Closed \({\text {G}}_{2}\)-structures on non-solvable Lie groups

  • Anna Fino
  • Alberto RafferoEmail author


We investigate the existence of left-invariant closed G\(_2\)-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a structure exists only when the semisimple part is isomorphic to \(\mathfrak {sl}(2,{\mathbb R})\) and the radical is unimodular and centerless. Moreover, we classify unimodular Lie algebras with non-trivial Levi decomposition admitting closed G\(_2\)-structures.


Closed \(\mathrm{G}_2\)-structure Non-solvable Lie group Levi decomposition 

Mathematics Subject Classification

53C10 53C30 



The authors would like to thank Fabio Podestà for useful conversations, and the two anonymous referees for their valuable comments. This work was done when A. R. was a postdoctoral fellow at the Department of Mathematics and Computer Science “U. Dini” of the Università degli Studi di Firenze.


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© Universidad Complutense de Madrid 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica “G. Peano”Università degli Studi di TorinoTurinItaly

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