Programa - Quantum 2013

RESÚMENES

Andruskiewitsch:  Álgebras de Hopf punteadas con grupo finito simple de tipo Lie

Resumen: (En colaboración con G. Carnovale y G. García). Se presentarán resultados recientes sobre la determinación de las álgebras de Hopf punteadas H cuyo grupo G(H) es isomorfo a un grupo finito simple de tipo Lie. Se espera que la única salvo isomorfismos sea el álgebra de grupo; se presentará evidencia en esta dirección.

Angiono: Producto tensorial de representaciones de álgebras de Hopf con traza

Resumen: Consideremos el álgebra envolvente cuantizada U_q(g) sobre el cuerpo de números complejos C, asociada a un álgebra de Lie semisimple g y un escalar no nulo q. Es conocido que la categoría de representaciones de dimensión finita de U_q(g) es esencialmente la de U(g), cuando q no es una raíz de la unidad. La situación es completamente diferente cuando q es una raíz de la unidad. Por ejemplo, dicha categoría de representaciones deja de ser semisimple, lo cual a la vez vuelve muy interesante su estudio. Se busca obtener, entre otros puntos, la descomposición del producto tensorial de dos representaciones irreducibles de U_q(g). Este problema fue abordado en [dCPRR], donde los autores se basaron en el hecho que U_q(g) posee una subálgebra de Hopf central Z, tal que U_q(g) es un Z-m\'odulo libre finito.

En esta charla daremos a conocer los resultados principales de [dCPRR]. Los principales puntos a tratar incluyen:

• introducir la definición de un álgebra de Hopf con traza,
• describir las propiedades que adquiere la categoría de representaciones de tales álgebras,
• aplicar esta teoría al caso U_q(g), q una raíz de la unidad, para obtener la descomposición del producto tensorial de una amplia familia de representaciones irreducibles.

Referencias:

[dCPRR] C. de Concini, C. Procesi, N. Reshetikhin, M. Rosso, Hopf algebras with trace and representations. Invent. Math. 161 (2005), 1-44.

Fantino: Método del levante

Resumen: Se expondrá de manera accesible el método del levante para clasificar álgebras de Hopf (de dimensión finita, o de crecimiento finito, o...) cuyo corradical es una subálgebra de Hopf. Se verán las nociones de álgebras y coálgebras filtradas, sus objetos graduados asociados, la filtración corradical y las relaciones con álgebras de Nichols.

Femic: Invertible module categories over the representation category of the Taft algebra

Resumen: We determine the tensor product of exact indecomposable bimodule categories over representation categories of finite Hopf algebras. We discuss the criteria to detect which of these bimodule categories are invertible. We show that the group of biGalois objects for a finite-dimensional Hopf algebra H embeds into the Brauer-Picard group of Rep(H). When H=T_q is the Taft algebra over an algebraically closed field of characteristic zero, where q is an n-th primitive root of unity, we classify all exact indecomposable Rep(H)-bimodule categories and present the advances of the computation of the Brauer-Picard group of Rep(H). This is a joint work with Martin Mombelli.

Ferrer Santos: Almost involutive Hopf algebras

Resumen: An involutive Hopf algebra is a Hopf algebra with the property that its antipode squared, is the identity map. We define an almost involutive Hopf algebra as a Hopf algebra whose antipode squared is the square of a Hopf automorphism. We give examples and basic properties of almost involutive Hopf algebras. In particular we show that compact quantum groups are examples.

Futorny: Representations of Weyl algebras and parabolic induction

Resumen: We will discuss  representations of infinite rank Weyl algebras and corresponding induced representations of Affine Kac-Moody algebras.  The talk is based on 2 recent papers with D.Grantcharov and V.Mazorchuk and with V.Bekkert, G.Benkart and I.Kashuba.

García: Álgebras de Hopf trenzadas, bosonización y álgebras de Hopf con una proyección

Resumen: En esta charla introduciremos la noción de álgebras de Hopf en una categoría trenzada, ilustrando la teoría con  algunos ejemplos específicos de ciertas categorías, como la de módulos de Yetter-Drinfeld sobre un álgebra de Hopf de dimensión finita H. Seguidamente mostraremos cómo obtener un álgebra de Hopf usual a partir de H y un álgebra de Hopf trenzada en esta categoría (bosonización) y finalizaremos con el enunciado y la demostración de un teorema de Radford que caracteriza este tipo de álgebras como álgebras de Hopf con una proyección.

Garcia Iglesias: A new approach to quantum groups

Resumen: Let U_t(g) be the quantum enveloping algebra associated to a Kac-Moody Lie algebra g. We will survey on the recent article [B] in which the author constructs the full quantum group U_t(g) starting from the Hall algebra H(A) of a suitable abelian category A. More explicitly, set q=t^2, let k be the finite field with q elements, and let Q be the quiver associated to the generalized Cartan matrix of g. Then A stands for the category of Z_2-graded complexes in Rep kQ, the category of finite-dimensional representations of the path algebra kQ. Then U_t(g) can be described as a subalgebra of H(A). Furthermore, these two algebras coincide

when the underlying graph of Q is a simply-laced Dynkin diagram. This work extends [G, R], in which the positive part U_t(n^+) of U_t(g) was described as a subalgebra of the Hall algebra of Rep kQ. The talk will be introductory, with special focus on the objects and theories involved in the construction.

References:

[B] Bridgeland, T., Quantum groups via Hall algebras of complexes, Ann. of Math. 177, 739--759 (2013).

[G] Green, J. A., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120, 361--377 (1995).

[R] Ringel, C. M., Hall algebras and quantum groups, Invent. Math. 101, 583--591 (1990).

Heluani: Mathieu's Moonshine

Resumen: Studying string theory with a K3 surface as background, physicists have found numerical evidence of a possible appearance of Mathieu's M24 group as a "hidden symmetry" of these surfaces. In this talk we will set up the problem in a rigurous mathematical framework and review recent progress. From a physicists' perspective the conjecture is simple to state: "There is an action of M24 on the chiral part of the N=2,2 supersymmetric sigma model with target a K3 surface commuting with the N=4 supersymmetry algebra". From the mathematics perspective it reads as "There exists a vertex operator algebra V structure on the elliptic genus of a K3 surface with an action of the N=4 superconformal vertex algebra and a commuting action of Mathiew's M24 sporadic group".  We plan to elucidate some of these terms as well as what the numerical evidence mentioned above is.

Kashuba: Variety of Jordan algebras

Resumen: Let k be an algebraically closed field, n be a positive integer and  A=A_{k}^{n^3} be a n^3-dimensional affine space. We consider a point c of A such that c={c_{ij}^h} gives a collection of structure constants defining a Jordan k-algebra. The set of all such points of A defines an algebraic subvariety Jor_n in A. The group GL_n acts on Jor_n by so-called "transport of structure" action, moreover GL_n-orbits of this action are in one-to-one correspondence with the isomorphism classes of n-dimensional Jordan algebras. An algebra J_2 is called a deformation of an algebra J_1 if  the orbit J_1^{GL_n} belongs to the Zariski-closure of the orbit J_2^{GL_n}. We will review the basic properties of deformations, describe the varieties Jor_n when n is 2,3,4 and also talk about asymptotics of dimension of Jor_n when n → \infty.

 Libedinsky: Hojas ligeras y conjetura de Lusztig

Resumen: La célebre conjetura de Lusztig predice los caracteres de las representaciones simples de los grupos algebraicos en característica positiva. Explicaremos como atacar esta conjetura con las hojas ligeras, unos nuevos objetos combinatorios apareciendo en el contexto de los bimódulos de Soergel y que dominan una gran parte de la combinatoria de Kazhdan-Lusztig. En particular las hojas ligeras juegan un rol fundamental en la demostración muy reciente de la conjetura de positividad de los polinomios de Kazhdan-Lusztig, que databa del año 79 y en la aparición de la emergente teoría de Hodge algebraica que que ya ha logrado demostrar algebraicamente teoremas profundos en algebras de Lie que históricamente sólo se podían atacar de manera geométrica.

Orosz: Súper álgebras de Lie contragradientes  de crecimiento finito y reflexiones impares

Resumen: Kac clasificó las superálgebras de Lie contragradientes de crecimiento finito sin raíces isotrópicas. Van de Leur clasificó las superálgebras de Lie contragradientes simetrizables.

Hoyt y Serganova terminaron la clasificación de las superálgebras de Lie contragradientes de crecimiento finito. Entre las herramientas fundamentales que utilizan para ello están las reflexiones impares.

En esta charla presentaremos algunos de los resultados importantes para entender la clasificación de Hoyt y Serganova, definiremos las reflexiones impares y a través de ejemplos

describiremos su utilidad en la clasificación de las superálgebras de Lie contragradientes de crecimiento finito con una raíz isotrópica.

Plavnik: Sobre categorías tensoriales y álgebras de Hopf

Resumen: En esta charla daremos definiciones y resultados sobre algunos conceptos básicos como los de categoría tensorial, categoría trenzada y álgebra de Hopf. También introduciremos ejemplos, como las categorías de módulos, comódulos y módulos de Yetter-Drinfeld sobre un álgebra de Hopf de dimensión finita, que ayudan a entender la conexión de estas nociones entre sí. Se dará la clasificación de los módulos de Yetter-Drinfeld sobre el álgebra de grupo de un grupo finito, sobre los complejos.

Pogorelsky: Representations of copointed Hopf algebras

Resumen: We investigate a family of copointed Hopf algebras of the Nichols algebra of the affine rack (F4; w). We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules.

Renz, Carolina Noele: Fusion Rules of Uq(sl2), q^m=1

Resumen: This presentation will be based on the same title article from Georg Keller and have the purpose to analyze the decomposition of multiple tensor products of finite-dimensional irreducible representations of Uq(sl2), when q is a root of 1.

Ronco: Algebraic structures associated to associahedra, permutohedra and other families of polytopes

Resumen: The goal of our talk is to describe algebraic structures related to associahedra like dendriform algebras, pre-Lie systems and brace algebras. The notion of pre-Lie system is equivalent to the non-symmetric algebraic operad one, we shall study the existence of similar structures associated to generalized associahedra, particularly to permutohedra. We shall describe the notion of permutad and relate it to shue operads dened by V. Dotsenko and A. Koroshkin.

Ryom-Hansen: Khovanov-Lauda-Rouquer-algebras and the Gram form on the Specht modules in positive characteristic

Resumen:  In the talk we present a continuation of the investigation published in [RH], that deals with the problem of finding the dimensions of the simple modules for the symmetric group in characteristic p. We showed in [RH] that the irreducible modules are generated by the Jucys-Murphy idempotents, suitably normalized. We also showed in that paper that the intertwiners from Brundan and Kleshchev’s seminal work [BK] can be realized within the classical theory for Jucys-Murphy elements. In the talk we explain how to combine these ingredients, together with the results from [BK], to obtain a very efficient algorithm for the Gram matrix on the Specht module, and hence for the dimension of the simple modules. In the same spirit we also address the problem of determining decomposition numbers.

References

[BK] Brundan, A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras. Invent. Math. 178 (2009) no. 3.

[RH] Ryom-Hansen, Steen, Young’s Seminormal Form and Simple Modules for Sn in Characteristic p, Algebras and Representation Theory, (2012), 1-23.

Shestakov: Basic super-rank of varieties of algebras

Resumen: The basic rank of a variety of algebras V is defined as the minimal cardinal number n such that V can be generated by an  n-generated algebra from V.   The varieties of all associative and all Lie algebras have basic rank 2, the variety of all alternative algebras has infinite basic rank. There are  varieties of associative algebras of infinite basic rank: for instance, the variety generated by the infinite-dimensional Grassmann algebra, or any non-matrix variety. We define the super-rank of the variety V as the minimal (anti-lexicografically) pair (m,n) such that V can be generated by the Grassmann envelope of a V-superalgebra with m even and n odd generators.  One of principal tools in Kemer's  solution of the Specht problem was the following result:  every variety of associative algebras of characteristic 0 has a finite basic super-rank.  Therefore, the basic super-rank is a more fine characteristic of a variety. We calculate super-rank for some varieties of infinite basic rank, construct examples of varieties of infinite basic super-ranks, and formulate open questions. This is a Joint work with A. Kuz'min.

Vay: Álgebras de Nichols
Resumen: Introduciremos la definición de un álgebra de Nichols y veremos diferentes maneras de construirla. Daremos ejemplos y sus propiedades más importantes.

Wilson: Root multiplicities of the indefinite type Kac-Moody algebra HD_n^(1)

Resumen: One of the fundamental problems in the theory of Kac-Moody algebras is that of finding the dimensions of the imaginary root spaces.  Although this problem is solved in the finite and affine cases, the indefinite case is still largely unknown.  In this talk we discuss recent work on the root multiplicity problem for the indefinite-type Kac-Moody algebra HD_n^(1).  It can be realized as a Z-graded Lie algebra in which the graded pieces are constructed out of certain representations for the embedded affine-type D_n^(1).  Using this construction, S.-J. Kang has given a formula for the root multiplicities of a class of Kac-Moody algebras which includes our case.  This use of this formula required the computation of weight multiplicities of D_n^(1) modules involved in the construction, which used the theory of crystal bases for the quantum group U_q(D_n^(1)).