论文标题
4维谎言代数的平行偏度对称张量
Parallel skew-symmetric tensors on 4-dimensional metric Lie algebras
论文作者
论文摘要
我们给出一个完整的分类，达到等轴测异构和缩放，为$ 4 $维度度量lie代数$（\ mathfrak {g} {g}，\ langle \ cdot，\ cdot \ rangle）$，该$承认非零相平行的skew-kew-Mectymmetricmmetric Intomorphism。特别是，我们区分那些承认这种内态的指标谎言代数，它不是复杂结构的倍数，对于它们中的每个代数，我们都会获得与相应的左不变型指标的相关相关谎言组的De Rham分解。另一方面，我们发现相关的简单连接的谎言组是不可记述的，因为那些公制的谎言代数是riemannian的歧管，其中每个平行偏斜的内态性是复杂结构的倍数。
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.
