Ideals in enveloping algebras of affineKac–Moody algebras

Let L be an affine Kac-Moody algebra, with central element c, and let .We study two-sided ideals in the central quotient U (L) := U (L)/(c) of the universal enveloping algebra of L and prove:(1) If = 0 then U (L) is simple.(2) The algebra U 0 (L) has just-infinite growth, in the sense that any proper quotient has polynomial growth.As an immediate corollary, we show that the annihilator of any nontrivial integrable highest-weight representation of L is centrally generated, extending a result of Chari for Verma modules.We also show that universal enveloping algebras of loop algebras and current algebras of finitedimensional simple Lie algebras have just-infinite growth, and prove similar results to the two results above for quotients of symmetric algebras of these Lie algebras by Poisson ideals.and is the unique (up to isomorphism) nontrivial central extension of the loop algebra g[t, t -1 ].To emphasize the relationship between L and g, we sometimes write L = g.