Partial co actions of hopf algebras and partial hopf galois theory authors. Representations of hopfore extensions of group algebras and. Any generalized cohomology theory, gives rise to a sequence of spaces, with the property that, the homotopy classes of mappings. All of the above examples have their hopf rings described with just a few generators and relations. Grunenfelder, constructing pointed hopf algebras by ore extensions, algebra, 225 2000, no. This categorical fact enables a definition of hopf monoids in some setups that do not. 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. Duality and rational modules in hopf algebras over commutative rings by. Susan montgomery usc dornsife university of southern california.
Hopf algebras and their actions on rings susan montgomery. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Hopf algebras and their actions on rings cbms regional. Bilinear maps and the tensor product of hopf algebras 24 6. On the stable green ring we define a new bilinear form which is more accurate to determine. Cbms regional conference series in mathematics publication year 1993. As for algebras, one can replace the underlying field k with a commutative ring r in the above definition the definition of hopf algebra is selfdual as reflected in the symmetry of the above diagram, so if one.
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. Hopf algebras and their actions on rings edition 1 by susan. For many but not all of the results presented here, the hopf. Hopf algebras, quantum groups and topological eld theory winter term 201920. This book brings together many of these recent developments from the viewpoint of the algebraic structure of. Dual constructions for partial actions of hopf algebras authors.
These are glued to the groupring itself to form new quantum groups by a radfordmajidconstruction, and often deformed by a doi twist to get a complete classification. In a hopf algebra, are the notions cobraided and coquasitriangular the same. Partial coactions of hopf algebras and partial hopfgalois theory authors. We study algebraic and homological properties of two classes of infinite dimensional hopf algebras over an algebraically closed field k of characteristic zero.
In this thesis, we explore the green rings of a class of hopf. This volume contains the proceedings of the international conference on groups, rings, group rings, and hopf algebras, held october 24, 2015 at loyola university, chicago, il, and the ams special session on groups, rings, group rings, and hopf algebras, held october 34, 2015, at loyola university, chicago, il. As an algebra, its product is defined by linear extension of the group composition in g, with multiplicative unit the identity in g. Cbms regional conference series in mathematics volume. Hopf algebras and their actions on rings download ebook. In these categories you have muchstudied braided hopf algebras resembling but maybe finitedimensional. Work on actions of hopf algebras has unified earlier results on group actions, actions of lie algebras, and graded algebras. Hopf algebras can be characterized among bialgebras by the fundamental theorem on hopf modules. Thus studying hopf algebras themselves as well as their actions on other structures is an important problem.
If represents a generalized homology theory and if there is a. Duality and rational modules in hopf algebras over. For instance, the existence of a tensor product allows for the existence of multiple particle configurations each particle labeled by their own representation of the hopf algebra. We nally introduce quasihopf algebras physically a nontrivial fmatrix. Integrals in hopf algebras are an essential tool in studying finite dimensional hopf algebras and their action on algebras. 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 rings extended selected. Sep 15, 2000 this study covers comodules, rational modules and bicomodules. Hopf algebras and their actions on rings edition 1 by. Dual constructions for partial actions of hopf algebras. Lecture course hopf algebras, quantum groups and topological field. Both constructions are connected via the drinfeld double and directly produce knot invariants. Bilinear pairings for hopf algebras with a lift of the verschiebung. On the green rings of pointed, coserial hopf algebras by.
Buy hopf algebras and their actions on rings cbms regional conference series in. Hopf algebras and their actions on rings, including automorphism groups of algebras, and quantum groups. Hopf algebras are a relatively new concept in algebra. In the 1960s, study of hopf algebras increased, perhaps spurred by the. A kalgebra is a kvector space with two linear maps m. Combinatorial results can then be seen as particular cases of general algebraic properties of hopf algebras e. Bulletin of the ams this book should be very useful as a comprehensive, up to date introduction to the algebraic aspects of hopf algebras. Hopf algebras and their representation categories 2. Hopf rings have a very rich algebraic structure, useful in two distinct ways. If is a multiplicative theory, then is a graded ring object in the homotopy category. Constructions of hopf algebras satisfying h\ in this section, we construct three families of hopf algebras which, together with two group algebras and two enveloping algebras, cover the hopf algebras classi. In this paper, we study the representation theory of hopfore extensions of group algebras and pointed hopf algebras of rank one over an arbitrary field k. The first class consists of those hopf k algebras that are connected graded as algebras, and the second class are those hopf k algebras that are connected as coalgebras.
Hopf algebra is nice object full of structure a bialgebra with an antipode. Their rst appearing was in drinfelds works relating deformations of lie. In this paper, ore extensions in the class of hopf algebras are studied. Groups, rings, group rings, and hopf algebras international conference in honor of donald s. 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 talk will explain how many of the properties of. Weak hopf algebras, or quantum groupoids, are generalizations of hopf algebras. The comultiplication then defines how the algebra acts on this tensored space. In other words, commutative hopf algebras are precisely the rings of regular functions on affine group schemes. These hopf algebras are pointed and coserial, meaning their simple comodules are 1dimensional, and their comodules possess unique composition series respectively. Over fields it has been shown by sweedler that the existence of. This ring is spanned by isomorphism classes of finitedimensional indecomposable representations that are added together via direct sums and multiplied via tensor products.
These are notes for a course held in summer semester 2010 on the ludwigmaximillian university munich. Hopf algebras and their actions on rings about this title. Eliezer batista, joost vercruysse submitted on 6 mar 2014 v1, last revised apr 2015 this version, v3. Hopf algebras are used for many different purposes in mathematics. Hopf algebras and their actions on rings ams bookstore. Hopf rings arise naturally in the study of the spectra associated with generalized cohomology theories. Hopf algebras over number rings connecting repositories. These respectively encode associativity and the unit element. 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 algebras and quantum groups.
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. Hopf algebras and their actions on rings cover image. Passmans 75th birthday groups, rings, group rings, and hopf algebras october 24, 2015 loyola university, chicago, il ams special session in honor of donald s. On the green rings of pointed, coserial hopf algebras. This work very nicely complements the other texts on hopf algebras and is a welcome addition to the literature. The course primarily constitutes an elementary, yet mathematically rigid introduction to the theory of hopf algebras.
This book brings together many of these recent developments from the viewpoint of the algebraic structure of hopf algebras and their actions and coactions. Jan 25, 2016 we study algebraic and homological properties of two classes of infinite dimensional hopf algebras over an algebraically closed field k of characteristic zero. In this thesis, we explore the green rings of a class of hopf algebras that form an extension of the taft algebras. Hopf algebras and their actions on rings pdf free download epdf. By an algebra r over k or simply an algebra we understand a unitary, associative kalgebra r, with identity 1 1r. Hopf algebras, quantum groups and topological eld theory. In this paper we give positive answers for number rings z ln. Kassel hopf algebras uses cobraided and montgomery has and their actions on rings uses the other. Montgomery, has contributed much to group actions on rings and their generalizations to hopf algebras especially finite dimensional. More generally, hopf algebras naturally occur as rings of functions of some kind on groups of some kind. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element.
Jun 16, 2014 hopf algebras are used for many different purposes in mathematics. In the sumless sweedler notation, this property can also be expressed as. This content was uploaded by our users and we assume good faith they have the permission to share this book. We prove that a noetherian hopf algebra of finite global dimension possesses further attractive. In this paper, we study the green ring and the stable green ring of a finite dimensional hopf algebra by means of bilinear forms. Get your kindle here, or download a free kindle reading app. This is morally the reason for their appearance in algebraic topology, e. Bilinear forms on the green rings of finite dimensional. Jul 31, 2017 their physics is inherited from topological quantum field theories, while their underlying mathematical structure is based on a class of hopf algebras. This site is like a library, use search box in the widget to get ebook that you want.
Hopf algebras generating fusion rings and topological invariants simon lentner abstract. Moreover hopf algebras provide a unifying framework for groups, lie algebras, and their recent quantum analogs. Their physics is inherited from topological quantum field theories, while their underlying mathematical structure is based on a class of hopf algebras. The computations are generally carried out using hopf ring techniques as well. Jan 28, 2015 in this paper, we study the representation theory of hopfore extensions of group algebras and pointed hopf algebras of rank one over an arbitrary field k. International workshop groups, rings, lie and hopf algebras. Bialgebras and hopf algebras university of chicago.
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. In this thesis, we explore the green rings of a class of hopf algebras that form an extension of. Kassel hopf algebras uses cobraided and montgomery has and their actions on rings uses the other both on page. If one could classify certain hopf algebras, it would help in describing these other structures. Montgomery, hopf algebras and their actions on rings, cmbs reg. Janssen submitted on 17 oct 2006 v1, last revised 9 dec 2008 this version, v2. The first class consists of those hopf kalgebras that are connected graded as algebras, and the second class are those hopf kalgebras that are connected as coalgebras. International workshop groups, rings, lie and hopf. A urry of activity in the area exploded in the 1990s and produced an ever growing zoo of interesting hopf algebras. Hopf algebras, fusion rings and topological invariants 5 topological spin. Passmans 75th birthday groups, rings, group rings, and hopf algebras october 34, 2015. Universal bilinear maps for hopf algebras over f p. 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.
A galois correspondence is exhibited between right coideals subalgebras of a finitedimensional pointed hopf algebra acting homogeneously and faithfully on a free associative algebra and free subalgebras containing the invariants of this action. A hopfgalois correspondence for free algebras sciencedirect. Click download or read online button to get hopf algebras and their actions on rings book now. Groups, rings, group rings, and hopf algebras icebe.