This thesis will focus respectively on a freecommutative basis, which produces. Hopf algebras series on knots and everything series by david e. In this thesis, we study a characterization under certain conditions of the hopf algebra structures on ore extensions of hopf algebras, following articles by panov and by brown, ohagan, zhang and zhuang. Characters of hopf algebras 353 dimension, then the hopf algebra and its dual cannot both be semisimple. We follow an axiomatic approach keeping as close as possible to the classical theory of hopf algebras. First, consider the case when h is an affine commutative hopf kalgebra, k as usual algebraically closed of characteristic 0. The emphasis is put on the new structure related to the presence of canonical subalgebras al and ar in any weak hopf algebra a that play the role of noncommutative numbers in many respects. On the structure of cofree hopf algebras 2 hence, as a consequence, we get a structure theorem for cofree hopf algebras.
We examine in detail the case when the hopf algebra is one of sweedler or tafts finitedimensional hopf algebras. The interest on tobas arises from problems of classification of pointed hopf algebras. In other words, commutative hopf algebras are precisely the rings of regular functions on affine group schemes. Certain hopf algebras give rise to invariants of knots and links. We will study a number of classical and interesting hopf algebras that appear in combinatorics. More generally, hopf algebras naturally occur as rings of functions of some kind on groups of some kind.
In part iii we turn to the conneskreimer hopf algebras of feynman graphs and, more generally, to incidence bialgebras. It is intended to be a graduate text as well as a research monograph. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys axiomatic characterization of it as a positive selfadjoint hopf algebra and its application to the representation theory of symmetric and briefly finite general linear groups. For physics, however, i have come to think that hopf algebras are too tied to perturbation theory and to feynman diagrams to get us out into a different conceptual playground. In mathematics, a hopf algebra, named after heinz hopf, is a structure that is simultaneously an unital associative algebra and a counital coassociative. One of our main focuses is the construction of hopf algebras through ore extensions of a preexisting hopf algebra. Characters of hopf algebras connecting repositories.
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. This one is more about the theory of hopf algebras than about drinfeldjimbo quantum groups. Connected hopf algebras and iterated ore extensions. Hopf algebras are named after heinz hopf who discovered them the last century in the context of algebraic topology hopf 1941. Finally, we use the fact that s2 is an algebra automorphism of h to deduce that. We discuss tobas from different points of view following ideas of lusztig, nichols and schauenburg. On the hopf algebra structure of perturbative quantum field theories dirk kreimer1 department of physics university of mainz staudingerweg 7, 55099 mainz, germany abstract we show that the process of renormalization encapsules a hopf algebra structure in a.
Kaplanskys type constructions for weak bialgebras and weak hopf algebras chebel, z and makhlouf, a, journal of. The conference focussed at new results on classical hopf algebras, with emphasis on the classification theory of finite di mensional hopf algebras, categorical aspects of hopf algebras, connections with mathematical physics, and the recent developments in the theory of corings and of quasi hopf algebras. In this thesis, we look at 4pdimensional hopf algebras over an algebraically closed field of characteristic zero. Get a printable copy pdf file of the complete article 460k, or click on a. Hopf algebras are a relatively new concept in algebra. Weak hopf algebras have quite interesting representation theory. The current version of these notes can be found under as a pdf file. This is morally the reason for their appearance in algebraic topology, e.
Hopf algebra structure of incidence algebras dieter denneberg, universitat bremen. Are mapped to two different function algebras and that each algebra is the function algebra. In the 1960s, study of hopf algebras increased, perhaps spurred by the. In the cocommutative case, bialgebras and hopf algebras can be viewed as monoids and groups in the symmetric monoidal category of cocommutative coalgebras. The separable subalgebras hl and hr coincide and are noncentral commutative algebras in this particular case the subalgebra of diagonal matrices. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys axiomatic characterization of it as a positive selfadjoint hopf algebra and its application to the representation theory of. New directions in hopf algebras msri publications volume 43, 2002 finite quantum groupoids and their applications dmitri nikshych and leonid vainerman abstract. In sections 5 and 6 we use our results in a detailed study of finitedimensional semisimple hopf algebras, paying particular attention to the antipode of such a hopf algebra. Each comes with an appropriate notion of a module and thus with an associated theory of representations. A 2associative bialgebra h is a vector space equipped with two associative.
We then concentrate on braided hopf algebras in the yetterdrinfeld category over h, where h is the group algebra of a non abelian finite group. In the case when h tv is the tensor algebra, we show that the invariant trc of the cokernel of the johnson homomorphism studied in 5 projects. Hopf algebras are algebras which possess a certain additional dual structure. The key nuance between the two hopf algebras is that where ht uses integers to label the taylor expansion, the hopf algebra hr. New directions in hopf algebras msri publications volume 43, 2002 pointed hopf algebras nicolas andruskiewitsch and hansj. The order of the antipode of finitedimensional hopf algebra. Involutory hopf algebras 2659 the preceding formula translates to the assertion that s2 t t. Bialgebras and hopf algebras university of chicago. This essentially consist of corrected and slightly reorganized. An extended version of a series of lectures given at bogota in december 2002. This is a survey on pointed hopf algebras over algebraically closed. In this article, we will give a general intrinsic framework for the duality of hopf c.
On posets and hopf algebras university of washington. In particular, it appeared as one of the fundamental algebraic structures in the profound work of connes and. The hopf algebra structure of a crossed product in a braided monoidal category alvarez, j. If h is a semisimple hopf algebra of dimension p2 over k, then h is isomorphic to kz p2 or kz. Introduction to hopf algebras and representations, spring 2011 lecturer. Its zeroth term, called the hopf coradical, is the subalgebra generated by the coradical. These respectively encode associativity and the unit element. Morphisms of hopf algebras are just bialgebra maps \preserving the antipode exercise 3. Since hopf power markov chains can exhibit very different behaviour depending on the structure of the underlying hopf algebra and its distinguished basis, one must restrict attention to certain styles of hopf algebras in order to obtain stronger results. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Hopf algebra structures in particle physics 3 l 1 l 2.
That is, any such f factors uniquely as p f for a morphism f. An element of the shu e algebra can be represented by a rooted tree without sidebranchings, as shown in the left gure. A primer of hopf algebras pierre cartier institut math. Indeed, the zeta polynomial of a poset is a hopf algebra homomorphism. J complete reducibility theorems for modules over pointed hopf algebras. Aug 30, 2004 an extended version of a series of lectures given at bogota in december 2002. Examples of finitedimensional hopf algebras over a field, whose antipodes have.
Thomas coding theory john asplund dalton state college graph theory. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Show full abstract algebras coacting on the quantum principal bundle. We provide a duality between subfactors with finite index, or finite dimensional semisimple hopf algebras, and a class ofc categories of endomorphisms. It was also shown by etingof, nikshych and ostrik that any fusion category is equivalent to a category of modules over a weak hopf algebra.
Red15 project next fellows first name last name employer research justin allman duke university geometry and combinatorics in representation theory sarah anderson university of st. For example, hopf algebras of dimension p over k are the group algebra kz p and hence there is only one type of hopf algebras of dimension p zhu94. Iwahorihecke algebras and schur algebras of the symmetric group. This study covers comodules, rational modules and bicomodules. Other readers will always be interested in your opinion of the books youve read.
Aof algebras which is universal with respect to maps of algebras f. We show that a nonsemisimple hopf algebra of dimension 4p with an odd prime p is pointed if, and only if, this hopf algebra contains more than two grouplike elements. The notion of 2associative bialgebra occuring in the theorem is as follows. Hopf algebras, from basics to applications to renormalization revised and updated version, march 2006 dominique manchon1 introduction these notes are an extended version of a series of lectures given at bogota from 2nd to 6th. Co 739 combinatorial hopf algebras and renormalization. The theme is the definition of algebraic struc tures by use of dual commutative diagrams. Hopf algebras series on knots and everything hardcover. A basic introduction to hopf algebras becca ebert april 18, 2014 abstract this paper o ers an introduction to hopf algebras. It is not a paper describing basic properties and applications of hopf algebras. Roughly speaking, a hopf c algebra is a c algebra a with a comultiplication. Beattie, an isomorphism theorem for ore extension hopf algebras, comm.
A kalgebra is a kvector space with two linear maps m. On the hopf algebra structure of perturbative quantum. This file cannot be posted on any website not belonging to the authors. Similarly, this relation between the theory of hopf algebras and the theory of. It consists in a presentation of some aspects of connes and kreimers work on renormalization in the context of general connected hopf algebras, in particular birkhoff decomposition and, in the graded case, the scatteringtype formula. The weak hopf algebra structure on this particular h is given by coproduct.
Hopf algebras and representations spring 2011 acknowledgements. Representations of hopfore extensions of group algebras and. Hopf algebras and their actions on rings, by susan montgomery. Braided hopf algebras over non abelian finite groups. Group algebras, enveloping algebras of lie algebras, and quantum groups are examples of hopf algebras. In particular, in the last part of the course i gave a complete proof from scratch of zhous theorem 1994. Hopf algebras, algebraic, formal, and quantum groups. Introduction to hopf algebras and representations, spring. Karen yeats outline we will be looking at combinatorial hopf algebras. Preliminaries we shall work over a commutative ground ring r. American mathematical society, providence, ri, 1999. Hopf algebras, renormalization and noncommutative geometry alain connes ihes dirk kreimer mainz univ. September 30, 2012 abstract the incidence algebra of a partially ordered set poset supports in a natural way also a coalgebra structure, so that it becomes a mweak bialgebra even a mweak hopf algebra with m. Hopf algebras and tensor categories international conference july 48, 2011 university of almeria almeria, spain nicolas andruskiewitsch juan cuadra.
The level of abstraction of hopf algebras is close enough to my math limits that getting them internalized has proved a challenge. The key nuance between the two hopf algebras is that where ht uses integers to label the taylor expansion, the hopf algebra hr uses rooted trees for labels. Some progress on classification problems for finite dimensional hopf algebras has been made recently. Free modified rotabaxter algebras and hopf algebras in mathematics and physics 1,3,14,17,19,21,22,24,25. This talk will be a short introduction to the semisimple hopf algebras over an algebraically closed field of characteristic 0 and their representation theories. Operads, algebras, modules, and motives 3 introduction there are many di erent types of algebra. Ore extensions provide a way of building new algebras from given ones. Hopf algebras, renormalization and noncommutative geometry. As we now recall, many, but not all, of the classically familiar connected hopf algebras are ihoes. The character table for a hopf algebra arising from the. We show that if h is a cocommutative hopf algebra, then there is a natural action of autfn on h n which induces an outfn action on a quotient h n. A b is a linear map which is both an algebra and a coalgebra homomorphism. There will be some emphasis but not an exclusive emphasis on how hopf algebras. The book provides a detailed account of basic coalgebra and hopf algebra theory with emphasis on hopf algebras which are pointed, semisimple, quasitriangular, or are of certain other quantum groups.
Hopf algebras, from basics to applications to renormalization. Rather, it is a paper providing the necessary structures in order to construct a hopf algebra. Hopf algebras of dimension 2 department of mathematics. This work very nicely complements the other texts on hopf algebras and is a welcome addition to the literature. Hopf algebras, quantum groups and topological field theory. It is intended to outline the main basic results about structure and known methods for the construction of semisimple hopf algebras. Thus the familiar concepts of algebra and module dualize to concepts of. The right gure shows a general rooted tree with sidebranchings.
This is easily seen by composing two hopf algebra homomorphisms. Commutative hopf algebras, lie coalgebras, and divided. Then in section11 we give a simple derivation of the properly combinatorial part of zimmermanns cancellationfree method, in. Co 739 combinatorial hopf algebras and renormalization winter 2020 instructor. Hopf algebras, algebraic, f ormal, and quan tum groups.
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