COLLOQUIUM ON ALGEBRAS AND REPRESENTATIONS - Quántum 17 - Universidad de Talca

Monday 20.03

Tuesday 21.03

Wednesday 22.03

 Thursday 23.03

Friday 24.03

10:00 - 11:00

V. Futorny

Gelfand-Graev continuation and new simple modules for gl(n) and U_q(gl(n))

C. Vay

On projective modules over finite quantum groups

G. A. García

Quantum subgroups of simple (twisted) quantum groups at roots of one

P. Sentinelli

Parabolic Temperley-Lieb modules and polynomials

S. Ryom-Hansen

Jantzen filtration for Soergel bimodules 

11:00 - 11:30

Coffee break

Coffee break

Coffee break

Coffee break

Coffee break

11:30 - 12:30

M. Mombelli

A Tannaka–Krein theorem  for monoidal categories

I. Angiono

A quantum version of the algebra of distributions of an algebraic group

S. Griffeth

W-exponentials, Schur elements, and the spherical representation of the rational Cherednik algebra

E. Herscovich

Using A-infinity-algebras to compute the algebraic structure of Hochschild (co)homology

M. Farinati

Set theoretical solutions of the Yang-Baxter equation and knot/link invariants

12:30 - 14:30

Lunch

Lunch

Lunch

Lunch

Lunch

14:30 - 15:20

Grupo de trabajo

Álgebras de Nichols

Tarde libre

Paseo:

Paso Pehuenche, Lago Colbún,

Casa Donoso

D. Bagio

Representations of the super Jordan plane

G. Burrull

 p-canonical basis for universal Coxeter groups

F. Fantino

Álgebras de Hopf copunteadas sobre grupos diedrales

15:20 - 15:50

Coffee break

Coffee break

Coffee break

Coffee break

15:50 - 16:20

 M.  E. Bernaschini

Group actions on 2-categories

A. Mejia

Deformaciones de algebras de Hopf semisimples de dimensión p³ y pq²

O.  Márquez

Representation of the Drinfeld double of Radford’s algebras

J. M. Jury Giraldi

Nichols algebras that are quantum planes

16:30 - 17:30

Grupo de trabajo

Álgebras de Nichols

Grupo de trabajo

Álgebras de Nichols

Grupo de trabajo

Álgebras de Nichols

                Abstracts

Iván Angiono

A quantum version of the algebra of distributions of an algebraic group

We construct a family of Galois extensions of the small quantum group of sl2, which mimic a family of (finite-dimensional) subalgebras of the algebras of distributions of SL2 over a field of positive characteristic. We prove a kind of Steinberg tensor product decomposition of simple representations. Finally we present some results about the general case, which is part of a work in progress.

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Dirceu Bagio

Representations of the super Jordan plane

Let z=xy+yx in the free associative algebra in generators x and y. Let B be the algebra generated by x and y with defining relations

x² and yz-zy-xz. The algebra B (which is graded by deg(x)=deg(y)=1) was introduced in [AAH1, AAH2] and is called the super Jordan plane. In fact, B is a Nichols algebra, GKdim(B)=2 and {x^a z^b y^c : a=0,1; b,c > -1} is a basis of B.

We prove in [ABFF] that the finite-dimensional simple B-modules are one-dimensional. Also, the indecomposable B-modules of dimension 2 and 3 are classified and two families of indecomposable B-modules of arbitrary dimension are presented.

[AAH1] N. Andruskiewitsch,  I. Angiono and I. Heckenberger. Liftings of Jordan and super Jordan planes. arXiv:1512.09271.

[AAH2] N. Andruskiewitsch,  I. Angiono and I. Heckenberger. On finite GK-dimensional Nichols algebras over abelian groups. arXiv:1606.02521.

[ABFF] N. Andruskiewitsch,  D. Bagio, S. Della Flora and Daiana Flores. Representations of the super Jordan plane. preprint.

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María Eugenia Bernaschini

Group actions on 2-categories

We define the notion of a group action on a 2-category and we introduce the corresponding equivariant 2-category. Some known constructions in tensor categories are encompassed in this setting. We prove that the center of the equivariant 2-category is monoidally equivalent to the equivariantization of a relative center.

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Gastón Burrull

p-canonical basis for universal Coxeter groups

We give a formula to calculate the intersection form I{w,x} of a reduced expression w at x in terms of the coefficients of the Cartan matrix of a realization. For this purpose we explore compositions of Libedinsky's light leaves modulo lower terms. Joint work with Paolo Sentinelli.

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Fernando Fantino

Álgebras de Hopf copunteadas sobre grupos diedrales

En esta charla se presentará la clasificación de las álgebras de Hopf de dimensión finita sobre un cuerpo algebraicamente cerrado de

característica cero cuyo corradical es el álgebra de funciones de un grupo diedral de orden 8t, t>2. Este es un trabajo en conjunto con G. A. Garcia y M. Mastnak.

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Marco Farinati

Set theoretical solutions of the Yang-Baxter equation and knot/link invariants

Given a set-theoretical solution of the Yang-Baxter equation on a set X, we define the notion of non-commutative 2-cocycle as a map XX->G (where G is a group, maybe noncommutative)  satisfying certain three conditions. As first part, from this data we generalize a construction of Carter-El Hamdadi-Graña and Saito, made for quandles, to general solutions of the Yang-Baxter equation, giving rise to knot/link invariants. But moreover, we define a group U_{nc}(X) that is the universal target of non-commutative 2-cocycles on X, hence, our invariant depends only on the set-theoretical solution X, and we do not "need" to find a group and cocycles. Given X, this group U_{nc}(X) is computable in the sense that for small X (say of cardinal less than 10) an easy algorithm gives a "small" set of generators and relations in a few seconds, so we get a lot of examples of non-commutative 2-cocycles, and the corresponding invariants. These ideas are also generalized to vitrtual knots and links.

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Vyacheslav Futorny

Gelfand-Graev continuation and new simple modules for gl(n) and U_q(gl(n))

Gelfand-Graev continuation method allows to extend the classical Gelfand-Tsetlin basis for finite dimensional modules of gl(n) to some simple infinite  dimensional modules. This method was revised by Lemire and Patera who conjectured when the Gelfand-Graev condition in fact defines a module. This conjecture was recently proved in a joint paper with L. E. Ramirez and J. Zhang, where also a new family of simple modules was constructed by extending the Gelfand-Graev and the Lemire-Patera methods. This approach is also applied to construct new simple modules for the quantum gl(n).

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Gastón Andrés García

Quantum subgroups of simple (twisted) quantum groups at roots of one

Let G be a connected, simply connected simple complex algebraic group. In this talk we will describe a method to determine all Hopf algebra quotients of some quantum groups associated with the coordinate function algebra O(G) of G, given by (multiparametric) deformations at roots of unity. As a byproduct of this construction one obtains families of examples of Hopf algebras with different properties that fit into central exact sequences. This talk is based in a series of papers [1], [2], [3] and [4] in collaboration with N. Andruskiewitsch and J. Gutiérrez.

[1] N. Andruskiewitsch and G. A. García. Finite subgroups of a simple quantum group. Compositio Math. 145, no. 2 (2009), 476--500.
[2] N. Andruskiewitsch and G. A. García. Extensions of finite quantum groups by finite groups. Tranformation Groups 14, no. 1 (2009), 1--27.
[3] G. A. García. Quantum subgroups of GLa,b(n). J. Algebra 324 (2010), 1392–1428
[4] G. A. García and J. A. Gutiérrez.
Quantum subgroups of simple twisted quantum groups at roots of one. Trans. of the AMS, to appear.

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Stephen Griffeth

W-exponentials, Schur elements, and the spherical representation of the rational Cherednik algebra

We explain how to compute the support of the spherical representation of the rational Cherednik algebra of a complex reflection group W in two ways: first, using a certain special function, the W-exponential function, deforming the usual exponential function, and second, using Schur elements for finite Hecke algebras. In case W is a real reflection group we recover the theorem of Etingof calculating the support in terms of the Poincare polynomial of W. Joint with Daniel Juteau.

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Estanislao Herscovich

Using A-infinity-algebras to compute the algebraic structure of Hochschild (co)homology

A-infinity-algebras were defined by J. Stasheff in 1963 within the realm of algebraic topology. In this talk I will explain how the A-infinity-algebra structure of the Yoneda algebra of a nonnegatively graded connected algebra allows to directly compute the algebra A-infinity-algebra structure of the Hochschild cohomology and the A-infinity-module structure of the Hochschild homology over the corresponding cohomology. In particular, this gives explicit expressions of the cup and cap products on the Hochschild cohomology and homology of any nonnegatively graded connected algebra, respectively.

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Eduardo Hoefel

Deformations of Homotopy Actions

An affine action of an associative algebra A on a vector space V is an algebra morphism A→V⋊End(V), where V is a vector space and V⋊End(V) is the algebra of affine transformations of V. The one dimensional version of the Swiss-Cheese operad, denoted sc1, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by (sc1)∞. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an A∞-morphism A→V⋊End(V), where A is an A∞-algebra. In this paper we prove a relative version of Deligne's conjecture. In other words, we show that the deformation complex of a homotopy affine action has the structure of an algebra over an SC2 operad. That structure is naturally compatible with the E2 structure on the deformation complex of the A∞-algebra.

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João Matheus Jury Giraldi

Nichols algebras that are quantum planes

Recently, in [GGi, Prop. 4.8, 4.9], braided vector spaces (V, c) of dimension 2 and non-diagonal type were found but such that the Nichols algebras are quantum planes. Consequently, the following question arises naturally: classify all the Nichols algebras (of rigid braided vector spaces) that are isomorphic to quantum linear spaces as algebras.

In this paper, we solve this question for quantum planes. More specifically, the classification of the solutions of the quantum Yang-Baxter equation had already been performed by J. Hietarinta when dim(V)=2 [Hi]. Thus, we consider the associated braided vector spaces and compute the quadratic relations. Therefore, we classify all these Nichols algebras that have at least one quadratic relation. This is a joint work with N. Andruskiewitsch [AG]

[AGi] N. Andruskiewitsch and J. M. J. Giraldi. Nichols algebras that are quantum planes. arXiv: 1702.02506.

[GGi] G. A. Garcíia and J. M. J. Giraldi. On Hopf Algebras over quantum subgroups. arXiv: 1605.03995.

[Hi] J. Hietarinta. Solving the two-dimensional constant quantum Yang-Baxter equation. J. Math. Phys. 34 (1993).

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Oscar Márquez

Representation of the Drinfeld double of Radford’s algebras

In this work, we consider the Drinfeld double D(R_{n,p}) of a n²p-dimensional Hopf algebra R_{n,p} over an algebraically closed field of characteristic zero whose coradical is not a subalgebra and describe its simple modules. The algebra R_{n,p} was  studied by Radford as an example of a noncommutative noncocommutative Hopf algebra whose Jacobson radical is not a Hopf ideal and with injective antipode. We describe D(R_{n,p}) by generators and relations and study the simple representation of D(R_{n,p})​ in order to describe the structure of the Yetter Drinfield modules over R_{n,p}.

The  case n=p=2 was studied by Garcia and Giraldi, so this work generalizes their results. Work joint with Dirceu Bagio and Gastón Garcia

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Luz Adriana Mejia Castaño

Deformaciones de algebras de Hopf semisimples de dimensión p³ y pq²

Es conocida la clasificación para álgebras de Hopf semisimples de dimensión p³ y pq² si p,q son primos diferentes. La primera familia fue clasificada por Akira Masuoka y la segunda fue clasificada por Sonia Natale. Si consideramos la categoría de corepresentaciones de cada una de esas álgebras, es lógico querer saber si esas categorías son equivalentes. En otras palabras si esas álgebras son monoidalmente Morita-Takeuchi equivalentes. Voy a presentar algunos resultados sobre clases de equivalencia de estas álgebras usando la clasificación de las categorías módulo sobre ciertas categorías de fusión group-theoretical, introducida por Victor Ostrik.

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Martin Mombelli

A Tannaka–Krein theorem for monoidal categories

I will present a result of M. Neuchl, concerning a reconstruction theorem for the 2-category of representations of a monoidal category.

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Steen Ryom-Hansen

Jantzen filtration for Soergel bimodules

Soergel introduced a category of bimodules in the nineties during his new proof of the Kazhdan-Lusztig conjectures. In the last decade, a diagrammatical version D of this category has been developed which is better behaved in positive characteristic than the original category.

Elias and Williamson proved that D is cellular, where the cellular basis is a diagrammatical version of Libedinsky's light leaves.

In the talk we give a formula for the determinant of the bilinear form on the cell modules associated with D. This gives rise to sum formulas on the Jantzen type filtrations on the cell modules. In certain cases these can be used to obtain decomposition numbers for algebraic groups.

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Paolo Sentinelli

Parabolic Temperley-Lieb modules and polynomials

For any Coxeter system (W,S) we study some modules over its Temperley-Lieb algebra, two for each quotient W^J, which have generators indexed by the fully commutative elements of W^J. Here J is any subset of the set S. These modules correspond to quotients of the two Hecke modules introduced by V. V. Deodhar to define a parabolic Kazhdan-Lusztig theory. Moreover, the existence, for any J, of a free Temperley-Lieb module (conjecturally isomorphic to one of the two quotients above mentioned), leads us to the realization of a Kazhdan-Lusztig-Stanley function for the poset of fully commutative elements in W^J with the Bruhat ordering. This function (or family of polynomials) reduces, for J empty, to the L-polynomials introduced by Green y Losonczy, which are the analogues of the Kazhdan-Lusztig polynomials in the Temperley-Lieb algebra.

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Cristian Vay

On projective modules over finite quantum groups

Let D be the Drinfeld double of the bosonization B(V)#kG of a finite-dimensional Nichols algebra B(V) over a finite group G. It is known that the simple D-modules are parametrized by the simple modules over D(G), the Drinfeld double of G. This parametrization can be obtained by considering the head L(x) of the Verma module M(x) for every simple D(G)-module x. In the present work, we show that the projective D-modules are filtered by Verma modules and the BGG Reciprocity holds for the projective cover P(x) of  L(x). We use graded characters to proof the BGG Reciprocity and obtain a graded version of it. As a by-product we show that a Verma module is simple if and only if it is projective. We also describe the tensor product between projective modules.

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