# Preprojective algebras and MV polytopes

@article{Baumann2010PreprojectiveAA,
title={Preprojective algebras and MV polytopes},
author={Pierre Baumann and Joel Kamnitzer},
journal={arXiv: Representation Theory},
year={2010}
}
• Published 13 September 2010
• Mathematics
• arXiv: Representation Theory
The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.
Modules with 1-Dimensional Socle and Components of Lusztig Quiver Varieties in Type A
• Mathematics
• 2011
We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they
Classifying tilting complexes over preprojective algebras of Dynkin type
• Mathematics
• 2015
We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding
A relation between Mirkovic-Vilonen cycles and modules over preprojective algebra of Dynkin quiver of type ADE
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie
Torsion pairs for quivers and the Weyl groups
• Mathematics
Selecta Mathematica
• 2020
We give an interpretation of the map $$\pi ^c$$ π c defined by Reading, which is a map from the elements of a Coxeter group to the c -sortable elements, in terms of the representation theory of
Quotient closed subcategories of quiver representations
• Mathematics
Compositio Mathematica
• 2014
Abstract Let $Q$ be a finite quiver without oriented cycles, and let $k$ be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in
Crystal Combinatorics and Geometric Satake
This is an REU paper written for the University of Chicago REU, summer 2017. The main purpose of this note is to collect some of the many combinatorial models for MV cycles that exist in the
On deformed preprojective algebras
• Mathematics
Journal of Pure and Applied Algebra
• 2022
Homological description of crystal structures on Lusztig's quiver varieties
Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite
Prospecies of algebras I: Basic properties
In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing

## References

SHOWING 1-10 OF 35 REFERENCES
Modules with 1-Dimensional Socle and Components of Lusztig Quiver Varieties in Type A
• Mathematics
• 2011
We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they
Geometric construction of crystal bases
• Mathematics
• 1996
We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As
Crystal bases and quiver varieties
Abstract. We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One can show that, as a crystal, it is isomorphic to the crystal base of
A polytope calculus for semisimple groups
We define a collection of polytopes associated to a semisimple group G. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain
The Preprojective Algebra of a Quiver
The preprojective algebra P k (Q) of a quiver Q plays an important role in mathematics. We are going to present some descriptions of these algebras and their module categories which seem to be
Mirkovic-Vilonen cycles and polytopes
We give an explicit description of the Mirkovie-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We
Total positivity in Schubert varieties
• Mathematics
• 1997
Abstract. We extend the results of [2] on totally positive matrices to totally positive elements in arbitrary semisimple groups.
Representation Theory of Artin Algebras: Contents
• Mathematics
• 1995
1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary
Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras
Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of
On the exceptional fibres of Kleinian singularities
We give a new proof, avoiding case-by-case analysis, of a theorem of Y. Ito and I. Nakamura which provides a module-theoretic interpretation of the bijection between the irreducible components of the