Hopf algebras constructed by a sequence of crossed products involving group algebras and their duals. Hopf algebras are an important topic of study in mathematics because of their wide range of applications. Bilinear pairings for hopf algebras with a lift of the verschiebung. There are some other ones which i know are out there, but i havent read. These are notes for a course held in summer semester 2010 on the ludwigmaximillian university munich. Montgomeryhopf algebras and their actions on rings. Duality and rational modules in hopf algebras over. On hopf 2algebras international mathematics research.
On the green rings of pointed, coserial hopf algebras by. Westreicha galoistype correspondence theory for actions of finitedimensional pointed hopf algebras on prime. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. Each of these areas of study have been quite active lately see 16, 9, 5. Hopf algebras and their generalizations from a category theoretical point of. So, there is a natural question of whether we could construct hopf algebras whose antipodal orders are arbitrary even integers at the level of commutative rings. We assume basic familiarity with hopf algebras, as found in the opening chapters of the book. Peter schauenburg, hopf modules and yetter drinfel. In other words, commutative hopf algebras are precisely the rings of regular functions on affine group schemes. This ring is spanned by isomorphism classes of finitedimensional indecomposable representations that are added together via direct sums and multiplied via tensor products.
This is morally the reason for their appearance in algebraic topology, e. This structure is also seen across diverse fields of physics, and so allows connections to be made between the kitaev models and topics as varied as quantum gauge theory and modified strong. Centered in the area of group rings and algebras, this volume contains a mixture of cutting edge research topics in group theory, ring theory, algebras and their representations, hopf. This work very nicely complements the other texts on hopf algebras and is a welcome addition to the literature. Hopf algebras generating fusion rings and topological invariants simon lentner abstract. Chapter 1 introduces algebras and coalgebras over a field k.
Hopf algebras were originally used in topology in the 1940s, but since then their applications and popularity as a topic of study have grown tremendously. We determine how the integral of the dual to a unimodular hopf algebra h changes under twisting of h. Janssen submitted on 17 oct 2006 v1, last revised 9 dec 2008 this version, v2. Bialgebras and hopf algebras university of chicago. Hopf algebras are used in combinatorics, category theory, homological algebra, lie.
In this paper we study the properties of drinfelds twisting for finitedimensional hopf algebras. Montgomery, has contributed much to group actions on rings and their generalizations to hopf algebras especially finite dimensional. This talk will explain how many of the properties of. Usc dornsife mathematics professor susan montgomery, along with one of her graduate students describe noncommutative algebras. Cbms regional conference series in mathematics publication year 1993. In this thesis, we explore the green rings of a class of hopf algebras that form an extension. As an algebra, its product is defined by linear extension of the group composition in g, with multiplicative unit the identity in g. Hopf algebras over number rings connecting repositories. Montgomery, hopf algebras and their actions on rings, 2nd revised.
Lecture course hopf algebras, quantum groups and topological field. Susan montgomery, hopf algebras and their actions on rings, cbms lecture notes 82, ams 1993, 240p. A primer of hopf algebras 5 heavily on the semisimplicity of the representations. On the green rings of pointed, coserial hopf algebras. Hopf algebras and their generalizations from a category. More generally, hopf algebras naturally occur as rings of functions of some kind on groups of some kind. Hopf algebras and their actions on rings, including automorphism. Hopf algebras and their actions on rings pdf free download epdf.
We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple quasitriangular finitedimensional hopf algebras are stable under twisting. Montgomery, hopf algebras and their actions on rings, cmbs reg. By an algebra r over k or simply an algebra we understand a unitary, associative kalgebra r, with identity 1 1r. Let h be a finitedimensional hopf algebra acting on an algebra a. Connections were made with many aspects of algebra. Bulletin of the ams this book should be very useful as a comprehensive, up to date introduction to the algebraic aspects of hopf algebras. These respectively encode associativity and the unit element. Kassel hopf algebras uses cobraided and montgomery has and their actions on rings uses the other both on page. Hopf algebras and their actions on rings, by susan montgomery. Hopf modules and the double of a quasihopf algebra, trans. Hopf algebras are a relatively new concept in algebra. Oriented hopf algebras and their actions springerlink.
Hopf algebras and their actions on rings extended selected. Combinatorial results can then be seen as particular cases of general algebraic properties of hopf algebras e. Chapter 3 discusses hopf algebras and chapter 4 consists of three applications of hopf algebras. The course primarily constitutes an elementary, yet mathematically rigid introduction to the theory of hopf algebras. Hopf algebras and their actions on rings susan montgomery.
An introduction crc press book this study covers comodules, rational modules and bicomodules. The green ring is a powerful mathematical tool used to codify the interactions between representations of groups and algebras. This one is more about the theory of hopf algebras than about drinfeldjimbo quantum groups. Hopf algebras and their actions on rings about this title. Integrals in hopf algebras are an essential tool in studying finite dimensional hopf algebras and their action on algebras. Buy hopf algebras and their actions on rings cbms regional conference. A fundamental concept in the first two cases has been the notion of inner action. Hopf algebras and their representation categories 2. Hopf algebras were seen as rings which were interesting in their own right. The group hopf algebra of g over k, denoted kg or kg, is as a set and a vector space the free vector space on g over k.
This content was uploaded by our users and we assume good faith they have the permission to share this book. Cartier 14 was able to reformulate the problem without the assumption of semisimplicity, and to extend the tannakakrein duality to an arbitrary algebraic linear group. Know that ebook versions of most of our titles are still available and may be downloaded immediately after purchase. A hopfgalois correspondence for free algebras sciencedirect. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Partial coactions of hopf algebras and partial hopfgalois theory authors. International workshop groups, rings, lie and hopf. In the 1960s, study of hopf algebras increased, perhaps spurred by the. This book brings together many of these recent developments from the viewpoint of the algebraic structure of hopf algebras and their actions and coactions. Hopf calgebras and their representations springerlink. International workshop groups, rings, lie and hopf algebras.
Weak hopf algebras, or quantum groupoids, are generalizations of hopf algebras. Their physics is inherited from topological quantum field theories, while their underlying mathematical structure is based on a class of hopf algebras. Another advantage is that hopf algebra and rota baxter theory allow for e. These hopf algebras are pointed and coserial, meaning their simple comodules are 1dimensional, and their comodules possess unique composition series respectively. For example, when hkg is a group hopf algebra, then a is an halgebra precisely when the group g acts by automorphisms on a. A urry of activity in the area exploded in the 1990s and produced an ever growing zoo of interesting hopf algebras. Irreducible representations of crossed products core. Then a is called an halgebra whenever h acts on products in a via the coproduct. Hopf algebras and their actions on rings cbms regional. One of the major purposes of this paper is to study inner actions of hopf algebras. First of all, we study the characters of representations, and show the orthogonal relations for characters of finite dimensional irreducible representations and the matrix coefficients of finite dimensional irreducible representations, these are generalizations of the results about. Universal bilinear maps for hopf algebras over f p.
Work on actions of hopf algebras has unified earlier results on group actions, actions of lie algebras, and graded algebras. Hopf algebras and their actions on rings ams bookstore. In this paper, we study the finite dimensional unitary representations of hopf calgebras. Skew littlewoodrichardson rules from hopf algebras. Hopf algebra is nice object full of structure a bialgebra with an antipode. In this thesis, we explore the green rings of a class of hopf algebras that form an extension of the taft algebras. Here lie is the category of lie algebras over the field k with or. Bilinear maps and the tensor product of hopf algebras 24 6. A kalgebra is a kvector space with two linear maps m. Galois extensions of the base ring with galois group f s aut g and. In this paper we give positive answers for number rings z ln. Hopf algebras generalize both lie groups and lie algebras, so the problem does not present itself in a hopf algebra formalism.
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