Lie algebras of an affinization of a generalized Cartan matrix

Abstract

Let A[1] be a 1-fold affinization of a generalized Cartan matrix A of finite or affine type so that A[1] has two identical columns and two identical rows. One can associate to A[1] a Lie algebra ga(A[1]) which is isomorphic to either the u.c.a. of a loop algebra or a 2-toroidal Lie algebra. On the other hand, there is also the radical free contragredient Lie algebra gc(A[1]) of A[1]. We show that there is a Lie algebra f so that both gc(A[1]) and ga(A[1]) are homomorphic images of f with finitely generated kernels.