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

Abstract

A Gröbner-Shirshov basis for the Lie algebra An, abstractly defined by generators hi, xi, yi, i=1,..., n and the Serre relations for the Cartan matrix An, over a field k of characteristic ≠2 is constructed. It consists of the Serre relations for An together with the following relations: [Formula: see text] with j≥1, i≥2, i+j≤n and the same relations for y1,…, yn, where by [z1z2…zm] we mean [z1[z2… zm]]. As an application we get a direct proof that An, as defined, is isomorphic to sℓn+1(k).