Hopf algebras and polynomial invariants of combinatorial structures.
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Hopf algebras and polynomial invariants of combinatorial structures. by Jeffrey Francis Green

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Published by University of Manchester in Manchester .
Written in English


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Edition Notes

Thesis (Ph.D.), - University of Manchester, Department of Mathematics.

ContributionsUniversity of Manchester. Department of Mathematics.
The Physical Object
Pagination134p.
Number of Pages134
ID Numbers
Open LibraryOL16571354M

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  These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive self-adjoint Hopf Cited by: category of combinatorial Hopf algebras with a multiplicative linear functional called a character. Their explicit map from any such algebra to QSymuni es many ways of assigning polynomial invariants to combinatorial objects, such as the chromatic polynomial of graphs and Ehrenboug’s quasisymmetric function of a ranked poset. Hopf Algebras in General and in Combinatorial Physics: a practical introduction by G.H.E. Duchamp, et al.. Publisher: arXiv Number of pages: Description: This tutorial is intended to give an accessible introduction to Hopf algebras. PDF | Contents Statement of Qualifications and Research 1 Background Binomial and Divided Power Hopf Algebras | Find, read and.

  A Combinatorial Hopf Algebra, in the sense of, is a pair (H, ζ) where H is a graded connected Hopf algebra and ζ: k → H is a character (a multiplicative linear map). The most central combinatorial Hopf algebra in this sense is the pair (Q S y m, φ 1) where Q S y m = ⨁ n ≥ 0 Q S y m n and Q S y m n is the k-span of {M α: α ⊨ n}. H!H0that is a morphism of Hopf algebras (i.e., a linear transformation that preserves the operations of a bialgebra) such that = 0. The binomial Hopf algebra. The binomial Hopf algebra is the ring of polynomials F[k] in one variable k, with the usual multiplicative structure; comultiplication de ned by (f(k)) = f(k 1 + 1 k) and (1) = 1 1;. for each degree and show that, after inverting ˘1, it becomes polynomial on a natural set of generators. Finally we note that, without inverting˘1, A ˜is far from being polynomial. 1. Introduction. The mod 2 dual Steenrod algebra, A, being a connected commutative Hopf algebra, has a canonical conjugation or anti-automorphism ˜. Thismapwas rst. Polynomial invariants of a semisimple and cosemisimple Hopf algebra based on braiding structures Michihisa Wakui (Kansai Univ.) August 31st, in La Falda, C´ordoba Colloquium of Hopf Algebras, Quantum Groups and Tensor Categories. Contents — a new family of monoidal Morita invariants of a.

A combinatorial Hopf algebra is a pair (H,ζ) where H = L n≥0 Hnis a graded con-nected Hopf algebra over a field kand ζ: H → kis a character (multiplicative linear functional), called its zeta function. A morphism α: (H′,ζ′) → (H,ζ) of combinato-rial Hopf algebras is a morphism of graded Hopf algebras such that ζ′ = ζ α. The.   The new results here are found by showing that the Martin polynomial is a translation of a universal skein-type graph polynomialP(G) which is a Hopf map, and then using the recursion and induction which naturally arise from the Hopf algebra structure to extend known properties. functions), the rigidity in the structure of a Hopf algebra can lead to enlightening proofs. One of the most elementary interesting examples of a combinatorial Hopf algebra is that of the symmetric functions. We will devote all of Chapter2to studying it, deviating from the usual treatments (such as in Stanley [, Ch. 7], Sagan [] and. Try the new Google Books. Check out the new look and enjoy easier access to your favorite features and the structure of cosemisimple Hopf algebras. extension group algebra grouplike element h e H H-comodule H-module algebra H-stable Hence Hopf Galois extensions Hopf modules Hopf subalgebra ideal invariants irreducible isomorphism left H.