Computação do corpo das invariantes de uma álgebra de Lie via folheações por curvas

Abstract

In this thesis, we present an algorithmic procedure for computing an algebraically independent generating set of the algebra of rational invariants of a nilpotent Lie algebra. We begin by introducing fundamental concepts of algebras. Next, we present the method of characteristics, an analytical tool used in solving partial differential equations, adapted to the algebraic context, and apply it to triangular derivations, obtaining an algebraically independent generating set of their kernels. We also address linear derivations, describing an effective procedure to compute an algebraically independent set that generates the kernels of these derivations. Finally, we use the developed theory to determine an algebraically independent generating set of the algebra of rational invariants of a nilpotent Lie algebra, demonstrating that, by considering a triangular basis of the Lie algebra, the adjoint derivation of an element of this basis, restricted to the kernel of the adjoint derivation of the previous element, results in a triangular derivation. Lastly, we present computational examples for solvable Lie algebras of dimensions up to four, using the techniques described for triangular and linear derivations.