Classification of minimal algebras over any field up to dimension 6

Bazzoni, Giovanni; Muñoz, Vicente

doi:10.1090/S0002-9947-2011-05471-1

Classification of minimal algebras over any field up to dimension $6$
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by Giovanni Bazzoni and Vicente Muñoz PDF

Trans. Amer. Math. Soc. 364 (2012), 1007-1028 Request permission

Abstract:

We give a classification of minimal algebras generated in degree $1$, defined over any field $\mathbf {k}$ of characteristic different from $2$, up to dimension $6$. This recovers the classification of nilpotent Lie algebras over $\mathbf {k}$ up to dimension $6$. In the case of a field $\mathbf {k}$ of characteristic zero, we obtain the classification of nilmanifolds of dimension less than or equal to $6$, up to $\mathbf {k}$-homotopy type. Finally, we determine which rational homotopy types of such nilmanifolds carry a symplectic structure.

References

Cerezo, A., Les algèbres de Lie nilpotentes réelles et complexes de dimension $6$, Prepublication J. Dieudonné, Univ. de Nice 27, 1983.
Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. MR 382702, DOI 10.1007/BF01389853
Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
Michel Goze and Yusupdjan Khakimdjanov, Nilpotent Lie algebras, Mathematics and its Applications, vol. 361, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1383588, DOI 10.1007/978-94-017-2432-6
Willem A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra 309 (2007), no. 2, 640–653. MR 2303198, DOI 10.1016/j.jalgebra.2006.08.006
Phillip A. Griffiths and John W. Morgan, Rational homotopy theory and differential forms, Progress in Mathematics, vol. 16, Birkhäuser, Boston, Mass., 1981. MR 641551
Stephen Halperin, Universal enveloping algebras and loop space homology, J. Pure Appl. Algebra 83 (1992), no. 3, 237–282. MR 1194839, DOI 10.1016/0022-4049(92)90046-I
Daniel Lehmann, Sur la généralisation d’un théorème de Tischler à certains feuilletages nilpotents, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 2, 177–189 (French). MR 535566
Gregory Lupton, The rational Toomer invariant and certain elliptic spaces, Lusternik-Schnirelmann category and related topics (South Hadley, MA, 2001) Contemp. Math., vol. 316, Amer. Math. Soc., Providence, RI, 2002, pp. 135–146. MR 1962160, DOI 10.1090/conm/316/05502
L. Magnin, Sur les algèbres de Lie nilpotentes de dimension $\leq 7$, J. Geom. Phys. 3 (1986), no. 1, 119–144 (French, with English summary). MR 855573, DOI 10.1016/0393-0440(86)90005-7
John W. Morgan, The algebraic topology of smooth algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 48 (1978), 137–204. MR 516917
Katsumi Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531–538. MR 64057, DOI 10.2307/1969716
Aleksy Tralle and John Oprea, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, vol. 1661, Springer-Verlag, Berlin, 1997. MR 1465676, DOI 10.1007/BFb0092608
S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2-3, 311–333. MR 1812058, DOI 10.1016/S0022-4049(00)00033-5
Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078