While exploring these applications is beyond the scope of this paper, this paper attempts to prepare the. Hopf algebras are named after heinz hopf who discovered them the last century in the context of algebraic topology hopf 1941. Crainic, t hat for group algebras and in general for hopf the results o f this pa per were announced in the ams meeting in t oronto, september 2324, 2000. A unique aspect of the book is its treatment of hopf algebras in a form accessible to physicists. Some progress on classification problems for finite dimensional hopf algebras has been made recently. If a is a cocommutative hopf algebra over k algebraically closed. To simplify the notation, i drop the distinction between a feynman graph and the evaluation of the graph. This book is an introduction to the basic theory of hopf algebras for the reader already familiar with the basic ideas of linear algebra and commutative algebra.
In the broadest sense of the term, an extension of hopf algebras is perhaps just some injective hopf algebra map k. 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. In the theory of hopf algebras, some of the rst fundamental results were concerned with the characteri. In this paper we show a bijective correspondence between normal hopf ideals and subhopf algebras of a commutative hopf algebra over a field k. Group algebras, enveloping algebras of lie algebras, and quantum groups are examples of hopf algebras. Hopf algebras from number theory, mathematical physics and algebraic topology. The associativity and unit constraints are just the usual isomorphisms of kmodules.
As an algebra this is the free commutative algebra on the 1particle irreducible graphs. This is an expanded version of the comment i made before. H or, dually, a hopf algebra surjection, which is the viewpoint taken in 35. Joyal duality 84 references 84 introduction hopf algebras have long been. Finite groups acting on rings by automorphisms, and groupgraded rings are instances of hopf algebras h acting on hmodule algebras a.
Hopf algebras from cooperads with multiplication 57 4. This study covers comodules, rational modules and bicomodules. The book may be used as the main text or as a supplementary text for a graduate algebra course. Another advantage is that hopf algebra and rota baxter theory allow for e. It turns out that kqc admits a graded hopf structure with length grading if and only if q is a hopf quiver. This book brings together many of these recent developments from the viewpoint. 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. Involutory hopf algebras 2659 the preceding formula translates to the assertion that s2 t t. An hhopf bimodulem is an hbimodule and simulta neously an hbicomodule such that the structure maps.
Frobenius algebras, and the algebras arising as dual of qcf coalgebras are entitled to be called generalized frobenius algebras, or rather generalized qf algebras. This is a study on pattern hopf algebras in combinatorial structures. We will exhibit explicitly this lie algebra in several cases, including the all b of rank 2. In 3 we will prove the existence of a counit and in 4 the existence of an antipode. We will be much stricter in our terminology by calling an extension only a short exact sequence k. Chases of congeries of exact sequences give the following comparison assertion. Hopf algebra, we look at a few simple examples at the end of this work. The last chapter chapter 4 discusses several applications of hopf algebras, some of which are further developed in the authors 2011 publication, an introduction to hopf algebras. These are the hopf algebra of goncharov for multiple zeta values, that of conneskreimer for renormalization, and a hopf algebra constructed by baues to study double loop spaces. In this thesis, we look at 4pdimensional hopf algebras over an algebraically closed field of characteristic zero. Incorporating antipodes sometimes called conjugations, we obtain the notion of a hopf. However, there are very few examples of nonsemisimple hopf algebras of these dimensions.
On june 6 7, 2009, the talks will be given in room sn2109. After introducing and discussing the basic properties of coalgebras, bialgebras and hopf algebras, the author treats the fundamental structure theorem of bimodules and sullivans. Hopf bimodules over group algebras let h be a hopf algebra. Lead by examples we introduce the notions of hopf algebra and. But there is a scarcity of treatments of hopf algebras at a level and in a manner that physicists are. Johns, nl, canada schedule abstracts of talks rooms all lectures and talks on june 3 5, 2009, will be delivered in room a1046 arts and administration building. It is not a paper describing basic properties and applications of hopf algebras. From hopf algebras to quantum groups mathematics stack. Hopf algebras of b by the universal enveloping algebra of a graded nilpotent lie algebra. We shall then explain the origin of the hopf algebra hr from the renormal.
The derived category of the heart and the 1minimal model 91 part v. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. In order to include a character theory for infinite dimensional hopf algebras, we must consider characters as elements of the hopf algebra which are associated. We show that these examples can be successively uni ed. A right comodule over c is a pair mm, where m is a kmodule and. His lectures were inspiring, and he kindly spent a lot of time discussing the subject with me, explaining things. In fact, there are no nonsemisimple hopf algebras over k of dimension 15, 21 or 35 cf. Hopf algebras are used in combinatorics, category theory, homological algebra, lie groups, topology, functional analysis, quantum theory, and hopfgalois theory.
Vercruysse syllabus bij wedwis12762 hopf algebras en quantum groepen hopf algebras and quantum groups master wiskunde vrije universiteit brussel en universiteit antwerpen. Hopf algebra structures in particle physics ful lls the rotabaxter relation. Hence qft amplitudes can be regarded as characters on this hopf algebra. The universal enveloping algebra of a lie algebra g is a coalgebra with the coproduct. Hopf algebras are algebras endowed with a linear map d. H are both hbimodulemaps, where thehmodulestructures ofh. These respectively encode associativity and the unit element. As 593 conditions for a frobenius algebra are independent of the choice of the sides, i. These turn out to have a wide range of applications to hopf algebras. It concerns solely the application of hopf algebras to renormalization in quantum field theory and the combinatorics of feynman diagrams. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity.
Lectures on hopf algebras, quantum groups and twists. In this paper we examine the propagators of the hopf algebra generated by the permutation group of three elements, cp 3. Aof algebras which is universal with respect to maps of algebras f. Hopf algebras and representations spring 2011 acknowledgements. Rather, it is a paper providing the necessary structures in order to construct a hopf algebra. Finally, we use the fact that s2 is an algebra automorphism of h to deduce that s2 ty. Roughly speaking, a hopf c algebra is a c algebra a with a comultiplication. Jul 31, 2017 semisimple hopf algebras and fusion categories. Bialgebras and hopf algebras university of chicago. In the 1960s, study of hopf algebras increased, perhaps spurred by the. Hopf algebras generalize both lie groups and lie algebras, so the problem does not present itself in a hopf algebra formalism. Bialgebras and hopf algebras gives rise to an exact sequence 0.
A kalgebra is a kvector space with two linear maps m. In addition, several conjectures of kaplansky have been solved, the most striking of which is a kind of lagranges theorem for hopf algebras. There is a closed formula for dx with x v1 vn 2v n, given in terms of p,n p shuf. Minimal algebras, 1minimal models, and colie algebras 77 3.
In the late 1970s rota introduced hopf algebras into combinatorics and there is. 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. In particular, i will explain how to prove that any haction on the weyl algebra and. A b is a linear map which is both an algebra and a coalgebra homomorphism. Many other concepts regarding algebras have their dual in coalgebras. Weak hopf algebras, or quantum groupoids, are generalizations of hopf algebras. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. Other applications of quantum groups and hopf algebras to lowdimensional physics etc. Certain hopf algebras give rise to invariants of knots and links. This action is similar to the previous one, for the map d.
What you will find related to hopf algebras in these articles is basicly the fact that with every compact quantum group, you can associate a canonical dense hopf algebra generated by the matrix elements of all finitedimensional unitary representations of your compact quantum group. From hopf algebras to quantum groups mathematics stack exchange. Hopf algebras, algebraic, f ormal, and quan tum groups. Twisting matrices and representations of colie algebras 85 6.
For other works on constructing hopf algebras via quivers see 2,3,6,9. The galois group algebra was replaced by a hopf algebra and a general hopf galois theory was eventually developed. The aim of this paper is to establish a close relation between the above hopf algebras hr and ht. Hopf algebras, renormalization and noncommutative geometry. We formulate conditions to have that the antipode is a bijection. Hopf algebras are a relatively new concept in algebra. Namely, the unit, hidden in the three examples gckt20, x1 by normalizations, will be given by the unit endomorphism of the monoidal unit 1 of the feynman category, viz. For these generalizations would hold for both these group algebras and restricted enveloping algebras. A correspondence between hopf ideals and subhopf algebras. Vercruysse syllabus bij wedwis12762 hopf algebras en quantum groepen hopf algebras and quantum groups master wiskunde vrije universiteit brussel en.
Andreas bauer, free university of berlin the hopf calgebraic quantum double models symmetries beyond group theory. Hopf algebra is nice object full of structure a bialgebra with an antipode. That is, any such f factors uniquely as p f for a morphism f. This categorical fact enables a definition of hopf monoids in some setups that do not. Hopf algebras are the group algebras generated by nite groups such as the permutation group of three elements, p 3. We can similarly define a right hopf algebra by the condition that b have a right antipode, i.
For nonsemisimple hopf algebras of dimension p2, the only known. Hopf algebras, rather than just bi algebras, one has to take quotients and require certain connectedness assumptions. Hopf algebras of dimension 2 iowa state university. Section 2 shows how commutative hopf algebras emerge from. Hopf algebras can be characterized among bialgebras by the fundamental theorem on hopf modules.
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