Symmetries of q-characters of representations of quantum affine algebras

In this thesis we address the q-characters of representations of quantum affine algebras, especially their symmetries under the Weyl group action defined recently by Frenkel and Hernandez. This subject gives us the opportunity to study the theory of characters in the classical setting of semisimple Lie algebras and the representation theory of quantum affine algebras. The dissertation is organized as follows. In Chapter 1 we collect the required notations to deal with simple Lie algebras and we prove the main results concerning their characters. In Chapter 2 we present the representation theory of quantized universal enveloping algebras and of quantum affine algebras, emphasizing the differences at level of category of finite-dimensional modules between the classical setting and the quantum setting. We give the definition of the q-character morphism and we prove many of its properties. In Chapter 3 we follow the construction by Frenkel and Hernandez of the action of the Weyl group on the target ring of q-characters, trying to make it clearer for a reader at a first approach to this theory. We deal especially with the proofs for Lie type A1 × A1 and A2. In Chapter 4 we present the proof of the result about the W-invariance of the image of q-characters. To do so, we introduce the screening operators. We also provide two appendices to complete the overview on characters. In Appendix A we address the formulas by Freudenthal, Konstant and Weyl to compute characters and multiplicities of g-modules. In Appendix B we construct the q-character using the notion of R-matrix. Throughout the thesis we provide examples and step-by-step computations to clarify the content. This work has the intent to be a self-contained guide to the studying of q-characters and to the reading of the article by Frenkel and Hernandez.