Hopf algebras and their representation theory have been outstanding in this latter period. Hopf algebras, quantum groups and topological eld theory winter term 201920 christoph schweigert hamburg university department of mathematics section algebra and number theory and center for mathematical physics as of. Ais an hmodule, and the multiplication and unit maps of aare hmorphisms. Request pdf oriented hopf algebras and their actions let h be a finitedimensional hopf algebra acting on an algebra a. Globalization of twisted partial hopf actions journal. Globalization of twisted partial hopf actions journal of. Brauer group of hdimodule algebras that is, algebras with an haction and an. These have been the focus of a great deal of research. In a 1975 publication kap kaplansky listed 10 conjectures on hopf algebras. For example that paper includes a generalization of incomparability for crossed products of finite groups. In the broadest sense of the term, an extension of hopf algebras is perhaps just some injective hopf algebra map k.
Jacobson radicals of invariants under hopf algebra actions. You must be prepared to devote at least 10 hours a week on this class, and have a solid knowledge of groups and rings. Herstein received april 17, 1984 finite groups acting on rings by automorphisms, and groupgraded rings are instances of hopf algebras h acting on module algebras a. This content was uploaded by our users and we assume good faith they have the permission to share this book. More generally one can consider crossed products a ffa h of an algebra a with a hopf algebra h, where the multiplication of the copy of h in a ffa h is twisted by a cocycle o, and their study is the other major purpose of this paper. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right hcomodules into a direct sum of indecomposables. Hopf algebras and their actions on rings, including automorphism.
Hopf algebras and their actions on rings pdf free download epdf. Bulletin of the ams this book should be very useful as a comprehensive, up to date introduction to the algebraic aspects of hopf algebras. Click download or read online button to get hopf algebras and their actions on r ings book now. K algebras with a seperable subalgebra whose centralizer satisfies a polynomial identity, comm. Hopf algebras can be characterized among bialgebras by the fundamental theorem on hopf modules.
Expanded version of ten lectures given at the cbms conference on hopf algebras and their actions on r ings, which took place at depaul university in chicago, august 1014, 1992preface. Susan montgomery, hopf algebras and their actions on rings, cbms lecture notes 82, ams 1993, 240p. Oriented hopf algebras and their actions request pdf. Cohen and montgomery used their construction to apply known theorems about crossed products of finite groups to prove results on groupgraded rings. Brauers theorem for kernels over group algebras is generalised to hopf algebras. Hopf algebras and their actions on rings download ebook pdf. Our particular focus will be on a notion ofrationality for primeideals ofathat takes thehaction into account. Hopf algebras and their actions on rings cover image. This notion is also defined for hopf algebra actions. We study the hopf algebra structures on the hopf algebra h1, q over it is shown that h1, q is a hopf algebra if and only if q 1 or q is a real number. Combinatorial results can then be seen as particular cases of general algebraic properties of hopf algebras e. Let h be an arbitrary pointed hopf algebra of rank one and g the group of grouplike elements of h. Montgomery, s hopf algebras and their actions on rings cbms regional conference series in mathematics 82, american mathematical society providence ri 1993 10. The book may be used as the main text or as a supplementary text for a graduate algebra course.
Montgomery, hopf algebras and their actions on rings, cbms regional. For every commutative ring r that is not a field, there exists a. My current interests are frobeniusschur indicators and cohomology rings of hopf algebras. Actions of a group or lie algebra on a given algebra a have proven a useful tool in analyzing its prime spectrum, speca. This categorical fact enables a definition of hopf monoids in some setups that do not. An introduction crc press book this study covers comodules, rational modules and bicomodules. Ocneanu a construction of ropf algebras wednesday, july 11, 1984 m. The representation theory of a hopf algebra is particularly nice, since the. Let k be a commutative ring, then mk denotes the category with as objects all right k. Dually, if h coacts on a, a subspace v a is hcostable if v is an hsubcomodule of a. Chevalley theorem for skew polynomial rings after a reasonable notion of a quasire. In other words, the hopf action does not factor through a smaller hopf quotient. Goodearl, homological aspects of noetherian pi hopf algebras and irreducible modules of maximal dimension, j. Gorenstein subrings of invariants under hopf algebra actions.
Hopf algebras and actions, part ii seattle workshop 2016. Buy hopf algebras and their actions on rings cbms regional conference series in mathematics on. Work on actions of hopf algebras has unified earlier results on group actions, actions of lie algebras, and graded algebras. Oriented hopf algebras and their actions springerlink. The pi will also study the possible actions of the taft hopf algebras on quantum polynomial rings. Usc dornsife mathematics professor susan montgomery, along with one of her graduate students describe noncommutative algebras. Hopf actions on algebras we say that a hopf algebra hacts on an algebra aif ais an hmodule algebra. A purely algebraic approach can be found in 2 and 3. Zhang, noetherian pi hopf algebras are gorenstein, trans. We then show that in order to generalize existing theories. Such a ring is not commutative if g is not abelian. H or, dually, a hopf algebra surjection, which is the viewpoint taken in 35.
A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. This enables us to describe the corepresentation ring of h in terms of generators and relations. Suppose that a is an algebra and we have algebra homomorphisms. Hopf modules and the double of a quasihopf algebra, trans. Hacts on ainner faithfully if there is not an induced action of hion afor any nonzero hopf ideal iof h.
Poisson algebrashave lately been playing an important role in algebra, geometry, mathematical physics and other subjects. Goodearl, lectures on algebraic quantum groups, birkh. Hopf algebras, quantum groups and topological field theory. Finite groups acting on rings by automorphisms, and groupgraded rings are instances of hopf algebras h acting on hmodule algebras a. Hopf algebras and their actions on r ings extended selected papers by susan montgomery, 9780821807385, available at book depository with free delivery worldwide. When the hopf algebra is finite dimensional and the action. Hopf algebras and their actions on rings, cbms regional conference series in mathematics, 82 american mathematical society, providence. The present notes were prepared for the lectures of my course introduction to hopf algebras and representations, given in the university of helsinki in the spring of 2011.
It was established long ago that there is no splitting theorem for the brauer. Namely, having a coproduct means that you can define the tensor product of representations and having an antipode means that you can define the dual of a. This is a continuation of the joint work with juan cuadra on nite dimensional hopf actions on weyl algebras arxiv. Hopf algebras and their actions on rings book, 1993. Department of mathematics section algebra and number theory and. Journal of algebra 100, 363379 1986 h opt algebra actions miriam cohen and da vida fishman ben gurion university of the negev, beer sheva, israel communicated hy l. The corepresentation ring of a pointed hopf algebra of.
Rieffel invariants of coiiiiiiutative rings ir group actions on c algebras ii ix. Hopf algebras and their actions on rings cbms regional. All the maps in the diagrams are algebra homomorphisms given our hy. The corepresentation ring of a pointed hopf algebra of rank. We will be much stricter in our terminology by calling an extension only a short exact sequence k. The hopf actions on vertex operator algebras are investigated. This site is like a library, use search box in the widget to get ebook that you want. Hopf algebras and their actions on rings download ebook. Then the hopf algebra structures on h1, q are classified up to the equivalence of hopf algebra structures. If a is a right comodule algebra, then it is also a left hmodule algebra, via fawd x a f. We will be much stricter in our terminology by calling an. Peter schauenburg, hopf modules and yetter drinfel. Hopf algebras, quantum groups and topological eld theory. Aug 07, 20 usc dornsife mathematics professor susan montgomery, along with one of her graduate students describe noncommutative algebras.
We then show that in order to generalize existing theories, we must consider actions of nontrivial hopf algebras, in particular, those that are not. This book brings together many of these recent developments from the viewpoint of the algebraic structure of hopf algebras and their actions and coactions. We begin with an introduction to the general study of hopf algebras and their basic properties, then explain why they are a natural choice to generalize the action of finite groups on rings. Hopf algebras download ebook pdf, epub, tuebl, mobi.
Finally the pi will study what happens to a hopf algebra and an algebra on which it acts when the hopf algebra is twisted by a cocycle. Thus studying hopf algebras themselves as well as their actions on other structures is an important. We begin with an introduction to the general study of hopf algebras and their basic properties, then explain why they are a natural choice to generalize the action of nite groups on rings. A possible strategy could be to try to adapt the methods from classical hopf. It is natural to consider extending the invariant theory from a group action to a hopf algebra action.
Hopf actions on vertex operator algebras request pdf. Know that ebook versions of most of our titles are still available and may be. The aim of this note is to develop galois theory for weak hopf algebras. Montgomery, has contributed much to group actions on rings and their generalizations to hopf algebras especially finite dimensional.
The last chapter chapter 4 discusses several applications of hopf algebras, some of which are further developed in the authors 2011 publication, an introduction to hopf algebras. Montgomery, susan 1993, hopf algebras and their actions on rings, regional conference series in mathematics, 82, providence, rhode island. Montgomery, hopf algebras and their actions on rings. I thank the participants of the course for making it a very enjoyable experience. A survey 369 in the case of an haction on a, we say a subspace v a is hstable if h v v. A general theory of hopf algebras began in the late 1960s swe. Hopf algebra is nice object full of structure a bialgebra with an antipode.
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 this paper, we examine the relationship between the jacobson radicals of rings r and their invariants rh under the actions of finite dimensional hopf algebras h. In the general case, we show that these kernels coincide with the categorical left and right hopf kernels of morphisms of hopf algebras defined in andruskiewitsch and devoto extensions of hopf algebras, algebra i analiz 7 1995, 2269. Then a is called an halgebra whenever h acts on products in a via the. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. 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. We begin with an introduction to the general study of hopf algebras and their basic properties, then explain why they are a natural choice to generalize the action of finite. If the action is semisimple, a schurweyl type decomposition is obtained. This text aims to provide graduate students with a selfcontained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras and hopf algebras. The nondegeneracy of this matrix allows one to define an action of the. Hopf algebras and their actions on rings, cmbs reg. The present article explores the more general situation where a hopf algebra h acts on a. Hopf algebras and their actions on rings extended selected. Hopf algebras and their actions on rings ams bookstore.