Exponential growth of Lie algebras of finite global dimension

Abstract:

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