseerx. One of the results in 3 is the fundamental theorem for hopf modules, opening the way to hopf galois theory. Finally in section 7 we consider a different approach to the galois correspondence via actions. A hopf galois structure on a finite extension of fields kk is.
A primer of hopf algebras pierre cartier institut math. Numerous and frequentlyupdated resource results are available from this search. Click download or read online button to get hopf algebras book now. Dec 23, 2016 brauer groups, hopf algebras and galois theory k. Then hopf galois descent theory for linear hopf categories, the hopf algebra version of a linear category, is developed. Commutative nilpotent algebras and hopf galois structures abstract. The concept of hopfgalois extensions for arbitrary eld extensions is due to chase and sweedler cs69. 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. 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.
Artins classic treatment of galois theory we know that l l. Childs, azumaya algebras which are not smash products. 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. 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. 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. Hopf categories as multiobjected versions of hopf algebras. This leads to the notion of hopf galois category extension. All the maps in the diagrams are algebra homomorphisms given our hy. Hopf galois structures on symmetric and alternating extensions. A hopf algebra h over the commutative unital ring k gives an example of a hopf algebroid. A hopf algebra also has representations which carry additional structure, namely they are algebras. Childs, on the hopf galois theory for separable eld extensions, comm. Commutative hopf galois module structure of tame extensions.
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. Hopf galois theory and galois module structure university of exeter, 2326 june 2015 abstracts of talks victor abrashkin. Pdf forms of hopf algebras and galois theory researchgate. Some papers consider hopf versions of classical topics, such as the brauer group, while others are closer to work in quantum groups. A survey susan montgomery we consider a hopf galois extension b. We supplement the study of galois theory for algebraic quantum groups started in the paper galois theory for multiplier hopf algebras with integrals by a.
A, for a a comodule algebra over the hopf algebra h with coinvariant algebra b. Let lkbe a galois extension of elds with galois group gan. The main aim of this paper is to develop hopf galois theory for hopf categories. Compatibility of hopf galois structures for separable. Our methods are based on the recently introduced galois theory for corings. A galois type correspondence theory for actions of finitedimensional pointed hopf algebras on prime algebras. Galois theory, hopf algebras, and semiabelian categories. 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. A possible strategy could be to try to adapt the methods from classical hopf.
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. 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. Buy hopf algebras and galois theory lecture notes in mathematics on free shipping on qualified orders. Hopf algebras, renormalization and noncommutative geometry alain connes ihes dirk kreimer mainz univ. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Pdf file 1551 kb djvu file 293 kb article info and citation. After giving a number of examples, we discuss galois extensions with additional properties, such as having a normal basis. The results, not previously published in book form, show that hopf algebras play a natural role in local galois module theory. 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. 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 module duality applied to xouter galois theory. Hopf algebras and galois theory lecture notes in mathematics. Depth two hopf subalgebra, hochschild complex and noncommutative galois theory. Galois theory for multiplier hopf algebras with integrals. On the galois correspondence theorem in separable hopf galois theory crespo, teresa, rio, anna, and vela, montserrat, publicacions matematiques, 2016. Pdf the theory of hopf algebras is closely connected with various applications, in particular to algebraic and formal groups. Let lkbe a eld extension, ha cocommutative k hopf algebra. Arithmetic structure of galois groups of pextensions of local elds with nilpotent class and m is a natural number. We focus on the situatation where the weak hopf algebra is a groupoid algebra or its dual. Pdf a galoistype correspondence theory for actions of. Hopf algebras and local galois module theory, 2000.
Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. 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. The aim of this note is to develop galois theory for weak hopf algebras. In 1969 chase and sweedler extended these ideas to coactions of hopf algebras acting on a commutative k algebra.
A hopfgalois correspondence for free algebras request pdf. We investigate the category amh b of relative hopf bimodules, and the morita equivalences between a and b induced by them. The structure of hopf algebras acting on galois extensions. Sweedler, hopf algebras and galois theory, lecture notes in math. However, formatting rules can vary widely between applications and fields of interest or study. 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. Galois and hopfgalois theory for associative salgebras. 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. New york journal of mathematics biskew braces and hopf. Descent and galois theory for hopf categories journal of. Galois extensions with galois group a frobenius group g h. 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. 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. We are interested in a systematic treatment of hopf algebras with the techniques of forms and descent.
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. Later, hopfgalois theory for separable extensions of elds was studied by greither and pareigis gp87. A purely algebraic approach can be found in 2 and 3. Hopf algebra, hopf galois theory, galois correspondence. Groups of order 4p, twisted wreath products and hopfgalois. Hopf algebroids, hopf categories, and their galois theories arxiv. 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. 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.
Here is a very brief indication of the origins of those connections. We intend the outline below to summarize how we achieve these goals. Twisted kummer and kummerartinschreier theories suwa, noriyuki, tohoku mathematical journal, 2008. We develop hopf galois theory for weak hopf algebras, and recover analogs of classical results for hopf algebras. 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. Hopf algebras download ebook pdf, epub, tuebl, mobi. It includes many illustrating exam ples some of which. Finally insection 7we consider a different approach to the galois correspondence via actions. 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. Pdf galois theory for multiplier hopf algebras with. Galois theories, hopf algebras, and semiabelian categories.
Hopf algebras and galois module theory keele university. The talk is based on the paper 3, so no proofs are given, but we. Then hopfgalois descent theory for linear hopf categories, the hopf algebra version of a linear category, is developed. Nonisomorphic hopf galois structures with isomorphic. Hopf galois objects were introduced by chase and sweedler 7, and was. Underwood department of mathematics and computer science auburn university at montgomery montgomery, alabamathe structure of hopf algebras acting on galois extensions 3. Hopf galois theory and elliptic curves jean gillibert abstract. This book studies hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. Hopf algebras can be generalized to hopf categories, see 3 for some examples. Childs, on hopf galois extensions of elds and number rings, in perspectives in ring theory, f.
Hopf algebras, renormalization and noncommutative geometry. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. We have a dual theory, where actions by dual linear hopf categories on linear categories are considered. Hopf galois theory and galois module structure university of exeter, 2326 june 2015 abstracts of talks. Hopf algebras and graded ring theory, held in brussels, september 1923, 2006. We focus on the situatation where the weak hopf algebra is a groupoid algebra or.
We consider a hopf algebra h and two h galois extensions a and b. 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. This leads to the notion of hopfgalois category extension. Hopfgalois extensions of algebras, the miyashitaulbrich. This site is like a library, use search box in the widget to get ebook that you want. Pdf we consider a hopf galois extension b a, for a a comodule algebra over the hopf algebra h with coinvariant algebra b. 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. Hopfgalois theory and elliptic curves jean gillibert abstract. 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. Groups of order 4p, twisted wreath products and hopf.
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