Résumé
In this thesis, we study the Lie algebra structure of the first Hochschild cohomology group H1(A,A) for a k-algebra A. This allows us also to examine the identity component of the algebraic group of outer automorphisms of A in characteristic zero.<br /><br />The first part is devoted to the study of the Lie algebra H1(A,A) of a monomial algebra A of finite dimension. This is done in terms of the combinatorics of the quiver of A, without any restriction on the characteristic of the field k. We show that the<br />semisimple Lie quotient of H1(A,A) by its radical is a product of Lie algebras pgl(n,k). Combinatoric criteria for the solvability, the (semi-)simplicity, the commutativity and the nilpotency are given.<br /><br />Next, we study the Lie algebra H1(kG,kG) of some group algebras for a field k of characteristic p>0. Thanks to a Morita equivalence given by Gabriel, we examine the case of finite groups admitting a normal cyclic Sylow p-subgroup. The Lie algebra H1(kG,kG) of finite abelian groups is studied using group cohomology. For p different from 2, the Lie algebra H1(kG,kG) is semisimple if and only if the Sylow p-subgroup of G is elementary. In this case, H1(kG,kG) is a product of Jacobson and Witt Lie<br />algebras. <br /><br />Finally, we consider the Lie algebra H1(TA,TA) of the trivial extension TA of an algebra A, in particular of a radical square zero algebra. In this case, the semisimple Lie quotient of H1(TA,TA) by its radical is a product of Lie algebras pgl(n,k) and so(2m,k). The Lie algebra H1(TA,TA) is never semisimple. This thesis ends with combinatoric criteria on the solvability and the commutativity of the Lie algebra H1(TA,TA).