# Low dimensional nilpotent Lie algebras

This is a reference request question. I would like to know more on the structure of low dimensional nilpotent lie algebras. I heard that up to dimension 6 there are only finitely many isomorphism classes, and every such algebra admits a gradation with only positive degrees (see http://en.wikipedia.org/wiki/Graded_Lie_algebra). Do you know of any source where I can find the corresponding proofs?

Classification of nilpotent Lie algebras in characteristic 0 is an old problem, with a lot of literature. For the dimensions up to 6 there is a finite list. Among the many relevant papers on MathSciNet, I'll list just a few:

MR2372566 (2009a:17027) 17B50 (17B20 17B30) Strade, H. (D-HAMBMI) Lie algebras of small dimension. Lie algebras, vertex operator algebras and their applications, 233–265, Contemp. Math., 442, Amer. Math. Soc., Providence, RI, 2007.

MR0498734 (58 #16802) 17B30 Skjelbred, Tor; Sund, Terje Sur la classification des algebres de Lie nilpotentes. (French. English summary) C. R. Acad. Sci. Paris S´er. A-B 286 (1978), no. 5, A241–A242.

MR855573 (87k:17012) 17B30 Magnin, L. (F-DJON-P) Sur les algebres de Lie nilpotentes de dimension 7. (French. English summary) [Nilpotent Lie algebras of dimension 7] J. Geom. Phys. 3 (1986), no. 1, 119–144.

MR1737529 (2001i:17010) 17B30 (17B05) Tsagas, Gr. (GR-THESS-DMP) Classification of nilpotent Lie algebras of dimension eight. J. Inst. Math. Comput. Sci. Math. Ser. 12 (1999), no. 3, 179–183.

EDIT: This is a somewhat random sample (I'm not a specialist), but these papers recall results for low dimensions and have many references to older literature. The reviews in Math Reviews (MathSciNet) are helpful to look at, if you have access. There is also a fairly modern book, which is very high-priced and probably difficult to access:

MR1383588 (97e:17017) Goze, Michel(F-HALS); Khakimdjanov, Yusupdjan(UZ-AOS) Nilpotent Lie algebras. Mathematics and its Applications, 361. Kluwer Academic Publishers Group, Dordrecht, 1996. xvi+336 pp. ISBN: 0-7923-3932-0 17B30 (17-02 17B40 17B56)

There is indeed lot of works devoted to classification of nilpotent Lie algebras of low dimension (those cited above and many more), with numerous mistakes and omissions. Even worse, all they are using different nomenclature and invariants to classify the algebras, and it is a nontrivial task to compare different lists. Luckily, Willem de Graaf undertook a painstaking task to make an order out of this somewhat messy situation in "Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2", J. Algebra 309 (2007), 640-653 (http://dx.doi.org/10.1016/j.jalgebra.2006.08.006 ); arXiv:math/0511668 . Even better, he provides an algorithm for identifying any given nilpotent Lie algebra with one in his list, and the corresponding code is available as a part of GAP package. He builds on earlier work of Skjelbred-Sund cited above and his own method of identification of Lie algebras by means of Groebner bases.

• This is a helpful reference, since de Graaf is well grounded in the subject matter and has a lot of computational expertise as well. Note that his arXiv preprint is posted only in v1, while the published paper has an expanded reference list and presumably a number of other changes (he thanks the "referees" for improving the exposition). Note too that the arXiv subject is math.RA, which unfortunately I don't routinely consult. This is one case where the 17B in MathSciNet works better for me; but no system is perfect. – Jim Humphreys Apr 15 '10 at 13:16
• It seems there are two schools of thought: those who think that anything Lie-algebraic belongs to math.RT (the majority), and those who think it belongs to math.RA (unless some heavy-duty representation theory is not involved indeed). – Pasha Zusmanovich Apr 15 '10 at 18:10
• @Pasha: Probably the arXiv labels will need tweaking over time. Maybe add an LT (Lie theory) option? But like MathSciNet it is tricky to devise permanent classifications, since those that fall into disuse can't be recycled like old telephone numbers without messing up searches. My problem with arXiv is that I don't have time or energy to browse daily the bigger lists (CO, GR, RA, QA, etc.) so I focus on a few labels and trust people to cross-reference carefully when they post. This blog invents lots of new tags, maybe too many to keep track of. – Jim Humphreys Apr 16 '10 at 17:23
• @Jim: LT would be a nice idea, actually. QA, on the other hand, looks to me as an (almost) complete nonsense: it's typical usage by many people seems to be "anything fashionable with an algebraic flavor". I, too, find it a tough task to choose a suitable list of arXiv subjects for daily browsing. – Pasha Zusmanovich Apr 16 '10 at 20:25
• An updated comment: J. Algebra and other Elsevier journals now make older articles like this freely available. See sciencedirect.com/science/article/pii/S0021869306005254 – Jim Humphreys Aug 11 '17 at 21:16

Many articles on classification of low-dimensional Lie algebras do contain mistakes. To the best of my knowledge, the full detailed proof is provided in the dissertation of Ming-Peng Gong:

where he classifies all algebras up to dimension 7 over algebraically closed fields of any characteristics except 2, and also over reals.

• This is a useful reminder that countless dissertations around the world remain formally unpublished. Many of these do lead to papers, but for example Ming-Peng Gong has only some papers up to 1997 listed by MathSciNet. Many sources also contain small or large errors, even after being refereed. List-making in mathematics is popular but especially error-prone: e.g., Killing was the first to find all the simple Lie algebras over $\mathbb{C}$, but thought there were two versions of $F_4$. Is there a 100% reliable published list of nilpotent Lie algebras of dim $\leq 6$? I don't know. – Jim Humphreys Apr 13 '10 at 13:23
• @Jim: In the article quoted by Pasha below, Willem de Graaf claims that M.-P. Gong lost a one-parameter family of 6-dimensional algebras. If this is true, then his results on 7-dimensional algebras may be also incomplete, unfortunately. – mathreader Apr 21 '11 at 4:00

The paper Invariants of real low dimension Lie algebras (journal link) lists all real Lie algebras of dimension $\leq 5$ and all nilpotent of dimension $\leq 6$ along with its invariants. It also contains references to the papers where the classifications are obtained: work of Mubarakzyanov and Morozov.

In his 1957 paper Dixmier computes the centres of the universal enveloping of all f-d complex Lie algebras up to dimension 5 and, in particular, lists them all.