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Franzen and M. Reineke - cf. Quiver Grassmannians are projective varieties parametrizing subrepresentations of quiver representations of a fixed dimension vector. The geometry of such projective varieties can be studied via the representation theory of quivers or of finite dimensional algebras.

Quiver Grassmannians appeared in the theory of cluster algebras. As a consequence of the positivity conjecture of Fomin and Zelevinsky, the Euler characteristic of quiver Grassmannians associated with rigid quiver representations must be positive; this fact was proved by Nakajima. We explore the geometry of quiver Grassmannians associated with rigid quiver representations: we show that they have property S meaning that: 1 there is no odd cohomology, 2 the cycle map is an isomorphism, 3 the Chow ring admits explicit generators defined over any field.

As a consequence, we deduce that they have polynomial point count. If we restrict to quivers which are of finite or affine type i.

Lie Superalgebra Day at Ben Gurion University

In this talk we review some features of the codimension growth of PI algebras, including the deep contribution of Giambruno and Zaicev on the existence of the PI-exponent, and discuss some recent developments in the framework of group graded algebras. In particular, a characterisation of minimal supervarieties of fixed superexponent will be given.

The last result is part of a joint work with O. Di Vincenzo and V. The class allows an analogue of twist deformations. The special subclass includes the quantum groupoids recently associated to the modular categories of type A. The talk is based on an ongoing joint work with Sebastiano Carpi and Sergio Ciamprone. Since then, many different and fundamental Hopf algebra structures were studied in connection to it.

Furthermore, the theory of pictures between partition diagrams is known to encode much of the combinatorics of symmetric group representations and related topics: it captures for example the Littlewood-Richardson formula, as already shown by Zelevinsky. In , this framework was extended to double posets pairs of orders coexisting on a given finite structure.

Recently we worked out a similar approach for finite preorders, which in turn are equivalent to finite topologies, and developed it from the point of view of combinatorial Hopf algebras, leading to new advances in the field. Friday, April 13th, h. Una congettura classica afferma che un risultato simile vale per schemi in gruppi piatti e finiti. The computation of Kronecker coefficients is in particular a very interesting problem which has many applications. Vergne and M.

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The algorithm does not only compute individual Kronecker coefficients, but also symbolic formulas that are valid on an entire polyhedral chamber. As a byproduct, it is possible to compute several Hilbert series. Friday, March 16th, h. Braden and MacPherson proved that the information contained in this moment graph is sufficient to compute the equivariant intersection cohomology of the variety.

Vera Serganova, Lecture I - 20 January 2015

In order to do this, they introduced the notion of a sheaf on moment graph whose space of sections stalks describes the local intersection cohomology. These results motivated a series of paper by Fiebig, where he developed and axiomatized sheaves of moment graphs theory and exploited Braden-MacPherson's construction to attack representation theoretical problems. In the talk we explain how to extend this theory of sheaves on moment graphs to an arbitrary algebraic oriented equivariant cohomology h in the sense of Levine-Morel e.

Moreover, we show that in the case of a total flag variety X the space of global sections of the respective h -sheaf also describes an endomorphism ring of the equivariant h -motive of X. This is a very recent joint work with Rostislav Devyatov and Martina Lanini. This theory serves as an algebraic analogue of the usual complex cobordism from algebraic topology of 60's similarly, the Chow group serves as an algebraic version of the usual singular cohomology.

Friday, February 23rd, h. This is part of a joint project with Andrea Iannuzzi, and this talk concludes his presentation of February 9. Namely, what in string theory is known as topological T -duality between K 0 -cocycles in type IIA string theory and K 1 -cocycles in type IIB string theory, or as Hori's formula, can be recognized as a Fourier-Mukai transform between twisted cohomologies when looked through the lenses of rational homotopy theory.

This is an example of topological T -duality in rational homotopy theory, which can be completely formulated in terms of morphisms of L -infinity algebras. In this context, the interplay of complex geometry and the Lie group structure of G C leads to an explicit realization of all the terms of such a structure.

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Time permitting, we compare the adapted context with the previous constructions. This is part of a joint project with Laura Geatti. More details on the adapted realization in the second talk. Friday, January 12th, h. In the simply laced case it is not too hard to prove that the relations defining U q are preserved by the Drinfeld coproduct which is then a well defined algebra homomorphism ; but in the other cases the expression of the Drinfeld coproduct applied to the Serre relations is very complicated, and till now the direct attempts to prove that it is zero failed.

In this talk a different strategy is presented: the bracket by the Drinfeld generators is deformed so to get a locally nilpotent derivation D on a suitable algebra V ; the study of the exponential of D , which is an algebra automorphism of D , provides a proof that the Drinfeld coproduct is well defined. This construction works for both the affine quantum algebras and the toroidal quantum algebras. I risultati presentati sono frutto di un lavoro in collaborazione con V.

Contributed talks

Kac e D. Friday, December 1st, h. Thus every supergroup has an associated pair given by its tangent Lie superalgebra and its maximal classical subgroup - what is called a "super Harish-Chandra pair" or "sHCp" in short : overall, this yields a functor F from supergroups to sHCp's. It is known that the functor F is an equivalence of categories: indeed, this was showed by providing an explicit quasi-inverse functor, say G , to F.

Koszul first devised G for the real Lie case, then later on several other authors extended his recipe to more general cases. In this talk I shall present a new functorial method to associate a Lie supergroup with a given sHCp: this gives a functor K from sHCp's to supergroups which happens to be a quasi-inverse to F , that is intrinsically different from G. In spite of different technicalities, the spine of the method for constructing the functor K is the same regardless of the kind of supergeometry i.

Lie superalgebra

Kiselman's semigroups are certain monoids, originally introduced in the context of convexity theory. Hecke-Kiselman monoids provide a generalization of both concepts. I will first address the finiteness problem for Hecke-Kiselman monoids, and then give a combinatorial description of Kiselman's semigroups - and possibly some of its quotients - by considering all possible evolutions of some special dynamical systems on a graph, called "update systems".

Friday, November 17th, h. These link homologies are categorifications of the link invariants defined by Reshetikhin-Turaev in case of the special linear group. We will discuss why functoriality is an important notion and how to show it. The latter will include the equivariant geometry of Grassmannians and partial flag varieties as well as higher representation theory.

Tuesday, September 26th, h. For Grassmannians, such degenerations can be obtained from birational sequences and the tropical Grassmannian.

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The first were recently introduced by Fang, Fourier, and Littelmann. They originate from the representation theory of Lie algebras and algebraic groups. In our case, we use a sequence of positive roots for the Lie algebra sl n to define a valuation on the homogeneous coordinate ring of the Grassmannian. Nice properties of this valuation allow us to define a filtration whose associated graded algebra if finitely generated is the homogeneous coordinate ring of the toric variety.

The second was defined by Speyer and Sturmfels and is an example of a tropical variety: a discrete object a fan associated to the original variety that shares some of its properties and in nice cases, as the one of Grassmannians, provides toric degenerations. In this talk, I will briefly explain the two approaches and establish a connection between them.

Monday, June 12th, h.

Lie superalgebra - Wikipedia

They owe their name to the well-known fact that this non-degeneracy is actually equivalent to the appearance of representations of the modular group P SL 2, Z by means of certain canonically defined matrices S , T also called modular data. I will explain their relevance in physics more precisely, in models of chiral CFT and comment on a trace formula for self-braidings by means of modular data, which can be used in the classification problem of MTCs.

We discuss the problem of combinatorial invariance in the parabolic setting. Wednesday, May 17th, h. Such an arrangement consists of parallel translates of collection of n hyperplanes in general position in C k which fail to form a generic arrangement in C k. In Falk showed that the combinatorial type of Discriminantal arrangement depends on the collection of n hyperplanes in general position in C k.