Incorporating antipodes sometimes called conjugations, we obtain the notion of a hopf algebra. Serban raianu this study covers comodules, rational modules and bicomodules. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. We explain a family of examples of such hopf algebras and their modules, and classify left, right, and.
Lectures on hopf algebras, quantum groups and twists. Morphisms of hopf algebras are just bialgebra maps \preserving the antipode exercise 3. Hp incidence hopf algebra of a poset p mod reduced congruence 0. Amy pang the exposition below is adapted from pan14, sec. In some cases particularly that of symmetric functions, the rigidity. It is intended to be a graduate text as well as a research monograph. Pdf a very basic introduction to hopf algebras researchgate. Many of these chains can be explicitly diagonalized using the primitive elements of the. Twococycles, pairings, and skew pairings of bialgebras. Hopf algebra, convolution h and examples conclusion in a nutshell recall. If you have a look at the definition, it looks almost exactly like the definition of a group object, but with tensor products instead of the usual products, and with compatible comultiplication. This categorical fact enables a definition of hopf monoids in some setups that do not. Perhaps you will be interested in the crossed product algebras.
For this reason, example ii is called a oneparameter deformation of example i. The present notes were prepared for the lectures of my course introduction to hopf algebras and representations, given in the university of helsinki in the spring of 2011. Intro and main background, zinbiel, and hopf algebra. Deformations of hopf algebras arising from species 561 chapter 17.
A pointed hopf algebra h is one for which all simple comodules are one dimensional. An introduction to hopf algebras, rcrayspokanecolloquium. By a hopf action we mean a hopf module algebra structure. Lets build up the definition of hopf algebra structure bitbybit. See paper by schmitt mentioned above for more details. Bialgebras and hopf algebras university of chicago.
In the cocommutative case, bialgebras and hopf algebras can be viewed as monoids and groups in the symmetric monoidal category of cocommutative coalgebras. The group of characters of a graded connected hopf algebra 4 2. This hopf algebra models the controllednot gate of qubit quantum information theory. The notion of inner linear hopf algebra is a generalization of the notion of discrete linear group. Hopf algebras download ebook pdf, epub, tuebl, mobi. The braid group b n on nstrands is the group with n 1 generators. Introduction to hopf algebras and representations aalto university. Hopf algebra is nice object full of structure a bialgebra with an antipode.
Click download or read online button to get hopf algebras book now. However, when we try to check that the algebra and coalgebra. A purely algebraic approach can be found in 2 and 3. In the late 1970s rota introduced hopf algebras into combinatorics and there is now a well established research. Aug 07, 20 usc dornsife mathematics professor susan montgomery, along with one of her graduate students describe noncommutative algebras. The notion of hopf algebra emerged slowly from the work of the topologists in the 1940s dealing with the cohomology of compact lie groups and their homogeneous spaces. An example of a nonconcrete category is as follows.
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 and their actions on rings, including automorphism. There are many examples of familiar combinatorial objects which can be given the structure of a hopf algebra, we. The cofree pointed irreducible hopf algebra on an algebra.
This terminology is motivated by the notion of a unimodular lie group, which is a lie group on which a left invariant haar measure is also right invariant, and vice versa. Usc dornsife mathematics professor susan montgomery, along with one of her graduate students describe noncommutative algebras. Integrals in quasihopf algebras mit opencourseware. Nonprojective modules may have projective tensor powers, and the variety of a tensor product of modules may not be contained in the intersection of their varieties. Rather, it is a paper providing the necessary structures in order to construct a hopf algebra. Summary this proposal is about combinatorial algebra, with a geometrical avor. To a large extent, we work inside the grothendieck groups gh.
The kreimer hopf algebra depends only on the topology of feynman diagrams. Hopf algebras, quantum groups and topological eld theory. This site is like a library, use search box in the widget to get ebook that you want. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. In the late 1970s rota introduced hopf algebras into combinatorics and there is. The universal enveloping algebra of the lie algebra g is the factor algebra of the tensor algebra tg by the ideal ig there is a bijective correspondence between left ug. Example 2 below explains his construction via the species of graphs. The theme is the definition of algebraic struc tures by use of dual commutative diagrams. A quick introduction to combinatorial hopf algebras.
Quasisymmetric and noncommutative symmetric functions 571 17. Notice that, in all these examples, at least one of the two algebras in duality is. Hopf algebras are important tools for studying a lot of examples of hopf algebras. The major goal of this thesis is the computation of several examples en route to a conjecture for an analogous result regarding the ring of invariants for some class of \nice algebras under nite dimensional hopf algebra actions. I thank the participants of the course for making it a very enjoyable experience.
Thus fung is a hopf algebra, it encodes the information on the group structure of g. A quick introduction to combinatorial hopf algebras c. Hopf algebra, we look at a few simple examples at the end of this work. The pointed hopf algebras of andruskiewitsch and schneider.
An introduction crc press book this study covers comodules, rational modules and bicomodules. Throughout, h denotes a finitedimensional hopf algebra over a field iof characteristic p g 0, and s is the antipode of h. The string in the above example illustrates and motivates the following definition. After the pioneer work of connes and kreimer1, hopf algebras have be come an established tool in perturbative quantum. Hopf coaction with trivial homological codeterminant. There are many examples of familiar combinatorial objects which can be given the structure of a hopf algebra, we look at a few simple examples at the end of this work. We thereby obtain a large variety of new examples of 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.
Hopf algebras in combinatorics math user home pages. Representations of hopf algebras 477 here are the main results of the article. In all examples considered, we show that the given hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. Before drinfelds work in the 1980s there was only marginal interest by mathematicians in general noncommutative, noncocommutative hopf algebras, so it could be difficult to honestly interest students in the hopf algebra axioms without mentioning quantum groups. More recently the study of these algbras has gained pace because of their applica. It turns out that many properties of ordinary hopf algebras possess an infinitesimal version.
Consider a hopf algebra and some algebra acting on it, then you can form a hopf algebra with underlying vector space of their tensor product. The aim of this note is to develop galois theory for weak hopf algebras. A graded, connected hopf algebra is a graded vector space h l 1 n0 h n equipped with two linear maps. In this paper, we prove two general results that enable us to enlarge the class of hopf algebras.
Jun 18, 20 a similar development works for any hopf algebra which is either a polynomial algebra as an algebra for instance, the algebra of symmetric functions, with generators e n, or is cocommutative and a free associative algebra as an algebra e. As a second example consider now the case where gis a group of matrices, a subgroup of glgiven by matrices ta b that satisfy some algebraic relation for example orthogonality conditions. Thus the familiar concepts of algebra and module dualize to concepts of. It is not a paper describing basic properties and applications of hopf algebras. Before drinfelds work in the 1980s there was only marginal interest by mathematicians in general noncommutative, noncocommutative hopf algebras, so it could be difficult to honestly interest students in the hopf algebra axioms without mentioning quantum groups on this view the place to start is drinfelds icm lecture and papers, or books on quantum groups, that discuss the eclectic. The major goal of this thesis is the computation of several examples en route to a conjecture for an analogous result regarding the ring of invariants for some class of ice algebras under nite dimensional hopf algebra actions. Suppose that a is an algebra and we have algebra homomorphisms. Also, resolution of overlapping divergences within this framework is subtle. Chains constructed in this way include widely studied methods of card shuffling, a natural rockbreaking process, and markov chains on simplicial complexes. If a is generated as an algebra by a set b, to prove that a is a bialgebra we only need to check the diagrams in 1. The operation of squaring coproduct followed by product in a combinatorial hopf algebra is shown to induce a markov chain in natural bases. This is in contrast to earlier work of etingof and walton, in which it was shown that most pointed hopf. We begin with an introduction to the general study of hopf algebras and their.
Schmitt memphis state university, memphis, tn 38152 1. In tak 1971 a hopf algebra is given where s is not bijective. We introduce bicrossproducts, quasitriangular infinites. That is why connes and kreimer proposed to define a hopf algebra structure directly on feynman diagrams. Together with students and collaborators i have been developing a diverse family of graded algebras and coalgebras, modules and comodules, often with hopf and di erential structures. As 593 conditions for a frobenius algebra are independent of the choice of the sides, i. If a is commutative or cocommutative s2 ida and therefore s is bijective. A basic introduction to hopf algebras becca ebert april 18, 2014 abstract this paper o ers an introduction to 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. A possible strategy could be to try to adapt the methods from classical hopf.
Thus particular attention is given to the properties of projective modules. Chen fliess series is an exponential lie series x w2z w w exp x b2b b b. A long series of structure theorems is summarized in. It is shown that invariance properties of an extensible connection on a bimodule over a hopf algebra are carried over to the extension. Yd is the nichols algebra b v associated to a yetterdrinfeld module v over k bv is the quotient of the tensor algebra tv by the largest homogeneous braided hopf ideal generated by homogeneous elements of degree at least 2.
Hopf algebras can be characterized among bialgebras by the fundamental theorem on hopf modules. The representations of some hopf algebras have curious behavior. In taft 1971 fd examples tn, where n 1, are given where s2 has. A hopf algebra is an example of an object called a hopf monoid, which you can define in any symmetric monoidal category. 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.
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