Kauffman department of mathematics, statistics and computer science, university of illinois at chicago. In chapter 4 we will give some self contained introduction to hopf algebras, starting with some elementary facts about lie algebras. Lethdenote a finitedimensional hopf algebra with antipodesover a field k. Journal of pure and applied algebra j5isviba journal of pure and applied algebra 100 1995 7392 hopf algebras and invariants of 3manifolds louis h. In order to do that we define the hochschild cohomology of an algebra in an abelian monoidal category. 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. Hopf algebras were introduced in mathematics in 1941 to describe in an unified manner similar aspects of groups and algebras 1.
In particular, it appeared as one of the fundamental algebraic structures in the profound work of connes and. Click download or read online button to get hopf algebras book now. New directions in hopf algebras download new directions in hopf algebras ebook pdf or read online books in pdf, epub, and mobi format. Pdf a very basic introduction to hopf algebras researchgate. To date these ideas do not seem to have penetrated computer science to any great. Click download or read online button to new directions in hopf algebras book pdf for free now. Indeed, the zeta polynomial of a poset is a hopf algebra homomorphism. Then we characterize those algebras which have dimension less than or equal to 1 with respect to hochschild. These respectively encode associativity and the unit element. Connected hopf algebras in positive characteristic temple.
The main goal of this paper is to investigate the structure of hopf algebras with the property that either its jacobson radical is a hopf ideal or its coradical is a subalgebra. Further we will use this biproduct to describe the majid bosonization j. Download pdf hopf algebras free online new books in. Representations of hopfore extensions of group algebras. Abstractwe develop the theory of semisimple weak hopf algebras and obtain analogues of a number of classical results for ordinary semisimple hopf algebras. Jacobson received march 22, 1980 introduction the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras.
A general classification of triangular hopf algebras is not known yet. On quantum algebras and coalgebras, oriented quantum. On posets and hopf algebras university of washington. Among the topics covered are results toward the classification of finitedimensional hopf algebras semisimple and nonsemisimple, as well as what is known about the extension theory of hopf algebras.
Radfords formula for bifrobenius algebras and applications. We study the quasitriangular structures for a family of pointed hopf algebras which is big enough to include tafts hopf algebras h n2, radfords hopf algebras. Pdf radfords formula for cofrobenius hopf algebras. The antipode of a finitedimensional hopf algebra over a field has finite order. In the theory of locally compact quantum groups, an analytical form of radford s formula can be proven in terms of bounded operators on a hilbert space. Q 1997 academic press introduction this article is a study of representations of finitedimensional hopf algebras h in the spirit of larsons characters of hopf algebras l, butwx with the emphasis on the nonsemisimple case. Hopf algebras by david e radford overdrive rakuten. Quasitriangular pointed hopf algebras constructed by ore. The formula still holds for multiplier hopf algebras with integrals algebraic quantum groups. Let a be a bialgebra such that its coradical is a hopf. By a theorem of radford ra, gr a is a biproduct gr a.
This is a survey on pointed hopf algebras over algebraically closed. Hopf algebras are named after heinz hopf who discovered them the last century in the context of algebraic topology hopf 1941. Introduction hopf algebras are named after heinz hopf who discovered them the last century in the context of algebraic topology hopf 1941. Examples of ribbon hopf algebras are the usual quantum groups uqg. On the irreducible representations of a class of pointed. The antipode of a finitedimensional hopf algebra over a. In mathematics, a braided hopf algebra is a hopf algebra in a braided monoidal category. It is straight forward to convert one notation into the other. Radfords home page university of illinois at chicago. Representations of finitedimensional hopf algebras. Journal of algebra 151, 111 1992 on the antipode of a quasitriangular hopf algebra david e. We prove a criterion for semisimplicity and analyze the square of the antipode s2 of a semisimple weak hopf algebra a.
Quasiquantum groups, knots, threemanifolds, and topological. Hopf algebras are algebras which possess a certain additional dual structure. For example, radford s formula wasrecentlygeneralizedfor weakhopf algebrasby nikshych in9,forbifrobeniusalgebrasintroducedin5. Radford s book contains at the end of each chapter a very useful set of chapter notes that discuss these references and therefore provide an entry point to the recent research literature, especially to the extensive literature on the classification of finitedimensional pointed hopf algebras, a topic not discussed in any of the other books. They take the role of quantum borel algebras in the classification of pointed hopf algebras, analogously to the classical lie algebra case. B of two hopf algebras to be a hopf algebra, and we show that in this case the antipode of the bicrossproduct is bijective if the antipodes of the factors are. Conference, syracuse university, syracuse, ny, april 9, 2010 pdf file. We outline a theory of quantum algebras and coalgebras and their resulting invariants of unoriented 11 tangles, knots and links, we. In chapter4, familiarity with representation theory of nite groups over c is assumed, along. Using this, we obtain the structure of quasi hopf algebras with a projection. Jan 28, 2015 in this paper, we study the representation theory of hopf ore extensions of group algebras and pointed hopf algebras of rank one over an arbitrary field k. Download pdf new directions in hopf algebras free online.
This note extends radford s formula for the fourth power of the antipode of a finitedimensional hopf algebra to cofrobenius hopf algebras and studies equivalent conditions to a hopf. On the antipode of a quasitriangular hopf algebra sciencedirect. Hopf algebras and representations spring 2011 acknowledgements. Representations of hopfore extensions of group algebras and. The structure of hopf algebras with a weak projection. We present an invariant of connected and oriented closed 3manifolds based on a coribbon weak hopf algebra h with a suitable leftintegral. The most common braided hopf algebras are objects in a yetterdrinfeld category of a hopf algebra h, particularly the nichols algebra of a braided vectorspace in that category. The hopf algebras we are considering arise in the work of n. New post fulltext search for articles, highlighting downloaded books, view pdf in a browser and download history correction in our blog.
The following definition will be key in the computations that follow. 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. A kalgebra is a kvector space with two linear maps m. As a consequence, this hopf algebra will be revealed isomorphic to the symmetric functions this is the famous frobenius correspondence between symmetric. We give a new proof of the fact, due toos, thathis a symmetric algebra if and only ifhis unimodular ands 2 is inner. This is easily seen by composing two hopf algebra homomorphisms. Hopf algebras download ebook pdf, epub, tuebl, mobi.
Biproducts and twococycle twists of hopf algebras by. Further examples are hopf algebras with a triangular decomposition, like all quantized enveloping algebras. New directions in hopf algebras msri publications volume 43, 2002 on quantum algebras and coalgebras, oriented quantum algebras and coalgebras, invariants of 11 tangles, knots and links david e. Weak hopf algebras unify the henningskauffman radford and. Note that the hopf algebras we consider are not semisimple hopf algebras. On the hopf algebra structure of perturbative quantum. In this paper, we study the representation theory of hopf ore extensions of group algebras and pointed hopf algebras of rank one over an arbitrary field k. Hopf algebras, quantum groups and topological field theory. Ifhis involutory and not semisimple, then the dimensions of all projectivehmodules are shown to be divisible by char k. On the classification of finitedimensional triangular hopf algebras. Click download or read online button to get on coquasitriangular hopf algebras and the quantum yang baxter equation book now. Radpord department of mathematics, university of illinois, chicago, illinois 60680 communicated by n. Download pdf hopf algebras free online new books in politics.
Hopf algebras series on knots and everything series by david e. This provides a categorical generalization of radford s s4 formula for finitedimensional hopf algebras presented in 1976, which was proved by nikshych 2002 for weak hopf algebras and by. Hopf algebras in combinatorics 3 the main prerequisite for reading these notes is a good understanding of graduate algebra3, in partic ular multilinear algebra tensor products, symmetric powers and exterior powers4 and basic categorical language5. Some papers consider hopf versions of classical topics, such as the brauer group, while others are closer to work in quantum groups. Moreover, in the last few years, this formula has been adapted for several types of generalized hopf algebras. On quantum algebras and coalgebras, oriented quantum algebras. Hopf algebra, we look at a few simple examples at the end of this work. In this paper, the groups of hopf algebra automorphisms for two classes of pointed hopf algebras are determined. Automorphism groups of pointed hopf algebras springerlink. Then we study properties of various modular elements of gwbf algebras. He thanks the university of almeria spain, and mount allison university canada for their warm hospitality. Locally compact quantum groups generalize hopf algebras and carry a topology. In an appendix, we improve a standard criterion for a bicrossproduct a. Radford department of mathematics, statistics, and computer science, university of illinois at chicago, chicago, illinois 60680 communicated by nathan jacobson received october 1, 1989 let a, r be a quasitriangular hopf algebra with antipode s in the category of vector spaces over a field k.
Group algebras, enveloping algebras of lie algebras, and quantum groups are examples of hopf al gebras. The algebra of all continuous functions on a lie group is a locally compact quantum group. Other references related to radford biproduct see 1, 68, 14. In the case where k is a splitting field forh, we give a formula for the rank of. Ribbon element on cofrobenius quasitriangular hopf algebras. We parameterize the finitedimensional irreducible representations of a class of pointed hopf algebras over an algebraically closed field of characteristic zero by dominant characters. It is intended to be a graduate text as well as a research monograph. While radford biproduct is one of the celebrated objects in the theory of hopf algebras, which plays a fundamental role in the classi. Certain hopf algebras give rise to invariants of knots and links. Jacobson received march 22, 1980 introduction the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1,5, and these notions can be viewed as. The structure of hopf algebras with a projection sciencedirect. Hopf algebras in the adscft correspondence diplomarbeit.
Group algebras, enveloping algebras of lie algebras, and quantum groups are examples of hopf algebras. Formulas in hopf algebras are illustrated either by tensor diagrams or algebraic expressions. 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 natural manner. Bialgebras and hopf algebras university of chicago. Journal of algebra 92, 322347 1985 the structure of hopf algebras with a projection david e.
The class of infinite dimensional cofrobenius hopf algebras includes cosemisimple hopf algebras, such as the group algebra of an infinite group. For a detailed treatment of hopf algebras, see, for instance, 27 32 33, etc. In this paper we introduce the notion of a generalized weak bifrobenius algebra or briefly gwbfalgebra which is designed to cover both weak hopf algebras and bifrobenius algebras. The group of hopf algebra automorphisms for a finitedimensional semisimple cosemisimple hopf algebra over a field k was considered by radford and waterhouse.
Graded hopf algebras are often used in algebraic topology. Radford, the structure of hopf algebras with a projection, journal of. Radford s s 4 formula for cofrobenius hopf algebras throughout this section, h will denote a cofrobenius hopf algebra, not necessarily finite dimensional. Hopf algebras and representation theory of hopf algebras faubox. This note extends radford s formula for the fourth power of the antipode of a finite dimensional hopf algebra to cofrobenius hopf algebras and studies.