Detalle, investigación

Líneas de investigación en detalle.

Grupos de transformaciones (W. Ferrer, I. Pan, A. Rittatore)

Teoría de invariantes relativos (A. Rittatore, W. Ferrer)

Given an action of a reductive algebraic group $G$ on a algebraic variety $X$, geometric invariant theory gives a reasonable good answer to the construction of $G$-stable open subsets $U\subset X$ such that the categorical or the geometric quotient of the restrictedaction exists. However, if $G$ is not reductive, One does not have a good answer to this problem. Work by Ferrer and Rittatore give conditions in order to garantee the existence of the categorical quotient for a given action $G\times X\to X$ of an affine algebraic group on an affine variety, related to the so-called category of $(\Bbbk[X], G)$-modules. A long term objective is to generalize the GIT construction to this context.

Esquemas en grupos y monoides: geometría y representaciones (A. Rittatore, W. Ferrer)

A classic result by Chevalley states that any algebraic group $G$ fits into a short exact sequence of algebraic groups $0\to G_{aff} \to G\to A \to 0$, were $A$ is the Albanese variety of $G$ and $G_{aff}$ is the smallet normal closed subgroup of $G$ such that the quotient $G/G_{aff}$ is proper then $G_{aff}$ is an affine algebraic group. This result (announced by Chevalley in the early 1950, proved by Barsotti, Rosenlicht, Sanchez and Chevalley) has been slightly generalized by Raynaud and Brion, to the context of groups schemes of finite type. On the other hand, Brion and Rittatore have generalized the Chevalley decomposition to the case of (normal) algebraic monoids. A main objective of this research line is to develop a theory of monoid schemes, including a generalization of this decomposition.

Variedades tóricas y esféricas (A. Rittatore)

When dealing with the study of the geometric properties of algebraic varieties (such as being smooth or Gorenstein- Fano), a very common
(and necessary procedure) is the construction of large families of algebraic varieties sharing the geoemtric property under study. For
this, the setting of toric --- or more generally spherical --- varieties, is a particuarly interesting one, since there exists (in
both cases) a dictionary between these familes and combinatorial objets: the so--called fans of $\mathbb R^n$ in the case of the toric
varieties and the colored fans in the case of spherical varieties. In a work in progress, P.-L. Montagard a A. Rittatore were able to classify the toric varieties associated to root systems that are Gorenstein-Fano. These varieties turn out to be in relationship with the lattice-regular convex polytopes (that were studied in a previous work by Montagard and Ressayre).

Acciones de esquemas en grupos (W. Ferrer, A. Rittatore)

In a recent work (in progress) by Ferrer, del Ángel and Rittatore, it is proposed a representation theory associated to an exact short sequence of group schemes $1\to H\to G\to A\to 0$, where $H$ is an affine group scheme, and $A$ an Abelian variety, such that the morphism $G\to A$ is a $H$ --torsor -- an \emph{affine torsor extension} of $A$. This representation theroy is supported on the homogeneous vector bundles
over $A$, and classifies equivalent affne torsor extensions. On the other hand, work by Brion on the structure of group schemes and their
actions on varieties,as well as work by Brion and Rittatore on algebraic monoids, show that the proposed category has many interesting applications on the study of actions of group schemes on varieties. It is proposed to develop such a study, focusing in problems such as the existence of quotients and its relationship with the categorical properties of the representation theory of the acting group scheme.

Álgebra Conmutativa (I. Pan)

Derivaciones en k-álgebras

Se estudian principalmente derivaciones simples de k-álgebras de tipo finito. En el caso de un dominio de integridad regular esto corresponde a foliaciones en variedades afines que no admiten subvariedades propias estables.

Geometría algebraica (I. Pan, A. Rittatore)

Transformaciones de Cremona (I. Pan, A. Rittatore)

El estudio de la geometría birracional es uno de los temas más importantes de la geometría algebraica. Las variedades tóricas juegan in rol importante pues constitiuyen una familia de ejemplos "fáciles" de manipular, con geometría birracional interesante, en el sentido que su grupo de transformaciones birracionales es lo más grande posible: es isomorfo al grupo de transformaciones de Cremona del espacio proyectivo. En el caso del espacio proyectivo, se estudia la estructura del grupo de Cremona y sus subgrupos, especialmente el llamando subgrupo de de Jonquières, así como la geometría asociada sus elementos, las transformaciones de Cremona.

Foliaciones (I. Pan)

Se estudian foliaciones en superficies lisas y en espacios proyectivos. Es particular se está interesado en el subgrupo de simetrías birracionales, que en el caso de variedades racionales corresponde a un subgrupo del grupo de Cremona.

Álgebras de Hopf (A. Abella, W. Ferrer, M. Haim)

Producto de Heisenberg

The constructions by Aguiar Ferrer Moreira in a joint work, can be generalized to a general categorical setting as have been recently done by Street and others. There is another product that can be put also in this set up, Drinfel'd product, that still has not been worked out, and it seems to be particularly relevant to deal with it as it is compatible with the original coproduct given in the Hopf algebra and the reasons for that compatibility are not fully understood.

Álgebras de Bi-Frobenius (W. Ferrer, M. Haim. M. Pereira)

Les a'lgebres de biFrobenius sont une g'en'eralisation des alg'ebres de Hopf de dimension finie: la condition de compatibilit'e entre l produit et le coproduit demand ́ee dans le cas d’une alg'ebre de Hopfest affaiblie, en n’imposant qu'une condition sur l'antipode. Ces objets ont'et'e' etudi ́es du point de vue de leur relation avec les structures d’alg'ebre de Hopf dans [Ha07] et [FH07].

Generalizaciones de grupos cuánticos (W. Ferrer, M. Haim)

Cette ligne de recherche a pour but l'étude de généralisations des Algèbres de Hopf ou groupes quantiques. Inspirés sur des travaux de Drinfel${}'$d et Joyal-Street, plusieurs mathématiciens ont établi l'étroite relation existente entre les algèbres de Hopf et la Théorie des Catégories. Ceci a mené naturellement a la considération de généralisations de la notion d'algèbres de Hopf, avec des objectifs à la fois méthodologiques et pratiques. Entre elles, les monades de Hopf, introduites et étudiées pendant ces dernières années par A. Bruguières et ses coauteurs, seront le centre de la recherche. Les algèbres de Hopf peuvent être généralisées dans une autre direction, comme celle qui a été initiée par Day, McCrudden er Street, et développée par I. L\'opez Franco.

Ore extensions, Ore monoids and Hopf algebras (W. Ferrer)

A new object, called $D$-structure, has been introduced recently by Cojuhari and Gardner. These structures generalize Ore extensions and
Ore Monoids and appear naturally in connections with certain gradings by monoids. They can be defined as follows. Given an algebra $A$ and
a monoid $G$ we denote the $D$ structure $A<G>$ as the set of sums $\sum_{g\in G}a_gg$. The addition is coefficient--wise and the
multiplication is based on the following rules: $$ ga=\sum_{g,h}\sigma_{g,h}(a)h.$$ where for $g,h\in g$, the maps $\sigma_{g,h}$ are additive endomorpisms of $A$. The associativity of the multiplication gives conditions that the maps $\sigma_{g,h}$ must satisfy. These structure are still new and a question that occur naturally is to study the case when in fact $A$ is
a Hopf algebra $H$. The comultiplication and counits will give constraints on the maps $\sigma_{g,h}$ and a first task would be to
determine them. Another task would be to analyze the ideal structure of these $D$ modules (this task has already started in a work Cojuhari and Gardner by but is far from complete). One red line to keep in mind is the case of iterated Ore extensions.

Mónadas y categorías monoidales múltiples

Grupos cuánticos compactos (A. Abella, W. Ferrer)

Les groupes quantiques compacts sont une g ́en ́eralisation au cas non-commutatif des groupes topologiques compacts. Dans les articles [An94], [AA02] et [AA99], A. Abella et N. Andruskiewitsch ́etudient d’une part l’unicit ́e de la structure compacte et d’autre part ils introduisent quelques familles d’exemples de cette construction. L’ ́equipe envisage de continuer cette ́etude en utilisant la th ́eorie de Fourier.