On the spectral geometry of 4-dimensional Lorentzian Lie group

The main focus of this paper is concern to the study on the point-wise osserman structure on 4-dimensional lorentzian lie group. in this paper we study on the spectrum of the jacobi operator and spectrum of the skew-symmetric curvature operator on the non-abelian 4-dimensional lie group g, whenever g equipped with an orthonormal left invariant pseudo-riemannian metric g of signature (-;+;+; +), i.e, lorentzian metric, where e1 is a unit time-like vector. the lie algebra structure in dimension four has key role in our investigation, also in this case we study on the classification of 1-stein and mixed ip spaces. at the end we show that g does not admit any space form and einstein structures.