Abstract
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∈Ns 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,⋅) 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))∼|G|E(−)(G,m).
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