Group algebras, enveloping algebras of lie algebras, and quantum groups are examples of hopf algebras. This study covers comodules, rational modules and bicomodules. In other words, commutative hopf algebras are precisely the rings of regular functions on affine group schemes. But there is a scarcity of treatments of hopf algebras at a level and in a manner that physicists are.
Ore extensions provide a way of building new algebras from given ones. Characters of hopf algebras 353 dimension, then the hopf algebra and its dual cannot both be semisimple. C hopf algebra hopf algebra map hopf module implies induces injective inverse ker f left ideal lemma let f let h lie algebra linear map map f maximal ideal morphism multiplication nsequence notation pointed irreducible coalgebra proposition proved rational. It concerns solely the application of hopf algebras to renormalization in quantum field theory and the combinatorics of feynman diagrams. 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. Syllabus bij wedwis12762 hopf algebras en quantum groepen hopf algebras and quantum groups. Some progress on classification problems for finite dimensional hopf algebras has been made recently. We show that these examples can be successively uni ed. These respectively encode associativity and the unit element.
We give an account of the conneskreimer renormalization in the context of connected graded. The theme is the definition of algebraic struc tures by use of dual commutative diagrams. Zelevinskys structure theory of positive selfdual hopf algebras. Unfortunately, this is not the case with all matrices. Representations of hopfore extensions of group algebras and. Pdf radfords formula for cofrobenius hopf algebras. Recall from basic linear algebra that for example any real symmetric matrix can be diagonalized. These descend to relations between graph polynomials on.
Hopf algebras from number theory, mathematical physics and algebraic topology. Hopf algebras download ebook pdf, epub, tuebl, mobi. Certain hopf algebras give rise to invariants of knots and links. This one is more about the theory of hopf algebras than about drinfeldjimbo quantum groups. Hopf algebras, renormalization and noncommutative geometry. We study their geometry and in particular their lie algebra of. Representations of hopfore extensions of group algebras. Introduction to hopf algebras and representations, spring.
Click download or read online button to get hopf algebras book now. A b is a linear map which is both an algebra and a coalgebra homomorphism. This site is like a library, use search box in the widget to get ebook that you want. Introduction to hopf algebras and representations, spring 2011 lecturer. More generally, hopf algebras naturally occur as rings of functions of some kind on groups of some kind. Quarks, leptons, and hopf algebra propagators received. Kaplansky conjecture says that if h is a finite dimensional hopf algebra, then the antipode has finite order. Hopf algebras over a field of characteristic zero is semiabelian. Victor reiner, hopf algebras in combinatorics math user home. Lead by examples we introduce the notions of hopf algebra and quantum group. Kauffman department of mathematics, statistics and computer science, university of illinois at chicago. This general question is addressed through the consideration of different classes of.
Roughly speaking, a hopf c algebra is a c algebra a with a comultiplication. It is not a paper describing basic properties and applications of hopf algebras. Walker department of mathematics, university of california riverside, ca 92521 usa october 16, 2010 abstract one problematic feature of hall algebras is the fact that the standard multiplication and comultiplication maps do. 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. The articles in this volume cover a wide variety of topics related to the theory of hopf algebras and. In particular, in the last part of the course i gave a complete proof from scratch of zhous theorem 1994.
Weak hopf algebras have quite interesting representation theory. A primer of hopf algebras pierre cartier institut math. From hopf algebras to quantum groups mathematics stack. Hopf algebras of dimension 2 department of mathematics. We provide a duality between subfactors with finite index, or finite dimensional semisimple hopf algebras, and a class ofc categories of endomorphisms. On differential hopf algebras by william browderi differential hopf algebras arise in several contexts in algebraic topology. But the special extra structure that makes them hopf algebras is ignored in the majority of papers dealing with these objects.
Hopf algebras, algebraic, f ormal, and quan tum groups. There are some other ones which i know are out there, but i havent read. Rather, it is a paper providing the necessary structures in order to construct a hopf algebra. In this thesis, we look at 4pdimensional hopf algebras over an algebraically closed field of characteristic zero. 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. Hopf algebras and representations spring 2011 acknowledgements. This essentially consist of corrected and slightly reorganized. One of our main focuses is the construction of hopf algebras through ore extensions of a preexisting hopf algebra. A spring 2012 course on hopf algebras and combinatorics at sfsu and. 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. J complete reducibility theorems for modules over pointed hopf algebras. Vercruysse syllabus bij wedwis12762 hopf algebras en quantum groepen hopf algebras and quantum groups master wiskunde vrije universiteit brussel en. We focus on the situatation where the weak hopf algebra is a groupoid algebra or its dual. Pdf representations of finitedimensional hopf algebras.
A kalgebra is a kvector space with two linear maps m. This is morally the reason for their appearance in algebraic topology, e. 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. Other applications of quantum groups and hopf algebras to lowdimensional physics etc. We obtain weak versions of strongly graded rings, and classical. Cocommutative hopf algebras are symmetric precisely if they are unimodular which is not generally the case cf. Moore the notion of hopf algebra has been abstracted from the work of hopf on manifolds which admit a product operation. A hopf algebra is defined as a bialgebra with an additional. Hopf algebras h over a principal ideal domain r the following are equivalent. Introduction to hopf algebras and representations aalto math. Supercharacters, symmetric functions in noncommuting.
Like hopf algebras, weak hopf algebras form a selfdual class of algebras. 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. A hopf algebra structure on hall algebras christopher d. When parameters such as mass, electric charge, acceleration, etc. Clay mathematics proceedings volume 12, 2009 renormalization in connected graded hopf algebras. Hopf algebras and their actions on rings, by susan montgomery. In this article, we will give a general intrinsic framework for the duality of hopf c.
In chapter4, familiarity with representation theory of nite groups over c is assumed, along. Lectures on hopf algebras, quantum groups and twists. Thus the familiar concepts of algebra and module dualize to concepts of. Characters of hopf algebras connecting repositories. Journal of algebra 17, 352368 1971 characters of hopf algebras richard gustavus larson department of mathematics, university of illinois at chicago circle, chicago, illinois 60680 communicated by d. Buchsbaum received december 8, 1969 in the development of the character theory of finite groups, one of the key facts used in proving the orthogonality relations is the invariance. Weak hopf algebras, or quantum groupoids, are generalizations of hopf algebras. A basic introduction to hopf algebras becca ebert april 18, 2014 abstract this paper o ers an introduction to hopf algebras. 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. In this paper i give a generalization of this result which needs only a minor restriction, which, for example, abvays holds if pitr 0 for the base ring r. The point of theorem 9 is that it can be used to show that the fact that one graph polynomial can be obtained as a specialisation of another follows from the fact that there is a hopf algebra morphism here usually projection between the corresponding hopf algebras. Hopf algebras, renormalization and noncommutative geometry alain connes ihes dirk kreimer mainz univ. In particular, i will explain how to prove that any haction on the weyl algebra and. Hopf algebras are generalizations of groups and their concepts are acquiring importance in the treatment of conformal field theories, noncommutative spacetimes, topological quantum computation and other important domains of investigation.
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