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Purely coclosed G\(_{\mathbf {2}}\)-structures on 2-step nilpotent Lie groups


We consider left-invariant (purely) coclosed G\(_2\)-structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by left-invariant purely coclosed G\(_2\)-structures. Then, we use them to determine the isomorphism classes of 2-step nilpotent Lie algebras admitting such type of structures. As an intermediate step, we show that every metric on a 2-step nilpotent Lie algebra admitting coclosed G\(_2\)-structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G\(_2\)-structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism.

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A.R. was supported by GNSAGA of INdAM and by the project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and Holomorphic Dynamics”. Part of this work was done during a visit of A.R. to the Laboratoire de Mathématiques d’Orsay of the Université Paris-Saclay. He is grateful to the LMO for the hospitality. This work was prepared while V.d.B was working at the Laboratoire de Mathématiques d’Orsay.

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Appendix A.: The classification of 7-dimensional 2-step nilpotent Lie algebras

Appendix A.: The classification of 7-dimensional 2-step nilpotent Lie algebras

The isomorphism classes of 7-dimensional nilpotent Lie algebras were determined in [14]. Here, we recall the classification of those that are 2-step nilpotent.

The notation we use is consistent with [14]: \(\mathfrak {n}_{n,t}\) or \(\mathfrak {n}_{n,t,\bullet }\) means that the Lie algebra has dimension n and derived algebra of dimension t, while different capital letters in the third argument are used to distinguish non-isomorphic Lie algebras whose derived algebras have the same dimension. We also denote by \(\mathfrak h _n\) the Heisenberg Lie algebra of dimension n and by \(\mathfrak {h}_3^{\mathbb {C}}\) the real Lie algebra underlying the complex Heisenberg Lie algebra.

For each Lie algebra \(\mathfrak {n}\), the structure equations are written with respect to a basis \(\{f^1,\ldots ,f^7\}\) of the dual Lie algebra \(\mathfrak {n}^*\).

\(\bullet \):

7-dimensional 2-step nilpotent Lie algebras \(\mathfrak n \) with \(\dim (\mathfrak n ')=1\):

$$\begin{aligned} \mathfrak {h}_3\oplus \mathbb {R}^4= & {} \left( 0,0,0,0,0,0,f^{12}\right) ,\\ \mathfrak {h}_5\oplus \mathbb {R}^2= & {} \left( 0,0,0,0,0,0,f^{12}+f^{34}\right) ,\\ \mathfrak h _7= & {} \left( 0,0,0,0,0,0,f^{12}+f^{34}+f^{56}\right) . \end{aligned}$$

The Heisenberg Lie algebra \(\mathfrak h _7\) is the only indecomposable one in the above list.

\(\bullet \):

7-dimensional 2-step nilpotent Lie algebras \(\mathfrak n \) with \(\dim (\mathfrak n ')=2\):

$$\begin{aligned} \mathfrak {n}_{5,2}\oplus \mathbb {R}^2= & {} \left( 0,0,0,0,f^{12},f^{13},0\right) ,\\ \mathfrak {h}_3\oplus \mathfrak {h}_3\oplus \mathbb {R}= & {} \left( 0,0,0,0,f^{12},f^{34},0\right) ,\\ \mathfrak {h}_3^{\mathbb {C}}\oplus \mathbb {R}= & {} \left( 0,0,0,0,f^{13}-f^{24},f^{14}+f^{23},0\right) ,\\ \mathfrak {n}_{6,2}\oplus \mathbb {R}= & {} \left( 0,0,0,0,f^{12},f^{14}+f^{23},0\right) ,\\ \mathfrak {n}_{7,2,A}= & {} \left( 0,0,0,0,0,f^{12},f^{14}+f^{35}\right) ,\\ \mathfrak {n}_{7,2,B}= & {} \left( 0,0,0,0,0,f^{12}+f^{34},f^{15}+f^{23}\right) . \end{aligned}$$

The only indecomposable Lie algebras in the above list are \(\mathfrak {n}_{7,2,A}\) and \(\mathfrak {n}_{7,2,B}\).

\(\bullet \):

7-dimensional 2-step nilpotent Lie algebras \(\mathfrak n \) with \(\dim (\mathfrak n ')=3\):

$$\begin{aligned} \mathfrak {n}_{6,3}\oplus \mathbb {R}= & {} \left( 0,0,0,0,f^{12},f^{13},f^{23}\right) ,\\ \mathfrak {n}_{7,3,A}= & {} \left( 0,0,0,0,f^{12},f^{23},f^{24}\right) ,\\ \mathfrak {n}_{7,3,B}= & {} \left( 0,0,0,0,f^{12},f^{23},f^{34}\right) ,\\ \mathfrak {n}_{7,3,B_1}= & {} \left( 0,0,0,0,f^{12}-f^{34},f^{13}+f^{24},f^{14}\right) \\ \mathfrak {n}_{7,3,C}= & {} \left( 0,0,0,0,f^{12}+f^{34},f^{23},f^{24}\right) ,\\ \mathfrak {n}_{7,3,D}= & {} \left( 0,0,0,0,f^{12}+f^{34},f^{13},f^{24}\right) ,\\ \mathfrak {n}_{7,3,D_1}= & {} \left( 0,0,0,0,f^{12}-f^{34},f^{13}+f^{24},f^{14}-f^{23}\right) . \end{aligned}$$

The only decomposable Lie algebra in the above list is \(\mathfrak {n}_{6,3}\oplus \mathbb {R}\).

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del Barco, V., Moroianu, A. & Raffero, A. Purely coclosed G\(_{\mathbf {2}}\)-structures on 2-step nilpotent Lie groups. Rev Mat Complut 35, 323–359 (2022).

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  • Purely coclosed G\(_2\)-structure
  • 2-Step nilpotent Lie algebra
  • Metric Lie algebra
  • G\(_2\)-Strominger system

Mathematics Subject Classification

  • 53C15
  • 22E25
  • 53C30