The seminar will have two talks, with the following schedule:
* 14.30-15.30: Seminar of Giovanna Carnovale,
* 15.30-16.00: Coffee at Sala de Matemática,
* 16.00-17.00: Seminar of Iván Darío Gomez.
More information about each talk is available below.
23/11/2017, 14:30 hs, aula 27 FaMAF
The Jordan stratification in Lie algebras and algebraic groups
Expositor: Giovanna Carnovale
Resumen: Semisimple Lie algebras and algebraic groups can be stratified in terms of so-called Jordan classes or decomposition classes. In the Lie algebra case they were introduced in Borho and Kraft’s work on sheets whereas the group analogue appeared in Lusztig’s construction of generalised Springer’s correspondence. Roughly speaking, Jordan classes are unions of adjoint orbits (or conjugacy classes) that are isomorphic as homogeneous space. We are interested in their geometry and in the geometry of the induced stratifications on the geometric quotients of the Lie algebra and of the group. We will show how some of these problems can be interpreted in terms of hyperplane arrangements.
It is based on a joint project with Francesco Esposito.
23/11/2017, 16:00 hs, aula 27 FaMAF
Extensiones modulares de categorías de fusión super-Tannakianas
Expositor: Iván Darío Gomez
Resumen: Let ℊ be Lie algebra over ℂ, the socle of the ℊ-module V is the unique maximal semi-simple ℊ-submodule of V and it is denote soc(V). A V it is called uniserial if the socle series is a composition series, i.e,
soc0(V)⊂ soc1(V)⊂ \hdots ⊂ socn(V)=V
is a composition series where soci(V)/soci-1(V)=soc(V/soci-1(V)) for 1≤ i≤ n.
In [C-S] it is obtenied the classification of the uniserial ℊ-modules when the Levi descomposition of ℊ is sl(2)⋉ V(m) for m≥ 1, where V(m) is a irreducible sl(2)-module of highest weight m.
These uniserial modules are the Z(a,l) (except for some special case) which as sl(2)-modules are
Z(a,l)=⨁i=0l V(a+im) and its respective dual Z(a,l)* with a,l ∈ ℕ∪ {0}.
In the first part of this talk will be discussed over the socle of the tensorial product of the uniserial sl(2)⋉ V(m)-modules, which we allow construct new modules and proof with m odd that Z(0,1)⊗ Z(b,1) is indecomposable if b≠ 0.
Remember that is V is a cyclic ℊ-module if V=U(ℊ)v for some v∈ V y where U(ℊ)v is the universal envelope of ℊ. In the second part of the talk, will be shown that certain tensorial products of uniserial sl(2)⋉ V(m)-modules are cyclics modules.
Bibliography:
[Ca] P. Casati, The classification of the perfect cyclic sl(n+1)⋉ ℂn+1, Journal of Algebra 476 (2017) 311-343.
[C-S] L. Cagliero and F. Szechtman, The classification of uniserial sl(2)⋉ V(m)-modules and a new interpretation of the Racah-Wigner 6j-symbol, J. of Algebra, Volume 386 (2013), 142-175.
[Pi] A. Piard, Sur des représentations indécomposables de dimension finie de \matfrak{SL}(2).R2, Journal of Geometry and Physics, Volume 3, Issue 1, 1986, 1–53.