Revista de la
Unión Matemática Argentina
Parallel skew-symmetric tensors on 4-dimensional metric Lie algebras
Andrea C. Herrera

Volume 65, no. 2 (2023), pp. 295–311    

Published online: November 7, 2023

https://doi.org/10.33044/revuma.2451

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Abstract

We give a complete classification, up to isometric isomorphism and scaling, of 4-dimensional metric Lie algebras $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ that admit a non-zero parallel skew-symmetric endomorphism. In particular, we distinguish those metric Lie algebras that admit such an endomorphism which is not a multiple of a complex structure, and for each of them we obtain the de Rham decomposition of the associated simply connected Lie group with the corresponding left invariant metric. On the other hand, we find that the associated simply connected Lie group is irreducible as a Riemannian manifold for those metric Lie algebras where each parallel skew-symmetric endomorphism is a multiple of a complex structure.

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