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?
[Math] Low dimensional nilpotent Lie algebras
lie-algebrasreference-request
Best Answer
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 alg`ebres 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 alg`ebres 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)