We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space H(g, k) for certain Lie algebras g. Among these Lie algebras are filiform CNLAs of dimension n ≤ 14. It turns out that there are many examples of CNLAs which admit a symplectic structure. A generalization of a sympletic structure is an affine structure on a Lie algebra.

The minimization control problem of quadratic functionals for the class of affine nonlinear systems with the hypothesis of nilpotent associated Lie algebra is analyzed. The optimal control corresponding to the first-, second-, and third-order nilpotent operators is determined. In this paper, we have considered the minimum fuel problem for the multi-input nilpotent control and for a scalar input...

We provide explicit formulas for the number of ad-nilpotent ideals of a Borel subalgebra of a complex simple Lie algebra having fixed class of nilpotence.

We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.

We associate to each nilpotent orbit of a real semisimple Lie algebra go a weighted Vogan diagram, that is a Dynkin diagram with an involution of the diagram, a subset of painted nodes and a weight for each node. Every nilpotent element of go is noticed in some subalgebra of go. In this paper we characterize the weighted Vogan diagrams associated to orbits of noticed nilpotent elements.

We consider the shape of balls for nilpotent Lie groups endowed with a left invariant Riemannian or sub-Riemannian metric. We prove that when the algebra is graded these balls are homeomorphic to the standard Euclidean ball. For two-step nilpotent groups we show that the intersections of the ball with the central cosets are star-shaped, and in special cases convex.

On the Automorphism Tower of Free Nilpotent Groups Martin Dimitrov Kassabov 2003 In this thesis I study the automorphism tower of free nilpo-tent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. Th...