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 $ {\frak g}_{a} (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 $ {\frak g}_{c}(A^{[1]}) $ of A [1]. We show that there is a Lie algebra $ {\frak k} $ so that both $ {\frak g}_{c}(A^{[1]}) $ and $ {\frak g}_{a}(A^{[1]}) $ are homomorphic images of $ \frak k $ with finitely generated kernels.