| In the present dissertation, we begin the study of the structure of the insertion elimination Lie algebras introduced by Alain Connes and Dirk Kreimer in their approach to the theory of renormalization of perturbative quantum field theories.; The insertion elimination Lie algebras are combinatorial Lie algebras that can be realized as Lie algebras of derivations of the Hopf algebra of rooted trees Hrt , or more generally, as derivations of the Hopf algebra of Feynman graphs. In the present work, we investigate the structure of a particular insertion elimination Lie algebra LL , called the ladder insertion elimination Lie algebra. This Lie algebra is the insertion elimination Lie algebra associated to the Hopf algebra HL generated by the ladder trees. LL is an infinite dimensional Z -graded Lie algebra, generated by Zn,m, n, m ≥ 0. It admits a triangular decomposition L+ ⊕ L0 ⊕ L- in terms of the positive, zero and negative degree components. The main result we prove concerns the structure of the Lie algebra LL : LL is a non-abelian extension by g l+(infinity) of a commutative Lie algebra. Moreover, we show that the center Z( LL ) is one dimensional, and it is the only abelian ideal, so that the quotient LL /Z( LL ) is a semisimple Lie algebra. In the final part, we report some results about the cohomology of LL . In particular, we prove that LL has infinitely many non equivalent central extensions. |
