Counting gradings on Lie algebras of block-triangular matrices

We study the number of isomorphism classes of gradings on Lie algebras of block-triangular matrices. Let G be a finite abelian group, for m is an element of Ns s we determine the number E ( - ) ( G, m) of isomorphism classes of elementary G-gradings on the Lie algebra UT (m) ( - ) of block-triangular matrices over an algebraically closed field of characteristic zero. We study the asymptotic growth of E ( - ) ( G, m) and as a consequence prove that the E ( - ) ( G, <middle dot>) determines G up to isomorphism. We also study the asymptotic growth of the number N ( - ) ( G, m) of isomorphism classes of G-gradings on UT (m) ( - ) and prove that N ( - ) ( G, m)) similar to|G|E(-)(G,m). G | E ( - ) ( G, m). (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.