Dual of a hopf algebra hopf algebras and combinatorics. This is the home page of algebraic combinatorics, an electronic journal of mathematics owned by its editorial board and editorsinchief it is dedicated to publishing highquality papers in which algebra and combinatorics interact in interesting ways. Some hopf algebras encode relevant combinatorial information and are quite common and useful in algebraic combinatorics. Schmitt department of mathematical sciences memphis state university memphis, tennessee 38152 abstract. Coalgebras andor hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures.
Hopf algebra structure on the tensor coalgebra tcr for a certain vector space. In the late 1970s rota introduced hopf algebras into combinatorics and there is now a well established research. Suppose that a is an algebra and we have algebra homomorphisms. In a combinatorial hopf algebra, the product and coproduct encode respectively how to combine. Hopf algebra is nice object full of structure a bialgebra with an antipode. Hopf algebras are a relatively new concept in algebra, first encountered by their namesake. We introduce a new family of noncommutative analogs of the halllittlewood symmetric functions. Hopf algebras in combinatorics astrophysics data system. Geometric combinatorial hopf algebras and modules 1. The other one is the recently introduced hopf algebra rqsym of weak quasisymmetric functions. This completes the hopf algebra structure of h, dual to the lie g roup structure of g. 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 representation theory of a hopf algebra is particularly nice, since the. Pdf a very basic introduction to hopf algebras researchgate. Index and overview of federico ardilas lectures on hopf algebras and combinatorics overview by sara billey this is an index for federico ardilas lectures on hopf algebras available on youtube. Stochastic mechanics of graph rewriting,accepted for publication inlics2016 nicolasbehr,vincentdanos,iliasgarnier,and tobias.
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. Introduction in 2008 several researchers made exciting advances in the application of algebra and combinatorics to particle physics. Section 1 is the introduction, and contains as well an elementary invitation to the subject. All the maps in the diagrams are algebra homomorphisms given our hy. Hopf algebras of combinatorial structures 1 citeseerx. Combinatorial hopf algebras and generalized dehnsommerville relations marcelo aguiar, nantel bergeron, and frank sottile abstract. Last time we looked at the definition of hopf algebras, using the group algebra as a motivating example. Hopf algebras in combinatorics mathematics stack exchange. Combinatorial hopf algebra may refer either to a hopf algebra coming from combinatorics and, in particular, to the study of various rings of. In a combinatorial hopf algebra, the product and coproduct encode respectively how to combine and split combinatorial objects. On the combinatorics of the hopf algebra of dissection diagrams springerlink. Combinatorial hopf algebras of simplicial complexes 951 1 3 4 2 1 3 4 2 fig. Combinatorial hopf algebras of simplicial complexes. If a is a hopf algebra, and m, n are amodules, then mk n is an amodule.
This time i want to look at ysym, a hopf algebra of binary trees. But im interested in knowing how hopf algebras are applied in solving combinatorial problem. Every character on a graded connected hopf algebra decomposes uniquely as a product of an even character and an odd character aguiar, bergeron, and sottile, combinatorial hopf algebra and generalized dehnsommerville equations, 2003, math. A quick introduction to combinatorial hopf algebras amy pangs. We obtain explicit formulas for the even and odd parts of the universal character on the hopf. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys axiomatic characterization of it. Much of the subject of combinatorial hopf algebras is fairly.
Ja966827 representations of finitedimensional hopf algebras martin lorenz department of mathematics, temple uni ersity, philadelphia, pennsyl ania 191226094 communicated by susan montgomery received april 3, 1995 let h denote a finitedimensional hopf algebra with antipode s over a field i. Combinatorial hopf algebra of superclass functions of type d. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. Hopf algebras in combinatorics darij grinberg and victor reiner contents introduction 2 1. This manuscript stands at the interface between combinatorial hopf algebra theory and renormalization theory. This categorical fact enables a definition of hopf monoids in some setups that do not. Pdf the hopf algebra of uniform block permutations. The grossmanlarson hopf algebras are hopf algebras with a basis indexed by forests of decorated rooted trees.
They are the universal enveloping algebras of free prelie algebras, seen as lie algebras. 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. When dealing with this divergence subtraction the subtraction of the socalled counterterms, an. H,\zeta, consisting of a hopf algebra h h over k k together with a character h k \zeta. If t t0, we can get from t0to tby removing a terminal edge, and fromtto t0by adding an edge and accompanying terminal vertex.
On the combinatorics of the hopf algebra of dissection. Discrete mathematics and theoretical computer science dmtcs vol. It is both free as an algebra and gradedcofree as a graded coalgebra, so is in some sense as far as possible from being either. Computers and internet mathematics algebra bifurcation theory research combinatorics mathematical research student aid student assistance programs student financial aid.
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 in combinatorics math user home pages. Pdf a combinatorial overview of the hopf algebra of. This universe of hopf algebras of trees is summarized below. We present an exact formula for enumerating such cycles of any length on any directed graph.
Apr 09, 2012 this is a frequent case to occur in algebraic combinatorics. In some sense, a hopf algebra is to a group what a vector space is to a set. In the 1960s, study of hopf algebras increased, perhaps spurred by the. In the cofreecoassociative general context a combinatorial hopf algebra is a. Hopf algebras can be characterized among bialgebras by the fundamental theorem on hopf modules. Usually the nth graded component of a combinatorial hopf algebra is spanned by things like graphs on n vertices partitions of n subsets of 1,2,n etc.
This is a frequent case to occur in algebraic combinatorics. Coalgebras, hopf algebras and combinatorics tu wien. 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. Combinatorial hopf algebra of superclass functions of type. In the late 1970s rota introduced hopf algebras into combinatorics and there. We conjecture it is cofree if \x1\ or x is not a root of unity. Combinatorics of rooted trees and hopf algebras 3797 be obtained by removing from t0a single terminal vertex and the edge into it.
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. 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 symmetric and briefly finite general linear groups. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including. This paper builds on two covering hopf algebras of the hopf algebra qsym of quasisymmetric functions, with linear bases parameterized by compositions. Are there examples of open problems in combinatorics which were finally solved using techniques from hopf algebras. Among the significant physical results of combinatorial physics, we may mention the reinterpretation of renormalization as a riemannhilbert problem, the fact that the slavnovtaylor identities of gauge theories generate a hopf ideal, the quantization of fields and strings, and a completely algebraic description of the combinatorics of. 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. Pdf a hopf algebra for counting simple cycles pierre.
A bialgebra is a kmodule a which is both an algebra and a coalgebra, such that the two structures are compatible i. 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. A combinatorial hopf algebra is a graded connected hopf algebra over a. Cycles, also known as selfavoiding polygons, elementary circuits or simple cycles, are closed walks which are not allowed to visit any vertex more than once. Hopf algebras are a relatively new concept in algebra. This tutorial is intended to give an accessible introduction to hopf algebras. A combinatorial noncommutative hopf algebra of graphs. 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 general and in combinatorial physics. To get some idea what it looks like, group itself is a hopf algebra, considered over a field with one element. Its characteristic is the smallest nonzero number p such that, for any a. Some hopf algebras encode relevant combinatorial information and are quite. There are many examples of familiar combinatorial objects which can be given the structure of a hopf algebra, we.
Heinz hopf in 1941 in the field of algebraic topology. More recently hopf algebras appeared also in the study of renormalization of qft, controlling its combinatorics. This work very nicely complements the other texts on hopf algebras and is a welcome addition to the literature. 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. 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. We use the version with a parameter x in the base field. Pdf we define a combinatorial hopf algebra as a hopf algebra which is free or cofree and equipped with a given isomorphism to the free algebra. Grossmanlarson hopf algebras sage reference manual v9. Hopf algebra, combinatorics, symmetric functions, quasisymmetric functions. If t t0, we can get from t0to tby removing a terminal edge, and fromtto t0by. Like hopf algebras, weak hopf algebras form a selfdual class of algebras.
A simplicial complex and its 1dimensional skeleton. Or where hopf algebra has been used to get alternative proofs of already known combinatorial results. The rest of part i, comprising sections 26, is devoted to the basics of hopf algebra theory and examples, in ascending. Citeseerx document details isaac councill, lee giles, pradeep teregowda. 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. Or where hopf algebra has been used to get alternative proofs. You must be prepared to devote at least 10 hours a week on this class, and have a solid knowledge of groups and rings. Combinatorial hopf algebras alessandro iraci october 25, 2017 1. Combinatorial realization of the hopf algebra of sashes.
Hopf algebras of combinatorial structures 1 william r. Certain hopf algebras arise in combinatorics because they have bases naturally parametrized by combinatorial objects partitions, compositions, permutations, tableaux, graphs, trees, posets, polytopes, etc. 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. Coalgebras, hopf algebras and combinatorics betreuer. Our construction relies upon tevlins bases and simple qdeformations of the.