A Characterization of Certain Loop Lie Algebras

In this paper, we study Z × Z-graded Lie algebras A = i, j Z Ai, j with dim Ai, j ≤ 1 satisfying• (I) dim = dim = 1, and is generated by , ;• (II) sl;• (III) [, ] = 0, and ad act faithfully on .We show that A is necessarily isomorphic to one of the two smallest loop algebras F[t, t − 1] sl2 and l(2)(3) with appropriate grading.