Many other concepts regarding algebras have their dual in coalgebras. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. This universe of hopf algebras of trees is summarized below. A simplicial complex and its 1dimensional skeleton. Hopf algebras in combinatorics mathematics stack exchange. Marcelo aguiar, swapneel mahajan, monoidal functors, species and hopf algebras, with forewords by kenneth brown, stephen chase and andre joyal. 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. Nowadays the major book on combinatorial hopf algebras is. Are there examples of open problems in combinatorics which were finally solved using techniques from hopf algebras. Hopf algebras of combinatorial structures 1 william r. 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. Other applications of quantum groups and hopf algebras to lowdimensional physics etc.
Hopf algebras in combinatorics math user home pages. 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. Your story matters citation novelli, jeanchristophe, thibon, jeanyves, and lauren williams. 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.
Schmitt department of mathematical sciences memphis state university memphis, tennessee 38152 abstract. Amy pang the exposition below is adapted from pan14, sec. Hopf algebras of combinatorial structures 1 citeseerx. If t t0, we can get from t0to tby removing a terminal edge, and fromtto t0by adding an edge and accompanying terminal vertex. Weak hopf algebras, or quantum groupoids, are generalizations of hopf algebras. You are encouraged to work together on the homework, but you must state who you worked with in each problem. Coalgebras andor hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures. A combinatorial hopf algebra is a graded connected hopf algebra over. For the basic intuition for hopf algebras in combinatorics see the chapter 5 of s. 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. If a is a cocommutative hopf algebra over k algebraically closed.
A quick introduction to combinatorial hopf algebras c. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including. Our construction relies upon tevlins bases and simple qdeformations of the. 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 self. The latter is closely related to other important hopf algebras in algebraic combinatorics, including the malvenutoreutenauer hopf algebra 20 and the hopf algebra of quasisymmetric functions 19, 24, 28. We will study hopf algebras, with an emphasis on their connections with combinatorics. Pdf hopf algebras and the combinatorics of connected. A quick introduction to combinatorial hopf algebras amy pangs. In chapter4, familiarity with representation theory of nite groups over c is assumed, along. A unital, associative and commutative algebra hendowed with a coproduct, a counit and an antipode s, satisfying all the properties listed above, is called a commutative hopf algebra. Combinatorial hopf algebras of simplicial complexes 951 1 3 4 2 1 3 4 2 fig. Let h be a hopf algebra that is a cofree graded coalgebra. I know that many examples of hopf algebras that come from combinatorics.
After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys axiomatic characterization of it. It follows from these descriptions that these associated graded hopf algebras are commutative. Geometric combinatorial hopf algebras and modules 1. Trees, functional equations, and combinatorial hopf algebras. Coalgebras, hopf algebras and combinatorics tu wien. Commutative combinatorial hopf algebras springerlink. Some hopf algebras encode relevant combinatorial information and are quite. This tutorial is intended to give an accessible introduction to hopf algebras. 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.
The duality between qsym and nsym is shown in the main diagonal of this diagram. Dual of a hopf algebra hopf algebras and combinatorics. 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. Introduction in 2008 several researchers made exciting advances in the application of algebra and combinatorics to particle physics. This is an expanded version of the comment i made before. These hopf algebras can be thought of as ktheoretic analogues of the by now classical square of hopf algebras consisting of symmetric functions, quasisymmetric functions, noncommutative symmetric functions and the.
Combinatorial hopf algebra may refer either to a hopf algebra coming from combinatorics and, in particular, to the study of various rings of. Some hopf algebras encode relevant combinatorial information and are quite common and useful in algebraic combinatorics. In a combinatorial hopf algebra, the product and coproduct encode respectively how to combine. 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. Combinatorial hopf algebras and khomology of grassmanians. Coalgebras, hopf algebras and combinatorics betreuer.
In conclusion, we just showed that if gis a topological group, and rg is a subalgebra of continuous functions on gsuch that rg. The crucial mathematical ingredients included hopf algebras and subalgebras, in regard. 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 over the. In the 1960s, study of hopf algebras increased, perhaps spurred by the. In a combinatorial hopf algebra, the product and coproduct encode respectively how to combine and split combinatorial objects. Motivated by work of buch on setvalued tableaux in relation to the ktheory of the grassmannian, we study six combinatorial hopf algebras. The first algebra is sym 2, the algebra of bisymmetric functions or symmetric functions of level 2. Combinatorial hopf algebras, noncommutative halllittlewood functions, and permutation tableaux the harvard community has made this article openly available. Functional equations associated with some generalizations of the hook length formula 4. Hopf algebras in general and in combinatorial physics. Combinatorial hopf algebras, noncommutative halllittlewood. Much of the subject of combinatorial hopf algebras is fairly. We introduce a new family of noncommutative analogs of the hall.
Combinatorial hopf algebras and generalized dehnsommerville relations marcelo aguiar, nantel bergeron, and frank sottile abstract. We conjecture it is cofree if \x1\ or x is not a root of unity. We introduce a new family of noncommutative analogs of the halllittlewood symmetric functions. A combinatorial hopf algebra is a graded connected hopf algebra over a. Combinatorial hopf algebras of simplicial complexes. Hopf algebras are spaces with a notion of multiplication and comultiplication, which satisfy certain compatibility relations. 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. The hopf algebra of permutations relates the rings of symmetric functions, quasisymmetric functions, and noncommutative symmetric functions, denoted sym, qsym, and nsym respectively, as depicted the following commutative diagram. Postnikovs identity and eisensteins exponential series. Hopf algebras are a relatively new concept in algebra.
Hopf algebras and the combinatorics of connected graphs in quantum field theory. We use the version with a parameter x in the base field. It concerns solely the application of hopf algebras to renormalization in quantum field theory and the combinatorics of feynman diagrams. Cartier ca describing hopf algebras in topology and group theory and applications in combinatorics. Cocommutative hopf algebras of permutations and trees.
Combinatorics of npoint functions via hopf algebra in quantum field theory. A quick introduction to combinatorial hopf algebras. The \1\ mathbbe\ transform in combinatorial hopf algebras journal of algebraic combinatorics, mar 2011 florent hivert, jeangabriel luque, jeanchristophe novelli, jeanyves thibon. 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. Heinz hopf in 1941 in the field of algebraic topology. Renormalization hopf algebras and combinatorial groups.
The paradigm here is the littlewoodrichardson rule, which becomes. You must write your solutions independently and in. Only in case you are interested in the third notion of dual i dont think it has much uses in combinatorics. More recently hopf algebras appeared also in the study of renormalization of qft, controlling its combinatorics. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Hopf algebras are a relatively new concept in algebra, first encountered by their namesake. A graded, connected hopf algebra is a graded vector space h l 1 n0 h n equipped with two linear maps. On the combinatorics of the hopf algebra of dissection.
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