Hopf galois theory pdf

Later, hopf galois theory for separable extensions of elds was studied by c. Let a be an object vector space, algebra, hopf algebra over k. Sweedler in 1969 and provides a generalisation of classical galois theory. Hopfgalois theory is a generalization of ordinary galois theory in several ways. Articles are suitable for graduate students and researchers, specifically those interested in galois theory and hopf algebras and their categorical unification. These are recovered as we specialize the hopf algebra h to one of a number of special types. Frlich received january 21, 1985 introduction it is well known that adjoining co 2 to q does not give a galois. On hopf galois structures and complete groups 101 d simplifying. A theory of galois or hopfgalois extensions for associative ring spectra is desirable. New york journal of mathematics biskew braces and hopf. We introduce galois theory for hopfgalois extensions proving existence of a galois connection between subalgebras of an hcomodule algebra and. We have a dual theory, where actions by dual linear hopf categories on linear categories are considered. However, for the hopf dense calois extensions, these properties are not true in general. Descent and galois theory for hopf categories journal of.

Galois theory gives a bijective correspondence between intermediate fields of a galois field extension normal and separable and subgroups of the galois group. Stephen urban, 1932hopf algebras and galois theory. Hopf galois coextensions in noncommutative geometry 5 let h be a hopf algebra and a a right h comodule with the coaction. Then hopf galois descent theory for linear hopf categories, the hopf algebra version of a linear category, is developed. Let f g be an isogeny between finite ndimensional formal groups defined over r, the valuation ring of some field extension k of q p. We focus on the situatation where the weak hopf algebra is a groupoid algebra or its dual.

They showed how to recast the problem of classifying all hopf galois structures on. 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. A survey 369 in the case of an haction on a, we say a subspace v a is hstable if h v v. The interrelation between c, s and the ring structure of qa is formalized by the concept of galois extension where h is a finite hopf algebra. This leads to hopf algebroid analogues of results for hopf algebra extensions by doi and, in the case of frobenius hopf algebroids, by cohen, fishman and montgomery. Galois and hopfgalois theory for associative salgebras. If a is a right comodule algebra, then it is also a left hmodule algebra, via fawd x a f. Titles in this series are copublished with the fields institute for research in mathematical sciences toronto, ontario, canada. Hopfgalois objects were introduced by chase and sweedler 7, and was. Galois theory for weak hopf algebras internet archive. Roque, on factorization systems and admissible galois structures 22. Hopfgalois extensions sanath devalapurkar let c n be a knalgebra in plocal stable homotopy theory.

Hopfgalois theory and elliptic curves let kq be a finite. Q3 v 2q moreover, one can take an extension lk which is galois with group. Galois theory, hopf algebras, and semiabelian categories. A workshop on hopfgalois theory and galois module structure took tuesday 23rd june to friday 26th june 2015 at the university of exeter. Depth two hopf subalgebra, hochschild complex and noncommutative galois theory lars kadison 25 july 2008. We focus on the situatation where the weak hopf algebra is a.

Our methods are based on the recently introduced galois theory for corings. An lform of a is a kobject b such that lqk bz lo, a. Martinsferreira, weak categories in additive 2categories with kernels 20. In fact, every galois extension with nonabelian galois group has at least two hopf galois structures, one by the group ring of the galois group, the other by the hopf algebra h l, where acts on lvia the galois action and on by conjugation a nontrivial ac. We first introduce the ideas of hopfgalois theory as an attempt to taming wild extensions with hopf algebras, in con nection with the problems of galois module. In the case of separable extensions it has a characterization of the hopf galois character in terms of groups. 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. This theory works well, but some of its aspects remain unsatisfactory, as was explained in the nal remark 8. Hopf algebras and galois theory lecture notes in mathematics 1969th edition by stephen u. To every modular crossed hmodule m we associate a cyclic object zh,m. One of the results in 3 is the fundamental theorem for hopf modules, opening the way to hopfgalois theory. Citeseerx document details isaac councill, lee giles, pradeep teregowda. A purely algebraic approach can be found in 2 and 3. 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.

Galois objects hopf algebras and galois theory galois objects and extensions of hopf algebras. Groups of order 4p, twisted wreath products and hopf. Theorem which lives on the crossroads of galois theory for. C n can then be considered as a wedge of the kns, i. We develop hopfgalois theory for weak hopf algebras, and recover analogs of classical results for hopf algebras. Hopf galois coextensions in noncommutative geometry. The concept of hopfgalois extensions was introduced by s. The main aim of this paper is to develop hopfgalois theory for hopf categories.

Of these tasks, checking regularity is the least natural. This was the starting point for the development of a galois theory for hopf categories in 10. By definition a modular crossed hmodule is a vector space m on which h acts and coacts in a compatible way. These notes give a concise exposition of the theory of.

This barcode number lets you verify that youre getting exactly the right version or edition of a book. Hopf galois theory of separable field extensions core. For such h, we classify galh, the group of hgalois objects. A possible strategy could be to try to adapt the methods from classical hopf. The morita theory, developed for corings by caenepeel, vercruysse and wang, is applied to obtain equivalent criteria for the galois property of hopf algebroid extensions. The purpose of the workshop was to bring together researchers from the uk, europe and the us interested in arithmetic aspects of hopfgalois theory and in other areas of ramification theory and galois module. We first introduce the ideas of hopfgalois theory as an attempt to taming wild extensions with hopf algebras, in connection with the problems of galois module structure in arithmetic. This is a research area in algebra and algebraic number theory which has led to a halfdozen doctoral theses at albany in the past 6 years. We introduce galois theory for hopfgalois extensions proving existence of a galois connection between subalgebras of an hcomodule algebra and generalised quotients of the hopf algebra h. This leads to the notion of hopf galois category extension. The concept of hopf galois extensions was introduced by s. Then hopfgalois descent theory for linear hopf categories, the hopf algebra version of a linear category, is developed. The cyclic homology of zh,m extends the usual cyclic homology of the algebra structure of h, and the relative cyclic homology of an hgalois extension.

Taylor and others, we explain how one obtains, from an elliptic curve having. The aim of this note is to develop galois theory for weak hopf algebras. Hopfgalois theory and galois module structure university of exeter, 2326 june 2015 abstracts of talks victor abrashkin. They showed how to recast the problem of classifying all hopfgalois structures on. This leads to the notion of hopfgalois category extension. Pdf hopf galois theory expands the classical galois theory by considering the galois property in terms of the action of the group algebra kg.

Let h be the rhopf algebra which arises from this isogeny. C n it is easy to see that any map c n v kn c n c n v. Arithmetic structure of galois groups of pextensions of local elds with nilpotent class pdf djvu. Galois module theory centers around the idea of understanding the ring of integers s of an extension l of an algebraic number field k with ring of integers r as an hmodule, where h is some appropriate order over r in a khopf algebra a. Hopf dense galois extensions with applications sciencedirect. Hopf algebras can be generalized to hopf categories, see 3 for some examples. Hopf galois structures on symmetric and alternating extensions. Hopf algebra, hopf galois theory, galois correspondence. Hopfgalois theory and elliptic curves 4 basis of zz into a basis of zgz, and ok is an hgaloisobject. Pdf galois theory of hopf galois extensions dorota. Biskew braces and hopf galois structures lindsay n. Pdf cyclic homology of hopf galois extensions and hopf. Hopf galois theory substitutes the galois group by a hopf algebra.