Simple and semisimple Lie algebras and codimension growth

Giambruno, Antonio; Regev, Amitai; Zaicev, Michail

doi:10.1090/S0002-9947-99-02419-8

Simple and semisimple Lie algebras and codimension growth
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by Antonio Giambruno, Amitai Regev and Michail V. Zaicev PDF

Trans. Amer. Math. Soc. 352 (2000), 1935-1946 Request permission

Abstract:

We study the exponential growth of the codimensions $c_ n^{L}(B)$ of a finite dimensional Lie algebra $B$ over a field of characteristic zero. In the case when $B$ is semisimple we show that $\lim _{n\to \infty } \sqrt [{n}]{c_ n^{L}(B)}$ exists and, when $F$ is algebraically closed, is equal to the dimension of the largest simple summand of $B$. As a result we characterize central-simplicity: $B$ is central simple if and only if $\dim B = \lim _{n\to \infty } \sqrt [{n}]{c_ n^{L} (B)}$.

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