Pith. sign in

REVIEW 3 cited by

Classification of almost abelian Lie groups admitting left-invariant complex or symplectic structures

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2406.06819 v2 pith:JZGHQ74U submitted 2024-06-10 math.DG math.RAmath.SG

Romina M. Arroyo , María L. Barberis , Verónica S. Diaz , Yamile Godoy , Isabel Hernández This is my paper · ORCID
classification math.DGmath.RAmath.SG
keywords complexmathbbmathfraksymplecticabelianclassificationstructuresadmits
verification ladder T0 review T1 audit T2 compute T3 formal

Signed reviews

No signed human review yet.

0 comments
abstract

We classify the almost abelian Lie algebras $\mathfrak g_A=\mathbb R e_0 \ltimes_A \mathbb R^{2n-1}$ admitting complex or symplectic structures. The matrix $A\in M(2n-1,\mathbb R )$ encodes the adjoint action of $e_0$ on the abelian ideal $\mathbb R^{2n-1}$, and the existence of complex or symplectic structures on $\mathfrak g_A$ imposes restrictions on the Jordan normal form of $A$. The classification essentially reduces to the case when $A$ is nilpotent, so we start by considering this case. It turns out that if $A$ is nilpotent and $\mathfrak g_A$ admits a complex structure, then $\mathfrak g_A$ necessarily admits a symplectic structure. This is not true in general when $A$ is non-nilpotent. Finally, several consequences of the classification theorems are obtained.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Torsion-free $H$-structures on almost Abelian solvmanifolds

    math.DG 2024-12 conditional novelty 8.0 of 10

    For almost Abelian Lie algebras, the existence of a torsion-free H-structure is characterized by a linear condition on the defining endomorphism f, and this condition is computed for many structure groups H.

  2. Symplectic solvmanifolds not satisfying the hard-Lefschetz condition

    math.DG 2025-05 conditional novelty 7.0 of 10

    On completely solvable almost abelian solvmanifolds, a symplectic form satisfies the hard-Lefschetz condition if and only if the defining action is semisimple.

  3. Almost abelian complex nilmanifolds

    math.DG 2025-02 accept novelty 7.0 of 10

    Complex structures on nilpotent almost abelian Lie algebras are unique up to isomorphism, giving full control of Dolbeault cohomology, Frölicher degeneration, and deformations of the corresponding nilmanifolds.

Pith tools