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Matrices de Cartan, bases distinguées et systèmes de Toda

Abstract : In this thesis, our goal is to study various aspects of root systems of simple Lie algebras. In the first part, we study the coordinates of the eigenvectors of the Cartan matrices. We start by generalizing the work of physicists who showed that the particle masses of the affine Toda field theory are equal to the coordinates of the Perron -- Frobenius eigenvector of the Cartan matrix. Then, we adopt another approach. Namely, using the ideas coming from the singularity theory, we compute the coordinates of the eigenvectors of some root systems. In the second part, inspired by Givental's ideas, we introduce q-deformations of Cartan matrices and we study their spectrum and their eigenvectors. Then, we propose a q-deformation of Toda's equations et compute 1-solitons solutions, using the Hirota's method and Hollowood's work. Finally, our interest is focused on a set of transformations which induce an action of the braid group on the set of ordered root basis. In particular, we study an orbit for this action, the set of distinguished basis and some associated matrices.
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Submitted on : Thursday, August 30, 2018 - 2:27:07 PM
Last modification on : Saturday, August 15, 2020 - 3:56:52 AM


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  • HAL Id : tel-01831978, version 2


Laura Brillon. Matrices de Cartan, bases distinguées et systèmes de Toda. Géométrie algébrique [math.AG]. Université Paul Sabatier - Toulouse III, 2017. Français. ⟨NNT : 2017TOU30077⟩. ⟨tel-01831978v2⟩



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