Combinatorial hopf algebras and generalized dehnsommerville relations marcelo aguiar, nantel bergeron, and frank sottile abstract. Section 3 gives our basic theorems, generalizing the two examples to polynomial hopf algebras and cocommutative, free associative hopf algebras. A quick introduction to combinatorial hopf algebras. Supercharacters, symmetric functions in noncommuting. Heinz hopf in 1941 in the field of algebraic topology. Coalgebras, hopf algebras and combinatorics tu wien. The roles in has of the associahedra and permutahedra as delineating the combinatorics of compositional and. The electrical selfinteraction results in a net force opposing the acceleration, proportional approximately to the acceleration. 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.
After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys axiomatic. They can loosely be thought of as generalizations of groups. 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. In a combinatorial hopf algebra, the product and coproduct encode respectively how to combine. In the late 1970s rota introduced hopf algebras into combinatorics and there. Im interested in understanding the role that noncrossing partitions play in hopf algebras monoids has as the components of the power series of the compositional inverse of formal power series or analytic functions at the origin, cf. An introduction to quasisymmetric schur functions hopf. Physics, combinatorics and hopf algebras 33 its interaction with the water. 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.
Even and odd subalgebras of a combinatorial hopf algebra 18 generalized dehnsommerville relations 21 6. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including. Naval academy mitcombinatoricsseminar april2011 michaele. Hopf algebras in combinatorics math user home pages. If t t0, we can get from t0to tby removing a terminal edge, and fromtto t0by adding an. Although in most familiar examples of hopf algebras the antipode has order 1 or 2, examples are known of infinite dimensional hopf algebras in which the antipode has infinite order or arbitrary. The first goal is to provide a reference text for the basic theory of hopf algebras, in particular the hopf algebras. Hopf algebras in general and in combinatorial physics. Are there examples of open problems in combinatorics which were finally solved using techniques from hopf algebras. Hopf algebras in combinatorics mathematics stack exchange. Commutative combinatorial hopf algebras springerlink. In conclusion, we just showed that if gis a topological group, and rg is a subalgebra of continuous functions on gsuch that rg.
Hopf algebras, and led to a vast generalization of the gennocchi numbers. Bialgebras and hopf algebras gives rise to an exact sequence 0. Renormalization hopf algebras and combinatorial groups. It follows from these descriptions that these associated graded hopf algebras are commutative. Hopf algebras emerged in combinatorics because the product and the coproduct can capture the actions of composing and decomposing combinatorial objects respectively. Chases of congeries of exact sequences give the following comparison assertion. Hopf algebras have important connections to quantum theory, lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics.
It is much more developed now than it was 25 years ago when i rst learned the subject with g. Nantelbergeron combinatorial hopf algebras ecco12,bogota multiplying the comultiplied tensors involves a twisting operation. 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. A kalgebra is a kvector space with two linear maps m. Hopf algebras of combinatorial structures 1 william r. Combinatorial hopf algebras and khomology of grassmanians. We try to compute a function on many variables, but in this case we can formulate the algorithm on a purely combinatorial level. In this paper we give a new generalization of halllittlewood functions, starting from tevlins bases. 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. This is an index for federico ardila s lectures on hopf algebras available on youtube.
We use the version with a parameter x in the base field. 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. We conjecture it is cofree if \x1\ or x is not a root of unity. Combinatorial hopf algebras from pros jeanpaulbultelandsamuelegiraudo abstract. The image of a hopf monoid under a bistrong monoidal functor is again a hopf monoid, in such a way that the antipode of the latter is the image of the antipode of the former proposition 3. This tutorial is intended to give an accessible introduction to hopf algebras. The \1\ mathbbe\ transform in combinatorial hopf algebras journal of algebraic combinatorics, mar 2011 florent hivert, jeangabriel luque, jeanchristophe novelli. I know that many examples of hopf algebras that come from combinatorics. Functional equations associated with some generalizations of the hook length formula 4. Cozinbiel hopf algebras in combinatorics phd thesis forest fisher. Some hopf algebras encode relevant combinatorial information and are quite.
For the basic intuition for hopf algebras in combinatorics see the chapter 5 of s. Research plan algebraic combinatorics in patterns and. Hopf algebras of combinatorial structures 1 citeseerx. An introduction to quasisymmetric schur functions is aimed at researchers and graduate students in algebraic combinatorics. The first algebra is sym 2, the algebra of bisymmetric functions or symmetric functions of level 2. The study of hopf algebras is a beautiful subject which is just coming into its prime in my opinion. Motivated by work of buch on setvalued tableaux in relation to the ktheory of the grassmannian, we study six combinatorial hopf algebras. Let h be a hopf algebra that is a cofree graded coalgebra. A quick introduction to combinatorial hopf algebras amy pangs.
Cocommutative hopf algebras of permutations and trees. Marcelo aguiar, swapneel mahajan, monoidal functors, species and hopf algebras, with forewords by kenneth brown, stephen chase and andre joyal. Coalgebras andor hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures. You are encouraged to work together on the homework, but you must state who you worked with in each problem. Nowadays the major book on combinatorial hopf algebras is. In the 1960s, study of hopf algebras increased, perhaps spurred by the. 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. At the heart of this effort lie hopf algebras and hopf monoids, objects in the interface of category theory and combinatorics. Even and odd subalgebras of qsym 23 canonical projections onto the odd and even subalgebras 26 7. Another reason is that combinatorial objects permutations, trees, graphs, posets, tableaux and so on have natural gradings which makes it easier to introduce hopf algebras on.
Postnikovs identity and eisensteins exponential series. Kreimer 7, 8, 9, as a powerful algebraic tool unveiling the combinatorial structure of renormalization. A primer of hopf algebras 3 basis, and the multiplication in gis extended to kgby linearity. These respectively encode associativity and the unit element. Many other concepts regarding algebras have their dual in coalgebras. 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.
Pdf a very basic introduction to hopf algebras researchgate. However, that reference like many other introductions to hopf algebras concentrates on nitedimensional hopf algebras, which are useful in representation theory as generalisations of group algebras. We will study hopf algebras, with an emphasis on their connections with combinatorics. Hopf algebras, combinatorics, renormalization, noncommutative geometry 1. After introducing coalgebras, bialgebras and hopf algebras in general, we study the hopf algebra of symmetric functions, including zelevinskys. Hopf algebras are a relatively new concept in algebra, first encountered by their namesake. The terminal object in the category of combinatorial hopf algebras is the algebra qsym of quasisymmetric functions, equipped with a canonical character. A combinatorial hopf algebra is a graded connected hopf algebra over. Much of the subject of combinatorial hopf algebras is fairly. Hopf algebras are a relatively new concept in algebra. This theory provides anatural framework for combinatorial invariants. Combinatorial physics or physical combinatorics is the area of interaction between physics and combinatorics overview combinatorial physics is an emerging area which unites combinatorial and discrete mathematical techniques applied to theoretical physics, especially quantum theory.
The duality between qsym and nsym is shown in the main diagonal of this diagram. 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. Along with the study of combinatorial hopf algebras, an area of algebra currently experiencing extraordinary growth is the study of cluster algebras and their combinatorics. Apr 09, 2012 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. Combinatorial results can then be seen as particular cases of general algebraic properties.
Hopf algebras of graphs have been introduced and studied by a. This universe of hopf algebras of trees is summarized below. Dual of a hopf algebra hopf algebras and combinatorics. Hopf algebras are spaces with a notion of multiplication and comultiplication, which satisfy certain compatibility relations. Pdf commutative combinatorial hopf algebras jeanyves. If t t0, we can get from t0to tby removing a terminal edge, and fromtto t0by adding an edge and accompanying terminal vertex. Its characteristic is the smallest nonzero number p such that, for any a. Only in case you are interested in the third notion of dual i dont think it has much uses in combinatorics. Again, this separates the analytics from the combinatorics.
Hopf algebras download ebook pdf, epub, tuebl, mobi. A charge qis added to the conductor, which is now subjected to acceleration. Combinatorics of rooted trees and hopf algebras 3797 be obtained by removing from t0a single terminal vertex and the edge into it. But im interested in knowing how hopf algebras are applied in solving combinatorial problem. 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.
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. The connections between the two areas are extensive, as illustrated by the work of reading on cambrian lattices and their congruences in the last three years 39, 37. A combinatorial noncommutative hopf algebra of graphs. Cozinbiel hopf algebras in combinatorics request pdf. 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. You must write your solutions independently and in your.
Combinatorial hopf algebras on generating trees and. These behave very di erently from the in nitedimensional hopf algebras that one uses in combinatorics. Contents i basic hopf algebra theory 3 1 why hopf algebras. Cozinbiel hopf algebras in combinatorics phd thesis. These hopf algebras tend to have bases naturally parametrized by combinatorial objects partitions, compositions, permutations, tableaux, graphs, trees, posets, polytopes, etc. In a combinatorial hopf algebra, the product and coproduct encode respectively how to combine and split combinatorial objects. A starting point is provided by the following result. If a is a cocommutative hopf algebra over k algebraically closed. Trees, functional equations, and combinatorial hopf algebras. Like hopf algebras, weak hopf algebras form a selfdual class of algebras. Weak hopf algebras, or quantum groupoids, are generalizations of hopf algebras. We introduce a general construction that takes as input a.
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