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Quantum groups at roots of unity and modularity
 J. Knot Theory Ramifications
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Lagrange’s theorem for Hopf monoids in species
, 2012
"... Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras, ..."
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Abstract. Following Radford’s proof of Lagrange’s theorem for pointed Hopf algebras,
GENERALIZED HOPF MODULES FOR BIMONADS
"... Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodulemonad ..."
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Abstract. Bruguières, Lack and Virelizier have recently obtained a vast generalization of Sweedler’s Fundamental Theorem of Hopf modules, in which the role of the Hopf algebra is played by a bimonad. We present an extension of this result which involves, in addition to the bimonad, a comodulemonad and a algebracomonoid over it. As an application we obtain a generalization of another classical theorem from the Hopf algebra literature, due to Schneider, which itself is an extension of Sweedler’s result (to the setting of Hopf Galois extensions).
HOPF MONOIDS IN THE CATEGORY OF SPECIES
"... Abstract. A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a selfcontained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We stu ..."
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Abstract. A Hopf monoid (in Joyal’s category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a selfcontained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and Dynkin idempotents and discuss the PoincaréBirkhoffWitt and CartierMilnorMoore theorems for Hopf monoids.
SKEW PIERI RULES FOR HALL–LITTLEWOOD FUNCTIONS
"... Abstract. We produce skew Pieri Rules for Hall–Littlewood functions in the spirit of Assaf and McNamara [3]. The first two were conjectured by the first author [6]. The key ingredients in the proofs are a qbinomial identity for skew partitions and a Hopf algebraic identity that expands products of ..."
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Abstract. We produce skew Pieri Rules for Hall–Littlewood functions in the spirit of Assaf and McNamara [3]. The first two were conjectured by the first author [6]. The key ingredients in the proofs are a qbinomial identity for skew partitions and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and the antipode. Let Λ[t] denote the ring of symmetric functions over Q(t), and let {sλ} and {Pλ(t)} denote its bases of Schur functions and Hall–Littlewood functions, respectively, indexed by partitions λ. The Schur functions (which are actually defined over Z) lead a rich life, making appearances in combinatorics, representation theory, and Schubert calculus, among other places. See [5, 9] for details. The Hall–Littlewood functions are nearly as ubiquitous (having as a salient feature that Pλ(t) → sλ under the specialization t → 0). See [8] and the references therein for their place in the literature. A classical problem is to determine cancellationfree formulas for multiplication in these bases, sλ sµ = ∑ c ν λ,µ sν and Pλ Pµ = ∑ f ν λ,µ (t) Pν. ν The first problem was only given a complete solution in the latter half of the 20th century, while the second problem remains open. Special cases of the problem, known as Pieri rules, have been understood for quite a bit longer. The Pieri rules for Schur functions [9, Ch. I, (5.16) and (5.17)] take the form sλ s1r = sλ er = ∑ sλ +, (1) with the sum over partitions λ + for which λ + /λ is a vertical strip of size r, and λ + ν sλ sr = ∑ sλ +, (2) with the sum over partitions λ + for which λ + /λ is a horizontal strip of size r. (See Section 1 for the definitions of vertical and horizontal strips.) The Pieri rules for Hall–Littlewood functions [9, Ch. III, (3.2) and (5.7)] state that Pλ P1r = Pλ er = λ +
Freeness of Hopf Algebras
, 2006
"... In 1975 Kaplansky discussed “Ten conjectures on Hopf algebras ” during a lecture at the University of Chicago. The first of these conjectures concerned freeness of a Hopf algebra as a module over a subHopfalgebra. Specifically he conjectured that “a Hopf algebra is free as a module over any subHopfa ..."
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In 1975 Kaplansky discussed “Ten conjectures on Hopf algebras ” during a lecture at the University of Chicago. The first of these conjectures concerned freeness of a Hopf algebra as a module over a subHopfalgebra. Specifically he conjectured that “a Hopf algebra is free as a module over any subHopfalgebra”. Although this was quickly shown to be false in the infinite dimensional case, the finite dimensional case turned out to be true, and was proven 14 years later by Nichols and Zoeller. This result is the heart of this paper. The NicholsZoeller freeness theorem states that a finite dimensional Hopf algebra is free as a module over any subHopfalgebra. We will prove this theorem, as well as the first significant generalization of this theorem, which was proven three years later. This generalization says that if the Hopf algebra is infinite dimensional, then the Hopf algebra is still free if the subHopfalgebra is finite dimensional and semisimple. We will also look at several other significant generalizations that have since been proven.
RMatrices, YetterDrinfel ′ d Modules and YangBaxter Equation
, 2013
"... In the first part we recall two famous sources of solutions to the YangBaxter equation – Rmatrices and YetterDrinfel ′ d (=YD) modules – and an interpretation of the former as a particular case of the latter. We showthat thisresultholds trueinthe moregeneral case ofweak Rmatrices, introduced her ..."
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In the first part we recall two famous sources of solutions to the YangBaxter equation – Rmatrices and YetterDrinfel ′ d (=YD) modules – and an interpretation of the former as a particular case of the latter. We showthat thisresultholds trueinthe moregeneral case ofweak Rmatrices, introduced here. In the second part we continue exploring the “braided ” aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD modules. The functoriality and several generalizations of this construction are studies using the original machinery of YD systems. As consequences, we get a conceptual interpretation of the tensor product structures for YD modules, and a generalization of the deformation cohomology of YD modules. The latter homology theory is thus included into the unifying framework of braided homologies, which contains among others Hochschild,
Quantum groups at roots . . .
, 2003
"... For each compact, simple, simplyconnected Lie group and each integer level we construct a modular tensor category from a quotient of a certain subcategory of the category of representations of the corresponding quantum group. We determine when at fractional levels the corresponding category is mod ..."
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For each compact, simple, simplyconnected Lie group and each integer level we construct a modular tensor category from a quotient of a certain subcategory of the category of representations of the corresponding quantum group. We determine when at fractional levels the corresponding category is modular. We also give a quantum version of the Racah formula for the decomposition of the tensor product. This work relies on developing the basic representation theory of quantum groups at roots of unity, including HarishChandra’s Theorem. It generalizes previous work which applied only to fractional levels or only to the projective form
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, 2013
"... une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles dirigée par Marc ROSSO Soutenue le 13 décembre 2012 devant le jury composé de: ..."
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une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles dirigée par Marc ROSSO Soutenue le 13 décembre 2012 devant le jury composé de:
Cofree compositions of coalgebras
"... Abstract. We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a onesided Hopf algebra. These c ..."
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Abstract. We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a onesided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes. Résumé.