English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

$k$-symplectic manifolds are a convenient framework to study classical fieldtheories and they are a generalization of polarized symplectic manifolds. Thispaper focus on the existence and the properties of left invariant$k$-symplectic structures on Lie groups having a bi-invariant pseudo-Riemannianmetric.  We show that compact semi-simple Lie groups and a large class of Lie groupshaving a bi-invariant pseudo-Riemannian metric does not carry any leftinvariant $k$-symplectic structure. This class contains the oscillator Liegroups which are the only solvable non abelian Lie groups having a bi-invariantLorentzian metric. However, we built a natural left invariant $n$-symplecticstructure on $\mathrm{SL}(n,\mathbb{R})$. Moreover, up to dimension 6, onlythree connected and simply connected Lie groups have a bi-invariantindecomposable pseudo-Riemannian metric and a left invariant k-symplecticstructure, namely, the universal covering of $\mathrm{SL}(2, \mathbb{R})$ witha 2-symplectic structure, the universal covering of the Lorentz group$\mathrm{SO}(3, 1)$ with a 2-symplectic structure, and a 2-step nilpotent6-dimensional connected and simply connected Lie group with both a 1-symplecticstructure and a 2-symplectic structure.

On the Existence and Properties of Left Invariant $k$-Symplectic Structures on Lie Groups with Bi-Invariant Pseudo-Riemannian Metric