in this paper, we classify the indecomposable non-nilpotent solvable lie algebras with $n(r_n,m,r)$ nilradical,by using the derivation algebra and the automorphism group of $n(r_n,m,r)$.we also prove that these solvable lie algebras are complete and unique, up to isomorphism.

Consider a diierential equation y 0 = A(t; y)y; y(0) = y0 with y0 2 G and A : R + G ! g, where g is a Lie algebra of the matricial Lie group G. Every B 2 g can be mapped to G by the matrix exponential map exp (tB) with t 2 R. Most numerical methods for solving ordinary diierential equations (ODEs) on Lie groups are based on the idea of representing the approximation yn of the exact solution y(t...

The Lie-group approach to the perturbative renormalization group (RG) method is developed to obtain an asymptotic solutions of both autonomous and non-autonomous ordinary differential equations. Reduction of some partial differetial equations to typical RG equations is also achieved by this approach and a simple recipe for providing RG equations is presented.

In this work we deal with coverings and actions of Lie groupgroupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-groupoids on Lie groups and the category of smooth coverings of Lie group-groupoids.

We construct a Polish group with an invariant metric in which Lie sums and Lie brackets do not exist. The construction of the group and the proof of the main theorem use some facts of combinatorial nature about the free group with two generators equipped with a Graev metric.

Let ‖| · ‖| be any give unitarily invariant norm. We obtain some exponential relations in the context of semisimple Lie group. On one hand they extend the inequalities (1) ‖|e‖| ≤ ‖|eReA‖| for all A ∈ Cn×n, where ReA denotes the Hermitian part of A, and (2) ‖|e‖| ≤ ‖|ee‖|, where A and B are n×n Hermitian matrices. On the other hand, the inequalities of Weyl, Ky Fan, Golden-Thompson, Lenard-Thom...