ALGEBRAIC METHODS IN DYNAMICAL SYSTEMS BANACH CENTER PUBLICATIONS, VOLUME 94 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2011 DRINFELD–SOKOLOV HIERARCHIES ON TRUNCATED CURRENT LIE ALGEBRAS PAOLO CASATI Dipartimento di matematica e applicazioni, Universit` a di Milano-Bicocca Via Cozzi 53, I-20125 Milano, Italy E-mail: paolo.casati@unimib.it Abstract. In this paper we construct on truncated current Lie algebras integrable hierarchies of partial differential equations, which generalize the Drinfeld–Sokolov hierarchies defined on Kac–Moody Lie algebras. 1. Introduction. Surely the discovery made by Drinfeld and Sokolov in their famous and celebrated paper [9], that to any affine Kac–Moody Lie algebra and to any vertex of its (extended) Dynkin diagram corresponds a hierarchy of completely integrable non- linear partial differential equations, is a hallmark in the theory of integrable systems. These hierarchies (usually called Drinfeld–Sokolov hierarchies) have been further inten- sively studied in the last twenty years by many mathematicians with the help of various techniques (see [8], [11], [1], [4], [5], [15], [3], [27], [10], [16]). The aim of this paper is to show how some generalizations of such hierarchies live on truncated current Lie algebras defined on affine Kac–Moody algebras. Roughly we shall proceed along the following lines. First we shall define, on the mentioned truncated current Lie algebras, Lax equations which are invariant with respect to a gauge action of a subgroup of the corresponding truncated current Lie groups. Then we shall show how such Lax equations give rise to integrable systems with an infinite number of conserved quantities. Apart from the intrinsic interest of the generalizations considered here, the main output of such a construction is to obtain coupled integrable systems, which in recent years, have received considerable attention (see [6], [7], [17], [18], [19], [20], [21], [22], [23], [24], [28]). 2010 Mathematics Subject Classification : Primary 17B69, 37K10; Secondary 37K30. Key words and phrases : Kac–Moody Lie algebras, partial differential equations. The paper is in final form and no version of it will be published elsewhere. DOI: 10.4064/bc94-0-9 [163] c  Instytut Matematyczny PAN, 2011