Classification of Minimal Algebras over any Field up to Dimension 6

2010/01/21 by Giovanni Bazzoni, Vicente Muñoz, Bazzoni, Giovanni +1 · 1 citation
Mathematics · #17B30 (Primary) #22E25 (Secondary) #55P62 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:17B30 #msc:22E25 #msc:55P62

paper · pdf · doi:10.48550/arxiv.1001.3860

19 pages. Fully revised version. To appear in Transactions of the AMS

openalex publication_date 2010/01/21 · arxiv created 2010/09/18 · arxiv updated 2010/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

Citations

• Complex structures on nilpotent Lie algebras
• On the Cohomology of Compact Homogeneous Spaces of Nilpotent Lie Groups

Cited by

• 8-dimensional 2-step nilpotent Lie algebras over algebraically closed fields of char ≠ 2, 3

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