Parallel skew-symmetric tensors on 4-dimensional metric Lie algebras

TL;DR

This paper classifies 4-dimensional metric Lie algebras that admit non-zero parallel skew-symmetric endomorphisms, distinguishing those with complex structure multiples and analyzing their geometric decompositions.

Contribution

It provides a complete classification of such Lie algebras up to isometric isomorphism and scaling, including their de Rham decompositions and irreducibility conditions.

Findings

Classification of 4D metric Lie algebras with parallel skew-symmetric endomorphisms

Identification of cases where endomorphisms are multiples of complex structures

De Rham decomposition and irreducibility results for associated Lie groups

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.