Exponential growth of Lie algebras of finite global dimension

Abstract

Let L L be a connected finite type graded Lie algebra. If dim L = ∞ L = \infty and gldim L > ∞ \, L>\infty , then log index L = α > 0 \, L=\alpha >0 . If, moreover, α > ∞ \alpha >\infty , then for some d d , ∑ i = 1 d − 1 \textrm {dim} L k + i = e k α k , \sum _{i=1}^{d-1} \mbox {\textrm {dim}}\, L_{k+i} = e^{k\alpha _k}\,,\,\, where α k → \alpha _k \to log index L L as k → ∞ . k\to \infty \,.