Resúmenes y presentaciones de las conferencias

• On Nichols algebras with standard braiding  
Iván Angiono
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The class of standard braided vector spaces, introduced by Andruskiewitsch and the author to understand the proof of a theorem of Heckenberger, is slightly more general than the class of braided vector spaces of Cartan type. We classify standard braided vector spaces with finite-dimensional Nichols algebra. For any such braided vector space, we give a PBW-basis, a closed formula of the dimension and a presentation by generators and relations of the associated Nichols algebra.

• Semisimple Hopf Algebras  
V.A. Artamonov
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Let H be a finiite dimensional semisimple Hopf algebra having as an algebra up to an equivalence only irreducible representation of dimension n > 1. Suppose that the the basic field is algebraically closed and its characteristic does not divide dimH.

It is shown that the order of the group G of group-like elements in H^¤ is a divisor of n² [1]. If the order of G is equal to n² then G is a noncyclic abelian group [4, 3]. Under this assumption there is given a classification of H up to an isomorphism in terms of projective irreducible representations of G of degree n [3]. If n > 2 then each algebra H is not self-dual [3].

References
[1] V.A. Artamonov, On semisimple finite dimensional Hopf algebras, Mat. Sbornik, { 198(2007), N 9, 3-28.
[2] V.A. Artamonov, I.A. Chubarov, Dual algebras of some semisimple finite dimensional Hopf algebras, Modules and Comodules Trends in Mathematics,65-85, 2008 BirkhÄauser Verlag Basel/Switzerland.
[3] V.A. Artamonov, I.A. Chubarov, Properties of some semisimple Hopf algebras, Contemp. Math., Proc. of the international conference dedicated to 60th anniversary of I.P.Shestakov.
[4] Tambara D., Yamagami S., Tensor categories with fusion rules of self-duality for finite abelian groups, J.Algebra 209(1998), 692-707. Department of Algebra, Faculty of Mechanics and Mathematics, Moscow State University

E-mail address: artamon@mech.math.msu.su

• Quantum Invariants, Hopf monads and the Double Construction  
Alain Bruguières
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Whe introduce the notion of Hopf monad, which generalizes the notion of Hopf algebra to a non-braided setting. Several aspects of the theory of Hopf algebras extend to Hopf monads, including Drinfeld's Double construction.

Our motivation for introducing Hopf monads is that they 'algebraize' the categorical center construction. More precisely, the categorical center of a (bounded, non-braided) tensor category C is the category of modules of a Hopf monad called the centralizer of C. Note that if C is braided, the categorical center of C is the category of modules on the coend of C, which is a Hopf algebra in C, but if C is not braided, Hopf algebras just don't make sense. The point of considering non-braided categories is that certain quantum 3-manifold invariants (of Turaev-Viro type) are naturally associated with such categories.

Using the notion of double of a Hopf monad, we generalize a theorem due to Mueger which asserts that the categorical center of a (non-braided) 'spherical' category is a modular category (hence gives rise to Reshetikhin-Turaev invariants). As a result we obtain new invariants of Reshetikhin-Turaev type associated with the center of a fusion category over an arbitrary ring. This is a joint work with Alexis Virelizier.

• New techniques for pointed Hopf algebras and applications  
Fernando Fantino
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Let G be a finite group and let   ^CG _CGYD be the category of Yetter-Drinfeld modules over CG. The most delicate of the questions raised by the Lifting Method for the classification of finite-dimensional complex pointed Hopf algebras H with G(H) isomorphic to G, is the determination of all V in ^CG _CGYD such that the Nichols algebra B(V ) is finite-dimensional. An approach for that is to discard irreducible Yetter-Drinfeld modules over CG with infinitedimensional Nichols algebra by subrack methods.

We will present new methods by abelian and non-abelian subracks, illustrating each of them with several examples. Finally, we will show some results for pointed Hopf algebras with G(H) belonging to the following families of non-abelian groups: symmetric, alternating, dihedral or Mathieu simple groups. These results are part of joint works with N. Andruskiewitsch.

• Products of Representations of the Symmetric Group and Non-Commutative Versions  
Walter Ferrer
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(joint work with M. Aguiar and Walter Moreira, Texas A & M)

We construct a new operation among representations of the symmetric group that interpolates between the classical internal and external products, which are defined in terms of tensor product and induction of representations.

Following Malvenuto and Reutenauer, we pass from symmetric functions to non-commutative symmetric functions and from there to the algebra of permutations in order to relate the internal and external products to the composition and convolution of linear endomorphisms of the tensor algebra.

The new product we construct corresponds to the Heisenberg product of endomorphisms of the tensor algebra. For symmetric functions, the Heisenberg product is given by a construction which combines induction and restriction of representations.

For non-commutative symmetric functions, the structure constants of the Heisenberg product are given by an explicit combinatorial rule which extends a well-known result of Garsia, Remmel, Reutenauer, and Solomon for the descent algebra.

• Normal Hopf subalgebras in cocycle deformations of finite groups  
César Galindo
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This is a report on a joint work with Sonia Natale. Let G be a finite group and let π: G --> G' be a surjective group homomorphism. Consider the cocycle deformation L = H^σ of the Hopf algebra H = k^G of k-valued linear functions on G, with respect to some convolution invertible 2-cocycle σ. The (normal) Hopf subalgebra k^G' of k^G corresponds to a Hopf subalgebra L' of L. Our main result is an explicit necessary and sufficient condition for the normality of L' in L.

• On a factorization of graded Hopf algebras using Lyndon words 
Matías Graña
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We find a generalization of the restricted PBW basis for pointed Hopf algebras over abelian groups constructed by Kharchenko. We obtain a factorization of the Hilbert series for a wide class of graded Hopf algebras. These factors are parametrized by Lyndon words, and they are the Hilbert series of certain graded Hopf algebras. This is a joint work with I. Heckenberger.

• Quadratic algebras extending the quantum coordinate algebra of a semisimple group  
Rodrigo Iglesias
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Let G be a connected complex semisimple Lie group, V a finite dimensional representation of G, and F(G,V) the quantized algebra of functions on G generated by the matrix coefficients of V. Finding a complete set of relations among these coefficients seems to be, in general, a hard computational problem. So we consider the algebra Q(G,V) defined only by the quadratic relations. We show that the quadratic algebra Q(G,V) is a good approximation of F(G,V) in the sense that, essentially, it is a finite extension of F(G,V).

• Clasificacion de las bialgebras de Lie reales de dimensiones 3 y 4  
Patricia Jancsa
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Este es un trabajo conjunto con Laura Barberis y Marco Farinati. Se expondran los resultados de clasificacion de las bialgebras de Lie reales de dimension 3 y el estado de avance de la clasificacion de las de dimension 4. Se expondran resultados generales que resultan utiles en la tarea de clasificacion.

• About the structure of slim double lie groupoids  
Jesús Ochoa
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In arXiv:math/0602497v2[math.CT] Andruskiewitsch and Natale has given a general description of a discrete double groupoid, that satisfy the filling condition, in terms of groupoid factorizations and groupoid 2-cocycles.
In this talk we will describe some results obtained in the project to carry the above description to the Lie groupoid setting. In fact, we will show that if in a diagram of groupoids (D,j,i) the maps i and j are transversal, then they give rise to a slim double Lie groupoid, and on the other hand, we will give an alternative presentation of the diagonal groupoid associated to a general slim double lie groupoid (B;V, H; P) in terms of an action of a the core groupoid E(B) on the manifold given by the fibration of V and H by α_V and β_H, the source and target projection of V and H respectively. These results are part of joint work in progress with N. Andruskiewitsch and A. Tiraboschi.

• Yetter-Drinfel'd modules under twists  
Mariana Pereira

In 2003, David Radford introduced a new method to construct all simple Yetter-Drinfel'd modules for certain finite-dimensional pointed Hopf algebras. These simple modules are in one-to-one correspondence with G(H^*) × G(H) and are realized as vector subspaces of the Hopf algebra H itself. We apply an explicit cocycle twist to these modules and give a precise correspondence between simple Yetter-Drinfel'd modules for H and simple Yetter-Drinfel'd modules for the twist H^σ. This result explains previous computational data obtained when computing all simple modules for the small two-parameter quantum groups u_r,s(sl₃) at roots of unity. This is joint work in progress with Georgia Benkart and Sarah Witherspoon.

• Bialgebras related to S-modules  
Maria Ronco
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An S-module is a family M(n)_{n ≥ 1} such that the symmetric group S_n acts on M(n), for n ≥ 1. The category S-Mod of S-modules is equipped with three different monoidal structures: the Hadamard product _H, the usual graded product , and the external plethysm º. An operad is a monoid in the category (S-Mod, º). Getzler and Jones defined the notion of Hopf operad for operads equipped with an extra structure of comonoid for the Hadamard product, verifying some compatibility relations. Patras and Livernet proved that any Hopf operad gives rise to a bialgebra object in the category (S-Mod, º), also called a twisted bialgebra. There exists a weaker notion, which is obtained by looking at monoids in the category (S-Mod, ) equipped with a graded coproduct, called shuffle bialgebras, some well-knwn combinatorial Hopf algebras like the Malvenuto-Reutenauer bialgebra, the Solomon-Tits algebra and the algebra of parking functions are examples of shuffle bialgebras. There exists also a notion of bialgebra in the category (S-Mod, º). Our purpose is to describe the relationships between these objects.

• Hopf algebras of dimension 16 
Cristian Vay
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We complete the classification of Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero. We show that a non-semisimple Hopf algebra of dimension 16, has either the Chevalley property or its dual is pointed. This work was conducted with Gastón Garcia.

• GAP and Nichols algebras 
Leandro Vendramin
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In this talk we will show how GAP can be used to study Nichols algebras over non-abelian groups. Also, we will give some interesting examples taken from a joint work with Nicolás Andruskiewitsch, Fernando Fantino and Matías Graña in which Nichols algebras over sporadic simple groups are studied.