Combinatorics of rooted trees and hopf algebras 3797 be obtained by removing from t0a single terminal vertex and the edge into it. In chapter4, familiarity with representation theory of nite groups over c is assumed, along. Last time we looked at the definition of hopf algebras, using the group algebra as a motivating example. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Algebraic and geometric methods in enumerative combinatorics. This did not really appear in print, nor did sweedlers hopf algebra colleagues sieze the subject and carry it fur. Algebraic methods in combinatorics poshen loh june 2009 1 linear independence. Matherne my research is at the intersection of algebra, geometry, and combinatorics. Hopf algebras, lie algebras, algebraic groups and their use in the renormalization in quantum field theory abstract. A hopf algebra over a field mathfmath is a set with a great. The group of characters of a graded connected hopf algebra 4 2. Hopf algebra is nice object full of structure a bialgebra with an antipode.
It turns out that is the terminal object in hopf algebra. We now turn to the existence of such a hopf algebra rg. 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. Coalgebras, hopf algebras and combinatorics tu wien. We introduce a complementary derivation called the pruning operator. Not too long ago, combinatorics mostly consisted of ad hoc methods and clever solutions to problems. The odd subalgebra is defined by certain canonical relations which we call the generalized dehnsommerville relations. Combinatorial hopf algebras and generalized dehnsommerville relations marcelo aguiar, nantel bergeron, and frank sottile abstract. Hopf algebras were seen as rings which were interesting in their own right. 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.
Computers and internet mathematics algebra functional equations research functions functions mathematics mathematical research permutations. 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. Hopf algebras are spaces with a notion of multiplication and comultiplication, which satisfy certain compatibility relations. In a combinatorial hopf algebra, the product and coproduct encode respectively how to combine and split combinatorial objects. Sections 26, is devoted to the basics of hopf algebra theory and examples, in ascending level of complexity. Combinatorial hopf algebras of simplicial complexes 951 1 3 4 2 1 3 4 2 fig. We use the version with a parameter x in the base field. Its characteristic is the smallest nonzero number p such that, for any a. Federico ardila, hopf algebras and combinatorics, 30 lectures, youtube list, with index. It is wellknown that this is the only hopf algebra structure on qv for which the algebra structure preserves the grading. 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.
This time i want to look at ysym, a hopf algebra of binary trees. The importance of this algebra is due to its applications in the representation theory of the symmetric group see e. We conjecture it is cofree if \x1\ or x is not a root of unity. Schmitt department of mathematical sciences memphis state university memphis, tennessee 38152 abstract. The last part is an introduction to the combinatorial.
On the combinatorics of the hopf algebra of dissection. 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. Hopf algebras in general and in combinatorial physics. A simple method to compute the standardization of a word w. I thank the participants of the course for making it a very enjoyable experience.
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. This categorical fact enables a definition of hopf monoids in some setups that do not. 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. A quick introduction to combinatorial hopf algebras. 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. Or where hopf algebra has been used to get alternative proofs of already known combinatorial results. Hopf algebras can be characterized among bialgebras by the fundamental theorem on hopf modules. 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. Certain hopf algebras arise in combinatorics because they have bases naturally parametrized by combinatorial objects partitions, compositions, permutations, tableaux, graphs, trees, posets, polytopes, etc. But im interested in knowing how hopf algebras are applied in solving combinatorial problem. All the maps in the diagrams are algebra homomorphisms given our hy. Suppose that a is an algebra and we have algebra homomorphisms. 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. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Algebraic combinatorics and symmetric functions9 7combinatorial hopf algebras can stick things together by combinatorics and you can also pull them apart. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. We show that the terminal object in the category of combinatorial hopf algebras is the algebra qsym of quasisymmetric functions. Bill schmitt, incidence hopf algebras, journal of pure and applied algebra 96 1994, 299330. We will study hopf algebras, with an emphasis on their connections with combinatorics. Hopf algebras in combinatorics math user home pages. A combinatorial noncommutative hopf algebra of graphs.
We study invariants of graphs, matroids and other combinatorial objects by considering certain associated hopf algebras. Combinatorial hopf algebra of superclass functions of type d. The study of cocommutative hopf algebras was pursued swe 1967, new. The algebra sym can also be used to encode equalities on generating functions via the notion of specialization of alphabet see e. A simplicial complex and its 1dimensional skeleton. 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. 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. If t t0, we can get from t0to tby removing a terminal edge, and fromtto t0by adding an edge and accompanying terminal vertex. A quick introduction to combinatorial hopf algebras amy pangs. Schmitt memphis state university, memphis, tn 38152 1. This work very nicely complements the other texts on hopf algebras and is a welcome addition to the literature. Part ii turns around the allimportant faa di bruno hopf algebra. Giancarlo rota, hopf algebra methods in combinatorics, problemes.
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. Over the last fty years, combinatorics has undergone a radical transformation. In the 1960s, study of hopf algebras increased, perhaps spurred by the. The first algebra is sym 2, the algebra of bisymmetric functions or symmetric functions of level 2. Combinatorial hopf algebras of simplicial complexes. 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. But here is the description that i am able to give. Section 8 gives applications of the faa di bruno algebra to quantum. Kreimer, on the hopf algebra structure of perturbative quantum field theories, adv. Lemma 1 let a be a square matrix over r, for which all nondiagonal entries are all equal to some t. The most studied example is the hopf algebra of symmetric functions sym. 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. Hopf algebras download ebook pdf, epub, tuebl, mobi. Hopf algebras of combinatorial structures 1 william r.
Combinatorial hopf algebras on generating trees and. These methods are a powerful tool to show that some algebras are free polynomial rings. H,\zeta, consisting of a hopf algebra h h over k k together with a character h k \zeta. 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 rigidity in the structure of a hopf algebra can lead to enlightening proofs, and many interesting. Combinatorial hopf algebras on generating trees and certain. We did not want to assume that the readership is familiar with all the notions of hopf algebra theory. Combinatorial hopf algebra of superclass functions of type.
Connections were made with many aspects of algebra. Hopf algebra, combinatorics, symmetric functions, quasisymmetric functions. In this case, the hopf algebra coincides with the usual renormalization hopf algebra for these graphs. This is why we start, in the footsteps of 20, with a motivational discussion that experts should probably skip. 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.
One can go online and find a more definitive answer than i am able to give, including various areas where hopf algebras are used. Hopf algebras in combinatorics mathematics stack exchange. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. 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. Much of the subject of combinatorial hopf algebras is fairly. 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. 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. A combinatorial hopf algebra is a graded connected hopf algebra over a. Are there examples of open problems in combinatorics which were finally solved using techniques from hopf algebras. First of all, lets try to understand what extra structure there is in a group with respect to a set. This tutorial is intended to give an accessible introduction to hopf algebras. Hopf algebra of permutation pattern functions extended.
Renormalization hopf algebras and combinatorial groups. We introduce a general construction that takes as input a socalled sti. Whenever you can pull things apart and stick them together, you can form a hopf algebra. Hopf algebras in combinatorics darij grinberg and victor reiner contents introduction 2 1. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys axiomatic characterization of it. In some sense, a hopf algebra is to a group what a vector space is to a set. When dealing with this divergence subtraction the subtraction of the socalled counterterms, an. Throughout the construction, we will use 3point feynman graphs with only vertex insertions in the scalar eld theory. Hopf algebra of permutation pattern functions extended abstract. Introduction in this paper we introduce a hopf algebraic framework for studying invariants of graphs, matroids, and other combinatorial structures. 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.
In the classical hopf algebra literature usually only one hopf algebra structure on qv is considered. We emphasize especially the category of unipotent and prounipotent algebraic groups, in connection with milnormoores theorem. Bill schmitt, hopf algebra methods in graph theory, journal of pure and applied algebra 101 1995, 7790. In a combinatorial hopf algebra, the product and coproduct encode. Di08 gives a unified method to build a supercharacter.
Section 2 turns around the allimportant faa di bruno hopf algebra. 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. 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. 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. If gis a nite group, then the largest such algebra is simply the linear dual rg fg cg of the group algebra. Combinatorial methods in algebra and geometry jacob p.
My dissertation work under pramod achar involved studying singularities of certain topological spaces via perverse sheaves. Combinatorial hopf algebras alessandro iraci october 25, 2017 1. 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. The representation theory of a hopf algebra is particularly nice, since the. Combinatorial hopf algebras from pros jeanpaulbultelandsamuelegiraudo abstract. A combinatorial hopf algebra is a graded connected hopf algebra over. We define a combinatorial hopf algebra as a hopf algebra which is free or cofree and. In order to prove our main theorem we construct a hopf algebra a of graphs and a homomorphism of hopf algebras. Hopf algebras and the penrose polynomial sciencedirect. Pdf a combinatorial overview of the hopf algebra of.