This thesis is about extended affine Lie algebras and extended affine Weyl groups. In Chapter I, we provide the basic knowledge necessary for the study of extended affine Lie algebras and related objects. In Chapter II, we show that the well-known twisting phenomena which appears in the realization of the twisted affine Lie algebras can be extended to the class of extended affine Lie algebras, in the sense that some extended affine Lie algebras (in particular nonsimply laced extended affine Lie algebras) can be realized as fixed point subalgebras of some other extended affine Lie algebras (in particular simply laced extended affine Lie algebras) relative to some finite order automorphism. We show that extended affine Lie algebras of type A₁, B, C and BC can be realized as twisted subalgebras of types A_§¤(l ¡Ã 2) and D algebras. Also we show that extended affine Lie algebras of type BC can be realized as twisted subalgebras of type C algebras. In Chapter III, the last chapter, we study the Weyl groups of reduced extended affine root systems. We start by describing the extended affine Weyl group as a semidirect product of a finite Weyl group and a Heisenberg-like normal subgroup. This provides a unique expression for the Weyl group elements which in turn leads to a presentation of the Weyl group, called a presentation by conjugation. Using a new notion, called the index, which is an invariant of the extended affine root systems, we show that one of the important features of finite and affine root systems (related to Weyl group) holds for the class of extended affine root systems. We also show that extended affine Weyl groups (of index zero) are homomorphic images of some indefinite Weyl groups where the homomorphism and its kernel are given explicitly.

Cette thèse traite du calcul et des applications des fonctions tau des hiérarchies de Drinfeld–Sokolov introduites en 1984. Les hiérarchies de Drinfeld–Sokolov sont des suites d’équations aux dérivées partielles intégrables que l’on associe à n’importe quelle algèbre de Lie semi simple. La fonction tau est une fonction associée à toute solution d’une hiérarchie donnée et qui contient toute l’information de la solution. Les fonctions tau sont au cœur des liens qui unissent les hiérarchies de Drinfeld–Sokolov et la géométrie algébrique. Au chapitre 3, nous établissons une transformation explicite entre les fonctions tau polynomiales de la hiérarchie de Korteweg–de Vries (associée à l’algèbre sl(2,C)) et les polynômes d’Adler–Moser (1978). Ces derniers forment une suite de polynômes satisfaisant une certaine relation de récurrence différentielle. Le chapitre 4 traite du calcul des fonctions tau polynomiales par les déterminants de Toeplitz ; une méthode introduite par Cafasso et Wu (2015). En collaboration avec Cafasso et Yang, nous avons obtenu une expansion de la fonction tau en une somme sur les partitions d’entiers. Nous en déduisons un critère de polynomialité de la fonction tau et donnons quelques exemples non triviaux. Au chapitre 5, en collaboration avec Paolo Rossi, nous confirmons la conjecture dite « DR/DZ forte » dans le cas de l’algèbre de Lie simple o(8,C) (D4). Elle prévoit l’équivalence, en particulier, entre les hiérarchies de Drinfeld–Sokolov et d’autres hiérarchies dites de « double ramification, » introduite par Buryak (2015) et construites à partir de la cohomologie de l’espace de modules des courbes complexes stables Mg,n. / This thesis deals with the computation and applications of tau functions of the Drinfeld– Sokolov hierarchies introduced in 1984. The Drinfeld– Sokolov hierarchies are sequences of integrable partial differential equations which one associates to any semisimple Lie algebra. The tau function is a function associated to any solution of a given hierarchy and which contains all the information of the solution. Tau functions are at the heart of the bonds between Drinfeld–Sokolov hierarchies and algebraic geometry. In Chapter 3, we establish an explicit transformation between the polynomial tau functions of the Korteweg–de Vries hierarchy (associated to the algebra sl(2,C)) and the Adler–Moser polynomials (1978). The latter form a sequence of polynomials satisfying a certain differential recursion relation. Chapter 4 is dedicated to the computation of tau functions via Toeplitz determinants; a method introduced by Cafasso and Wu (2015). In collaboration with Cafasso and Yang, we obtained an expansion of the tau function as a sum over all integer partitions. It follows a simple criterion for the polynomiality of the tau function; we give some nontrivial examples. In Chapter 5, in collaboration with Paolo Rossi, we confirm the so-called ‘strong DR/DZ conjecture’ for the algebra o(8,C) (D4). The latter states an equivalence between, in particular, Drinfeld–Sokolov hierarchies and another kind of hierarchies called ‘the double ramification hierarchies’ introduced by Buryak (2015) and constructed from the cohomology of the moduli spaces of stables complex curves Mg,n.