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| Groupes et Représentations Reims 2012 |
Organisers
Partners
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Speakers
Practical information
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Programme |
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Click on the title for an abstract of the talk. |
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| Wednesday 5 December | |
| 13:30, | Accueil |
| 14:00-14:50, | H. Kraft (Basel) |
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Varieties characterized by their endomorphisms.
I will explain the following astonishing result: If two varieties \(X\) and \(Y\)
have isomorphic endomorphism semigroups and if one of them is affine and contains a copy
of the affine line, then \(X\) and \(Y\) are isomorphic up to base change.
The proof is based on some classical results of Dick Palais and uses tools from
algebraic geometry and algebraic transformation groups. (Joint work with Raffael Andrist)
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| 15:00-15:50, | S. Launois (Kent) |
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From quantum algebras to total positivity.
In recent publications, the same combinatorial description has
arisen for three separate objects of interest: non-negative cells in the
real Grassmannian (Postnikov, Williams); torus orbits of symplectic leaves
in the classical Grassmannian (Brown, Goodearl and Yakimov); and torus
invariant prime ideals in the quantum Grassmannian (Launois, Lenagan and
Rigal). The aim of this talk is to present some of these results and explore the
reasons for this coincidence.
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| Coffee break | |
| 16:20-17:10, | B. Leclerc (Caen) |
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Nakajima varieties and orbit closures of Dynkin quivers.
The aim of the talk is to show that orbit closures for
A-D-E quivers are particular cases of graded Nakajima varieties.
This geometric result was suggested by the representation theory
of quantum enveloping algebras. In a joint work with David Hernandez,
we have introduced certain tensor categories of representations of
the quantum loop algebra \(U_q(Lg)\), whose quantum Grothendieck ring
is isomorphic to the positive part of the quantum enveloping algebra
\(U_v(g)\). Comparing the canonical bases of these algebras which have
been described geometrically by Nakajima nad Lusztig, one is lead
to the above result.
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| Thursday 6 December | |
| 9:00-9:50, | A. Premet (Manchester) |
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Multiplicity free primitive ideals and derived subalgebras of centralisers.
Let \(g\) be a finite dimensional complex simple Lie algebra. In my talk I am
going to discuss a way to classify the primitive ideals \(I\) of \(U(g)\) with the property
that the associated variety \(VA(I)\) occurs with multiplicity \(1\) in the characteristic
cycle \(AC(I)\). The classification relies on the detailed knowledge of the quotients
\(g_e/[g_e, g_e]\) where \(g_e\) is the centraliser of a nilpotent element \(e\) in \(g\).
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| 10:00-10:50, | S. Goodwin (Birmingham) |
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Principal \(W\)-algebras for \(\mathrm{gl}_{m|n}\).
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| Coffee break | |
| 11:20-12:10, | O. Schiffmann (IMJ, Paris) |
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Algèbres de Cherednik et algèbres W affines.
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| Lunch | |
| 14:00-14:50, | M. Duflo (Paris 7) |
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Frobenius Lie subalgebras of simple Lie algebras.
A Frobenius Lie algebra is a Lie algebra for which the coadjoint
action has an open orbit. I present results on Frobenius Lie subalgebras
of a simple complex Lie algebra which contain a Cartan subalgebra,
with a special emphasis on the "Ooms spectrum" (the Ooms spectrum is
an equivalent for Frobenius Lie algebras of the set of exponents of a
simple Lie algebra).
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| 15:00-15:50, | A. Pasquale (Metz) |
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Representation-theoretic aspects of the cosine transform.
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| Coffee break | |
| 16:20-17:10, | A. Alekseev (Geneva) |
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The Duflo Isomorphism Theorem and the Kashiwara-Vergne Conjecture: Hard or Soft?
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| Conference dinner | |
| Friday 7 December | |
| 9:10-10:00, | M. Rosso (Paris 7) |
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Quantum quasi-symmetric algebras and quantum groups.
The "positive part" of quantum groups is known to be a quantum symmetric algebra associated with a particular braided
vector space. From this point of view, one can realize the finite dimensional representations of the whole quantum group
inside a larger quantum symmetric algebra, leading to new character formulas.
We shall introduce quantum multibrace algebras (which provide all Hopf algebra structures on cofree cotensor Hopf algebras) and the particularly interesting and manageable subclass of quantum quasi symmetric algebras (roughly, "quasi" means that, compared to the quantum symmetric algebra situation, we have (and we use!) an extra algebra structure on the underlying braided vector space). We shall show that the whole quantum group is a (mild) quotient of a suitable quantum quasi symmetric algebra. This allows again to give a new realization of the representations, and also of the representations of the double of the quantum groups. |
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| 10:10-11:00, | D. Hernandez (Paris 7) |
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Fractions rationnelles de Drinfeld et catégorie \(\mathcal{O}\).
Nous introduisons une catégorie \(\mathcal{O}\) de représentations d'une algèbre de Borel,
associée à une algébre affine quantique. Nous en construisons les représentations
fondamentales comme limites de modules de représentations de dimension finie, ce qui permet d'établir
une formule de caractères explicite et uniforme pour ces représentations. Nous démontrons
que les modules simples de cette catégorie sont paramétrés par des \(n\)-uplets de fractions
rationelles en une variable, sans pôle ni racine à l'origine.
(travail en commun avec M. Jimbo) |
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| Coffee break | |
| 11:30-12:20, | A. Joseph (Weizmann) |
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Simple Modules for Relative Yangians.
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| Lunch | |
Participants |
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A. Alekseev J. Alev R. Berger S. Chemla A. De Goursac |
M. Duflo X. Fang F. Fauquant-Millet Y. Fittouhi L. Foissy |
V. Gayral S. Goodwin J. Haut D. Hernandez A. Jago |
A. Joseph S. Korvers H. Kraft S. Launois B. Leclerc |
J. Lin A. Mansuy M. Mansuy M. Medina J.-Ph. Michel |
S. Morier-Genoud A. Ooms A. Pasquale M. Pevzner A. Premet |
R. Rentschler L. Rigal M. Rosso G. Sadaka E. Sarycheva |
O. Schiffmann F. Spinnler R. Yu H. Zhang |
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