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: [[x_i+jx_i+j-1 ⋯ x_i-1]x_i+j-1], [[x_i+j ⋯ x_i] [x_i+j ⋯ x_ix_i-1]] 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).