Geometry of Umbrella Matrices in Lorentz Space

In this study, umbrella matrices in Lorentz space are examined for the first time in the context of Lie groups and Lie algebra. The relationship between the Euclidean and Lorentz Umbrella Lie groups is presented through a different approach, and a new dual transformation is defined between their Lie Algebra. In addition, the characterization of double-umbrella matrices is obtained using a different Lorentzian metric.