Resumo

The classification of finite-dimensional semisimple Lie algebras in characteristic \(0\) represents one of the significant achievements in algebra during the first half of the 20th century. This classification was developed by Killing and by Cartan. According to the Killing–Cartan classification, the isomorphism classes of simple Lie algebras over an algebraically closed field of characteristic zero correspond one-to-one with irreducible root systems. In the infinite-dimensional case the situation is more complicated, and the so-called algebras of Cartan type appear. It is somewhat surprising that graded identities for Lie algebras have been relatively few results to that extent. In this presentation, we will discuss some of the results obtained thus far and introduce an algorithm capable of generating a basis for all graded identities in Lie algebras with Cartan gradings. Specifically, over any infinite field, we will apply this algorithm to establish a basis for all graded identities of \(U_1\), the Lie algebra of derivations of the algebra of Laurent polynomials \(K[t, t^{-1}]\), and demonstrate that they do not admit any finite basis. The findings discussed in this presentation are joint works with P. Koshlukov (UNICAMP).