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247
ON THE VASSILIEV KNOT INVARIANTS
, 1995
"... The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful a ..."
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Cited by 139 (0 self)
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The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful as those polynomials. As invariants of finite type are much easier to define and manipulate than the quantum group invariants, it is likely that in attempting to classify knots, invariants of finite type will play a more fundamental role than the various knot polynomials.
On the nonexistence of elements of Hopf invariant one
 Ann. of Math
, 1960
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Cited by 69 (0 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics.
Combinatorial Hopf algebras and generalized DehnSommerville relations
, 2003
"... A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the u ..."
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Cited by 65 (16 self)
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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the ubiquity of quasisymmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra (H, ζ) possesses two canonical Hopf subalgebras on which the character ζ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized DehnSommerville relations. We show that, for H = QSym, the generalized DehnSommerville relations are the BayerBillera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that QSym is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the MalvenutoReutenauer Hopf algebra of permutations, the
Elliptic spectra, the Witten genus and the theorem of the cube
 Invent. Math
, 1997
"... 2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7 ..."
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Cited by 63 (16 self)
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2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7
HopfAlgebraic Structures of Families of Trees
 J. Algebra
, 1987
"... this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to e#ciently compute certain di#erential operators. Given ..."
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Cited by 63 (17 self)
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this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to e#ciently compute certain di#erential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivations on an algebra, one expects to find a Hopf algebra of some sort. We were pleased to find that not only was there a Hopf algebra associated with these data structures, but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees. Previous work applying Hopf algebras to combinatorial objects (such as [10], [13] or [14]) has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets, rather than on trees themselves. # The first author is a National Science Foundation Postdoctoral Research Fellow
On modules associated to coalgebraGalois extensions
 J. Algebra
, 1999
"... For a given entwining structure involving an algebra A, a coalgebra C, and an entwining map ψ: C ⊗A → A ⊗C, a category of right (A,C,ψ)modules is defined and its structure analysed. It is shown that in the case of the canonical entwining structure of a CGalois extension A of an algebra B this cate ..."
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Cited by 53 (14 self)
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For a given entwining structure involving an algebra A, a coalgebra C, and an entwining map ψ: C ⊗A → A ⊗C, a category of right (A,C,ψ)modules is defined and its structure analysed. It is shown that in the case of the canonical entwining structure of a CGalois extension A of an algebra B this category is equivalent to the category of left Bmodules if and only if A is faithfully flat as a left Bmodule. Left modules E and right modules Ē associated to a CGalois extension A of B are defined. These can be thought of as objects dual to fibre bundles with coalgebra C in the place of a structure group, and a fibre V. Crosssections of such associated modules are defined as module maps E → B or Ē → B. It is shown that they can be identified with suitably equivariant maps from the fibre to A. Also, it is shown that a CGalois extension is cleft if and only if A = B ⊗ C as left Bmodules and right Ccomodules. The relationship between modules E and Ē is studied in the case when V is finitedimensional and in the case when the canonical entwining map is bijective. 1.
Structure of the MalvenutoReutenauer Hopf algebra of permutations
 Adv. Math
"... Abstract. We analyze the structure of the MalvenutoReutenauer Hopf algebra SSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product ..."
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Cited by 51 (15 self)
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Abstract. We analyze the structure of the MalvenutoReutenauer Hopf algebra SSym of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasisymmetric functions. In addition, we describe the structure constants of the multiplication as a certain number of facets of the permutahedron. As a consequence we obtain a new interpretation of the product of monomial quasisymmetric functions in terms of the facial structure of the cube. The Hopf algebra of Malvenuto and Reutenauer has a linear basis indexed by permutations. Our results are obtained from a combinatorial description of the Hopf algebraic structure with respect to a new basis for this algebra, related to the original one via Möbius inversion on the weak order on the symmetric groups. This is in analogy with the relationship between the monomial and fundamental bases of the algebra of quasisymmetric functions. Our results reveal a close relationship between the structure of the MalvenutoReutenauer Hopf algebra and the weak order on the symmetric groups.
Construction of Field Algebras with Quantum Symmetry from Local Observables
, 1996
"... It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reco ..."
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Cited by 49 (8 self)
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It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....
Vassiliev Homotopy String Link Invariants
, 1995
"... . We investigate Vassiliev homotopy invariants of string links, and nd that in this particular case, most of the questions left unanswered in [3] can be answered armatively. In particular, Vassiliev invariants classify string links up to homotopy, and all Vassiliev homotopy string link invariants co ..."
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Cited by 48 (2 self)
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. We investigate Vassiliev homotopy invariants of string links, and nd that in this particular case, most of the questions left unanswered in [3] can be answered armatively. In particular, Vassiliev invariants classify string links up to homotopy, and all Vassiliev homotopy string link invariants come from marked surfaces as in [3], using the same construction that in the case of knots gives the HOMFLY and Kauman polynomials. Alongside, the Milnor invariants of string links are shown to be Vassiliev invariants, and it is reproven, by elementary means, that Vassiliev invariants classify braids. Contents 1. Introduction 1 2. Vassiliev invariants of string links 3 2.1. A brief review of [3] 3 2.2. Vassiliev invariants of string links 4 3. Vassiliev homotopy string link invariants 5 4. Vassiliev invariants classify braids 9 4.1. Braids 9 4.2. Braids with double points 11 5. On the Milnor invariants 13 5.1. Vassiliev invariants classify string links up to homotopy 13 5.2. The Milnor ...
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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