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I'm working out some computations on Lie algebras $L$ of low dimensions (by which I mean $3, 4$ or $5$). For my purposes, it is convenient to choose an orthonormal basis $\{e_1, e_2, \ldots, e_n\}$, for which one has $$[e_i, e_j] = \sum_k \alpha_{ijk}e_k,$$ and then prescribe some nonzero $\alpha_{ijk}$ satisfying antisymmetry and Jacobi. However, I would like to make sure that I'm not doing my computations on the same Lie algebra twice when I decide to pick different constants. Hence my question:

Q: What are some easy-to-compute invariants that one can use to distinguish two Lie algebras when the structure constants $\alpha_{ijk}$ are nice, e.g., are all $\pm 1$ or zero?

As an example of what I'm looking for, if $L$ is a Lie algebra, the dimension of $[L, L]$ is fairly easy to compute if you have the structure constants, although it fails to distinguish quite a few examples.

L..
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  • The dimension of the center, the signature of the Killing form... there's lots of stuff to do depending on what kind of computations you're willing to do and how much structure theory of Lie algebras you know. Also, I don't know what you mean by "orthonormal" here. – Qiaochu Yuan Apr 24 '15 at 03:22
  • @QiaochuYuan I have a metric on my Lie algebra as well. That's why the orthonormal. – L.. Apr 24 '15 at 15:12

1 Answers1

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There are a lot of useful invariants, also if the structure constants are not so "nice". The Killing form is already quite useful. If it is non-degenerate, then the Lie algebra $L$ is semisimple, so that we know that $\dim(L)=3$, since the dimensions $4$ and $5$ are impossible then. If $L$ is nilpotent, then the Killing form is identically zero.
Furthermore there are several invariants for low-dimensional Lie algebras, which are used in the theory of degenerations and contractions (from physics); i.e., the maximal dimension of an abelian subalgebra, the dimension of the derivation algebra, and Kostrikin's covariants $$ c_{ij}(L)=\frac{tr ({\rm ad}(x))^i \cdot tr({\rm ad}(y)^j)}{tr({\rm ad}(x)^î{\rm ad}(y)^j)} $$ For more invariants, and a full discussion for all complex $4$-dimensional Lie algebras, see the paper here.

Dietrich Burde
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  • Thank you Dietrich. Do all the invariants you mention work for the real case as well? This is the case I had in mind, sorry for forgetting to specify. – L.. Apr 24 '15 at 15:15
  • Yes, no problem. If you want a reference for real Lie algebras, see here, and other papers by the authors. – Dietrich Burde Apr 24 '15 at 15:24