2021/12/15 by Rekha Biswal, Biswal, Rekha, Susan J. Sierra +1
Mathematics · Physics and Astronomy · #16D30 #16P90 #17B10 #17B67 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Primary: 16S30 #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary 17B65
paper · pdf · doi:10.48550/arxiv.2112.08334
openalex publication_date 2021/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let L be an affine Kac-Moody algebra, with central element c, and let λ∈ \mathbb C. We study two-sided ideals in the central quotient Uλ(L):= U(L)/(c-λ) of the universal enveloping algebra of L, and prove: Theorem 1. If λ≠ 0 then Uλ(L) is simple. Theorem 2. The algebra U0(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 finite-dimensional simple Lie algebras have just-infinite growth, and prove similar results to Theorems 1 and 2 for quotients of symmetric algebras of these Lie algebras by Poisson ideals.