A combinatorial hopf algebra is a graded connected hopf algebra over. Sections 26, is devoted to the basics of hopf algebra theory and examples, in ascending level of complexity. Introduction in this paper we introduce a hopf algebraic framework for studying invariants of graphs, matroids, and other combinatorial structures. In order to prove our main theorem we construct a hopf algebra a of graphs and a homomorphism of hopf algebras. These notes originating from a onesemester class by their second author at the university of minnesota survey some of the most important hopf algebras appearing in combinatorics. Are there examples of open problems in combinatorics which were finally solved using techniques from hopf algebras. We did not want to assume that the readership is familiar with all the notions of hopf algebra theory. This did not really appear in print, nor did sweedlers hopf algebra colleagues sieze the subject and carry it fur. Whenever you can pull things apart and stick them together, you can form a hopf algebra. Citeseerx document details isaac councill, lee giles, pradeep teregowda. H,\zeta, consisting of a hopf algebra h h over k k together with a character h k \zeta. Hopf algebras, renormalization and noncommutative geometry.
Schmitt department of mathematical sciences memphis state university memphis, tennessee 38152 abstract. Over the last fty years, combinatorics has undergone a radical transformation. Schmitt memphis state university, memphis, tn 38152 1. We define a combinatorial hopf algebra as a hopf algebra which is free or cofree and. Combinatorial hopf algebras and generalized dehnsommerville relations marcelo aguiar, nantel bergeron, and frank sottile abstract. Combinatorial hopf algebra of superclass functions of type. A combinatorial hopf algebra is a graded connected hopf algebra over a. Not too long ago, combinatorics mostly consisted of ad hoc methods and clever solutions to problems. Combinatorial hopf algebras alessandro iraci october 25, 2017 1. Combinatorial hopf algebra of superclass functions of type d. Bill schmitt, hopf algebra methods in graph theory, journal of pure and applied algebra 101 1995, 7790.
A quick introduction to combinatorial hopf algebras. Hopf algebra, combinatorics, symmetric functions, quasisymmetric functions. In this case, the hopf algebra coincides with the usual renormalization hopf algebra for these graphs. In algebra, the malvenutopoirierreutenauer hopf algebra of permutations or mpr hopf algebra is a hopf algebra with a basis of all elements of all the finite symmetric groups s n, and is a noncommutative analogue of the hopf algebra of symmetric functions. Matherne my research is at the intersection of algebra, geometry, and combinatorics.
The group of characters of a graded connected hopf algebra 4 2. Or where hopf algebra has been used to get alternative proofs of already known combinatorial results. It is wellknown that this is the only hopf algebra structure on qv for which the algebra structure preserves the grading. Hopf algebra of permutation pattern functions extended. Hopf algebras in combinatorics math user home pages. A quick introduction to combinatorial hopf algebras amy pangs. If t t0, we can get from t0to tby removing a terminal edge, and fromtto t0by adding an edge and accompanying terminal vertex. We use the version with a parameter x in the base field. Moreover, when q2 this hopf algebra is a realization of a wellknown combinatorial hopf algebra, namely, the hopf algebra of symmetric functions in noncommuting variables see 11, 17. The last part is an introduction to the combinatorial. In chapter4, familiarity with representation theory of nite groups over c is assumed, along. This is an example of a combinatorial hopf algebra cha, meaning a hopf algebra defined over some combinatorial object such as partitions, compositions, permutations, trees. 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. Part ii turns around the allimportant faa di bruno hopf algebra.
Pdf a combinatorial overview of the hopf algebra of. A simplicial complex and its 1dimensional skeleton. Hopf algebras in general and in combinatorial physics. Throughout the construction, we will use 3point feynman graphs with only vertex insertions in the scalar eld theory. Section 2 turns around the allimportant faa di bruno hopf algebra.
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. Hopf algebras in combinatorics mathematics stack exchange. Last time we looked at the definition of hopf algebras, using the group algebra as a motivating example. Suppose that a is an algebra and we have algebra homomorphisms. In the classical hopf algebra literature usually only one hopf algebra structure on qv is considered. A combinatorial noncommutative hopf algebra of graphs. Section 8 gives applications of the faa di bruno algebra to quantum. Bill schmitt, incidence hopf algebras, journal of pure and applied algebra 96 1994, 299330.
The algebra sym can also be used to encode equalities on generating functions via the notion of specialization of alphabet see e. 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. Hopf algebras are spaces with a notion of multiplication and comultiplication, which satisfy certain compatibility relations. Hopf algebra is nice object full of structure a bialgebra with an antipode. Renormalization hopf algebras and combinatorial groups.
The first algebra is sym 2, the algebra of bisymmetric functions or symmetric functions of level 2. Combinatorial hopf algebras of simplicial complexes. Algebraic combinatorics and symmetric functions9 7combinatorial hopf algebras can stick things together by combinatorics and you can also pull them apart. First of all, lets try to understand what extra structure there is in a group with respect to a set. One can go online and find a more definitive answer than i am able to give, including various areas where hopf algebras are used. Federico ardila, hopf algebras and combinatorics, 30 lectures, youtube list, with index. Kreimers hopf algebra h k admits a derivation called the growth operator, which is important in describing the relation of this algebra to another hopf algebra studied earlier by a. We will study hopf algebras, with an emphasis on their connections with combinatorics. The odd subalgebra is defined by certain canonical relations which we call the generalized dehnsommerville relations. In mathematics, a hopf algebra, named after heinz hopf, is a structure that is simultaneously an unital associative algebra and a counital coassociative coalgebra, with these structures compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. We introduce a complementary derivation called the pruning operator.
As a matter of fact, in a hopf algebra structure is provided on the graded vector space sc of superclass characteristic functions over u n q, for n. Pros are particular pros that can be described by generators and relations with precise conditions. The study of cocommutative hopf algebras was pursued swe 1967, new. Lemma 1 let a be a square matrix over r, for which all nondiagonal entries are all equal to some t.
All the maps in the diagrams are algebra homomorphisms given our hy. The rigidity in the structure of a hopf algebra can lead to enlightening proofs, and many interesting. In the 1960s, study of hopf algebras increased, perhaps spurred by the. Its characteristic is the smallest nonzero number p such that, for any a.
In this article, we give a combinatorial overview of the hopf algebra structure of the macmahon symmetric functions relying on the construction of a hopf algebra from any alphabet of neutral. A hopf algebra over a field mathfmath is a set with a great. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics, showing that in this latter. When dealing with this divergence subtraction the subtraction of the socalled counterterms, an.
We study invariants of graphs, matroids and other combinatorial objects by considering certain associated hopf algebras. This talk will present an alternative description obtained in collaboration with marcus berg of the hopf algebraic methods used by connes and kreimer in their reformulation of quantum field renormalization. But here is the description that i am able to give. Di08 gives a unified method to build a supercharacter. I thank the participants of the course for making it a very enjoyable experience. Hopf algebras were seen as rings which were interesting in their own right. Algebraic methods in combinatorics poshen loh june 2009 1 linear independence. Combinatorial hopf algebra may refer either to a hopf algebra coming from combinatorics and, in particular, to the study of various rings of symmetric functions, but also to the now standard notion of a pair, consisting of a hopf algebra over together with a character, the zeta function of a hopf algebra. Combinatorial hopf algebras of simplicial complexes 951 1 3 4 2 1 3 4 2 fig. Algebraic and geometric methods in enumerative combinatorics. Combinatorial hopf algebras on generating trees and. In a combinatorial hopf algebra, the product and coproduct encode respectively how to combine and split combinatorial objects. These methods are a powerful tool to show that some algebras are free polynomial rings. Furthermore, the transformation from the m ij to the f ij is a special case of the general transformation involving additive invariants mentioned above, and we thus obtain formulas for m ij and f ij in terms of doubly connected subgraphs, whose coe.
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. But im interested in knowing how hopf algebras are applied in solving combinatorial problem. We introduce a general construction that takes as input a socalled sti. Combinatorial hopf algebras from pros jeanpaulbultelandsamuelegiraudo abstract. Hopf algebras can be characterized among bialgebras by the fundamental theorem on hopf modules. Certain hopf algebras arise in combinatorics because they have bases naturally parametrized by combinatorial objects partitions, compositions, permutations, tableaux, graphs, trees, posets, polytopes, etc. 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. We emphasize especially the category of unipotent and prounipotent algebraic groups, in connection with milnormoores theorem. We show that the terminal object in the category of combinatorial hopf algebras is the algebra qsym of quasisymmetric functions. Combinatorial hopf algebra may refer either to a hopf algebra coming from combinatorics and, in particular, to the study of various rings of symmetric functions, but also to the now standard notion of a pair, h. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. This work very nicely complements the other texts on hopf algebras and is a welcome addition to the literature.
On the combinatorics of the hopf algebra of dissection. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics, showing that in this latter case the axioms of hopf algebra arise naturally. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys axiomatic characterization of it. A simple method to compute the standardization of a word w. Hopf algebras download ebook pdf, epub, tuebl, mobi. Kreimer, on the hopf algebra structure of perturbative quantum field theories, adv.
Connections were made with many aspects of algebra. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The aim of our paper is to show that we can define three versions of the rbell polynomials in three combinatorial hopf algebras in the same way. This is why we start, in the footsteps of 20, with a motivational discussion that experts should probably skip. Combinatorial hopf algebras on generating trees and certain. Hopf algebras, lie algebras, algebraic groups and their use in the renormalization in quantum field theory abstract. Giancarlo rota, hopf algebra methods in combinatorics, problemes. Combinatorial methods in algebra and geometry jacob p. Much of the subject of combinatorial hopf algebras is fairly. This categorical fact enables a definition of hopf monoids in some setups that do not.
This time i want to look at ysym, a hopf algebra of binary trees. We conjecture it is cofree if \x1\ or x is not a root of unity. Coalgebras, hopf algebras and combinatorics tu wien. Moreover, rg is cocommutative if and only if gis abelian. In some sense, a hopf algebra is to a group what a vector space is to a set. My dissertation work under pramod achar involved studying singularities of certain topological spaces via perverse sheaves. The notes then continue with the quasisymmetric and the noncommutative symmetric functions, some hopf algebras formed from graphs, posets and matroids, and the malvenutoreutenauer hopf algebra of permutations. 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. If gis a nite group, then the largest such algebra is simply the linear dual rg fg cg of the group algebra.
Computers and internet mathematics algebra functional equations research functions functions mathematics mathematical research permutations. We now turn to the existence of such a hopf algebra rg. The most studied example is the hopf algebra of symmetric functions sym. In a combinatorial hopf algebra, the product and coproduct encode. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Hopf algebra of permutation pattern functions extended abstract. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. This tutorial is intended to give an accessible introduction to hopf algebras. The importance of this algebra is due to its applications in the representation theory of the symmetric group see e. Hopf algebras in combinatorics darij grinberg and victor reiner contents introduction 2 1. It turns out that is the terminal object in hopf algebra. The representation theory of a hopf algebra is particularly nice, since the. 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. A combinatorial noncommutative hopf algebra of graphs 357 all the integer scales associated to the internal edges are higher then the edges associated to the external edges see again vincent rivasseaus book 28 for details.