On symplectic exact Lie groups

A finite dimensional Lie algebra $\mathcal{G}$ with Lie bracket $[.,.]$ over $\mathbb K,$ is called quasi-Frobenius or symplectic, if it possesses a nondegenerate skew-symmetric closed 2-form $\omega.$ If there exists a linear 1-form $\alpha$ on $\mathcal{G}$, such that $\omega (x,y)= \partial \alpha (x,y) := - \alpha ([x,y]),$ for all $x,y$ on $\mathcal{G}$, then $\mathcal{G}$ is called a Frobenius (or exact symplectic) Lie algebra and $\alpha$ a Frobenius functional. Quasi-Frobenius Lie algebras are also endowed with a natural left symmetric algebra (LSA) structure Frobenius Lie algebras appear in many research areas such as Poisson Geometry and Hamiltonian systems, invariant affine geometry, symplectic and Kähler geometry, homogeneous domains, contact geometry, deformation quantization, etc.

$\bullet$ We explore the geometric properties of principal elements of general Frobenius Lie algebras, from the viewpoint of invariant affine geometry, namely, by exploiting the induced structure of left symmetric algebra.

$\bullet$ A square matrix is called non-derogatory (or cyclic) if its (unitary) characteristic and minimal polynomials coincide. A non-derogatory matrix gives rise to a $2$-step solvable Lie algebra endowed with an exact symplectic structure (hence a family of locally isomorphic $2$-step solvable Lie groups possessing left invariant exact symplectic structures). We investigate the geometry of such Lie groups via their Lie algebras.