SERRE RELATIONS AND GRÖBNER-SHIRSHOV BASES FOR SIMPLE LIE ALGEBRAS I

A Gröbner-Shirshov basis for the Lie algebra A_n, abstractly defined by generators h_i, x_i, y_i, i=1,..., n and the Serre relations for the Cartan matrix A_n, over a field k of characteristic ≠2 is constructed. It consists of the Serre relations for A_n together with the following relations:

with j≥1, i≥2, i+j≤n and the same relations for y₁,…, y_n, where by [z₁z₂…z_m] we mean [z₁[z₂… z_m]]. As an application we get a direct proof that A_n, as defined, is isomorphic to sℓ_n+1(k).