Programa - Quantum 2012

Programa - Quantum 2012
Viernes 30/03
Sábado 31/03
Lunes 02/04
9:50
Inauguración
10.00 10.50
Flavio Coelho
Agustín García Iglesias
Cristian Vay
10.50 11.10
Coffee Break
Coffee Break
Coffee Break
11.10 12.00
Julia Plavnik
Gastón García
Irina Kashuba
12.00 12.10
Coffee Break
Coffee Break
Coffee Break
12.10 13.00
Barbara Pogorelsky / Pablo Román
Slava Futorny
Ivan Shestakov
13.00 15.30
Lunch
Lunch
Lunch
15.30 16.20
María Ronco
Iván Angiono
16.20 16.40
Coffee Break
Coffee Break
16.40 17.30
Steen Ryom-Hansen
Nicolás Andruskiewitsch
RESÚMENES
Andruskiewitsch: From Hopf algebras to tensor categories: Towards fusion categories associated to finite dimensional Nichols algebras of diagonal
type. Part II
Angiono: From Hopf algebras to tensor categories: Towards fusion categories associated to finite dimensional Nichols algebras of diagonal type.
Part I
Coelho: Homological dimensions in representation theory of algebras
Futorny: Gelfand-Tsetlin representations for quantum gl(n)
García: Deformation by cocycles of pointed Hopf algebras over non-abelian groups
Abstract: we will introduce a method to construct multiplicative 2-cocycles for bosonizations
of Nichols algebras over Hopf algebras with bijective
antipode. These cocycles arise as liftings of invariant e-biderivations defined on the Nichols algebras. Using this construction, we will show that all known
finite dimensional pointed Hopf algebras over the dihedral groups Dm with m=4t > 11, over the symmetric group S3 and some families over S4 are cocycle
deformations of bosonizations of Nichols algebras, by constructing explicitly the 2-cocycles. This talk will be about a joint work with M. Mastnak, St. Mary's
University, Halifax, Canada. Preprint: arXiv:1203.0957v1.
Garcia Iglesias: Liftings of Nichols algebras over some affine racks
Kashuba: Induced modules for Affine Kac-Moody algebras
Abstract. We will discuss induced modules with arbitrary multiplicities over affine Lie algebras. In particular, we consider generalized loop modules induced
from some parabolic subalgebras. Our main result shows that the induction functor preserves irreducible modules when the central charge is non-zero.
These results generalize similar reductions in particular cases previously considered by Futorny,Koenig and Mazorchuk, and also by Dimitrov, Futorny and
Penkov.
Plavnik: On fusion categories with few irreducible degrees
Abstract: In this talk we shall consider the general problem of understanding the structure of a fusion category C after the knowledge of the set cd(C) of
Frobenius- Perron dimensions of its simple objects.
For a finite group G, the knowledge of the set cd(G) = cd(kG) gives in some cases substantial information about the structure of G. It is known, for instance,
that if |cd(G)| is at most 3, then G is solvable.
In particular, we consider the case where cd(C) = {1, p}, with p a prime number. We shall show various structural results regarding nilpotency and
solvability, in the sense introduced by Etingof, Gelaki, Nikshych and Ostrik.
The talk is based on joint work with S. Natale (preprint: arXiv:1103.2340v2).
Pogorelsky: Right coideal subalgebras of quantum groups of type G2
Román: Matrix-valued orthogonal polynomials related to spherical functions
Ronco: Tubings sobre grafos finitos simples y álgebras de Hopf
Abstract: En [1], M. Carr and S. Devadoss introducen la noción de tubing sobre un grafo finito y simple G, y asocian a cada grafo un polítopo cuyas caras
corresponden a los tubings del grafo. Cuando G es el grafo lineal Ln, con n vértices, el polítopo KLn es el polítopo de Stasheff o asociaedro, cuando se
considera el grafo completo C_n se obtiene el permutoedro, y el polítopo asociado al grafo con n vértices y sin aristas es el n-1 simple estándar Δn-1. Por otra
parte, es bien conocido que existen sobre los espacios vectoriales generados por el conjunto de caras del permutoedro y del asociaedro, sendas estructuras
de álgebras de Hopf. Estas estructuras permiten interpretar esos espacios como objetos libres para teorías algebraicas más finas (álgebras dendriformes y
sistemas pre-Lie en el caso del asociaedro, y álgebras shuffle en el caso del permutoedro), que permiten además reconstruir el borde de estos polítopos en
términos de derivaciones para estas estructuras.
Nuestro objetivo es:
1. Describir estructuras de álgebras de Hopf en el espacio generado por el conjuntos de tubings sobre todos los grafos simples finitos conexos,
a partir de una operación de conexión de grafos. Estudiar la restricci’on de esta estructuras a ciertas subfamilias de grafos; que en los casos
particulares de las familias de grafos completos y de grafos lineales recuperar las álgebras de Hopf ya conocidas.
2. Describir un orden parcial en el conjunto de tubings de un grafo simple, que generalice el orden de Tamari en el conjunto de vértices del
asociaedro. Para algunas familias de grafos, este orden induce el producto asociativo en el espacio vectorial generado por el conjunto de
tubings de todos los grafos de la familia, mencionado en el punto anterior.
3. Estudiar algunos ejemplos no conocidos de álgebras de Hopf en este contexto, relacionadas con el multipliedro y el stelloedro.
Si es posible, trataremos de introducir la noción de álgebra shuffle como monad en la categoría de funtores sobre los espacios vectoriales graduados
(ver [1]); y su posible extensión a otros tipos de graph asociaedra.
Bibliografía
[1] M. Carr , S. Devadoss: Coxeter complexes and graph associahedra, Topology and its Applications, 153 (1-2) 2155--2168, 2006.
[2] S. Forcey, D. Springfield: Geometric combinatorial algebras: cyclohedron and simplex, J. Algebraic Comb., 32, 597--627, 2010.
[3] J.-L. Loday, M. Ronco: Hopf algebra of the planar binary trees, Adv. Math.,139 (2), 293--309, 1998.
[4] J.-L. Loday , M. Ronco: Order structure and the algebra of permutations and of planar binary trees, J. of Algebraic Combinatorics, 15 N* 3, 253--270, 2002.
[5] J.-L. Loday , M. Ronco: Permutads, Preprint 2011, arXiv.
[6] C. Malvenuto , C. Reutenauer: Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177 (3), 967--982, 1995.
Ryom-Hansen: Gradings on the blob algebra
Abstract: This is an account of joint work with David Plaza. We show that the blob algebra is a graded cellular algebra, thus explaining the fact that the
decomposition numbers are given by polynomials.
Shestakov: The universal enveloping algebra of the traceless octonions is rigid
Abstract: We prove that, over fields of characteristic zero, any bialgebra deformation of the universal enveloping algebra of the traceless octonions that
satisfies the dual of the left and right Moufang identities must be coassociative and cocommutative.
It is a joint work with Prof. J.M.Pérez Izquierdo, Universidad de La Rioja, Logroño, Spain.
Vay: Representations of copointed Hopf algebras over S3