Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Classification of minimal algebras over any field up to dimension $ 6$

Authors: Giovanni Bazzoni and Vicente Muñoz
Journal: Trans. Amer. Math. Soc. 364 (2012), 1007-1028
MSC (2010): Primary 55P62, 17B30; Secondary 22E25
Published electronically: September 15, 2011
MathSciNet review: 2846361
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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.

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Giovanni Bazzoni
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolas Cabrera 13-15, 28049 Madrid, Spain

Vicente Muñoz
Affiliation: Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain

Keywords: Nilmanifolds, rational homotopy, nilpotent Lie algebras, minimal model.
Received by editor(s): May 28, 2010
Received by editor(s) in revised form: September 16, 2010
Published electronically: September 15, 2011
Additional Notes: This research was partially supported by Spanish grant MICINN ref. MTM2007-63582.
Article copyright: © Copyright 2011 American Mathematical Society