[Math] Complete classification of six dimensional non-semi simple Lie algebra

lie-algebraslie-groupsroot-systemsrt.representation-theory

I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$).

In this paper, in page 7, it stated that: "There is no complete classification of the six-dimensional real Lie algebras. However, all nilpotent six-dimensional Lie algebras are known." I also found this paper: J.Math.Phys. 17 (1976) 986, which lists nilpotent six-dimensional Lie algebras in Table III, p.991.

My question, again, is that: whether complete classification of 6 dimensional non-semi simple Lie algebra is known; for both real/complex non-semi simple Lie algebra of dimension 6? What is the most complete result? (In which paper/ref can I find a table of their Lie algebra?) Is that Table III a complete or incomplete for all real/complex non-semi simple Lie algebra of dimension 6?

Deep appreciation to whom concern and reply. (My question is motivated by a problem in topological field theory.)

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Best Answer

I am considering the complex case.

The list you wrote is complete for the NILPOTENT Lie algebras and was first obtained by Morozov in 1958 (paper in Russian). To obtain the calssification of ALL Lie algebras of dimension 6 one has to consider two more case:

  • non solvable

  • solvable but not nilpotent

As for the non solvable case there are only three possibilities:

1) ${\mathfrak sl}_2(\mathbb C)\oplus {\mathfrak sl}_2(\mathbb C)$

2) ${\mathfrak sl}_2(\mathbb C)\ltimes \mathbb C^3$ where the semidirect product is given by the adjoint action;

3) ${\mathfrak sl}_2(\mathbb C)\ltimes {\mathfrak n}_3(\mathbb C)$ where the semidirect product is by a sutable action on the Heisenberg -Lie algebra ${\mathfrak n}_3(\mathbb C)$ (3-dimensional, nilpotent).

Then you need to classify the solvable ones. This was achieved by Yamaguchi Mem. Fac. Sci. Kyushu Univ. Ser. A 35, pages 341-351 (1981). Some more informations are contained in a paper by Goze-Ancochea Bermudez (1985) where the variety of Lie algebras is studied as an algebraic variety for dimension up to 7. The relevant information connected to Clark's comment, is the number of irreducible components.

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