arXiv subjects

Shane Rankin

Publications and source records attributed to Shane Rankin.

4 recordsLinked to original sources

arXiv2026

The curvature of left-invariant magnetic systems

We compute the magnetic curvatures (in the sense of Assenza) of left-invariant magnetic systems on Lie groups and explore relations between curvature properties and algebraic properties, extending to the magnetic setting a number of results from Milnor's classic 1976 paper on the geometry of left-invariant metrics. We also discuss the existence of non-trivial bi-invariant magnetic systems and exhibit the resulting curvature formulas in some concrete examples, including the Heisenberg group.

math.DG↗

arXiv2026

Algebroid Desingularizable Poisson Structures

We introduce algebroid desingularizable Poisson manifolds, a class of Poisson manifolds induced by symplectic Lie algebroids with almost-injective anchors, generalizing structures including log-symplectic, $b^m$-symplectic, $E$-symplectic manifolds, and hypersurface algebroids. We give an infinitesimal obstruction to the existence of such an algebroid for a general Poisson manifold, and then characterize the linear case by showing that the dual of a real finite-dimensional Lie algebra, equipped with the KKS Poisson structure is desingularizable if and only if it possesses an abelian ideal of dimension $\dim(\mathfrak{g})-r$, where $2r$ is the maximal coadjoint orbit dimension.

math.DG↗

arXiv2025

The Hard Lefschetz Theorem on K\"ahler Lie Algebroids

Compact K\"ahler manifolds classically satisfy the Hard Lefschetz Theorem, which gives strong control on the underlying topology of the manifold. One expects a similar theorem to be true for K\"ahler Lie Algebroids, and we show for a certain class of them that this is indeed true, with an added ellipticity requirement. We provide examples of Lie Algebroids satisfying this, as well as an example of a K\"ahler Lie Algebroid that does not meet this Ellipticity requirement, and consequently fails to satisfy the Hard Lefschetz condition.

math.DG↗

arXiv2024

Symplectic Hodge Theory on Lie Algebroids

We explore the natural analogues of the Brylinksi condition, Strong Lefschetz condition, and $d\delta$-lemma in Symplectic Geometry originally explored by Brylinksi, Mathieu, Yan, and Guillemin in the Symplectic Lie Algebroid case. The equivalence of the three conditions is re-established as a purely algebraic statement along with a primitive notion of the $d\delta$-lemma established by Tseng, Yau, and Ho. We then apply this algebraic theory to the desired geometric setting to show that these analogues hold, and finally provide a few classes of examples.

math.SG↗