Deformations of the five dimensional Heisenberg Lie algebra
Alice Fialowski, Ashis Mandal
TL;DR
The paper classifies all deformations of the 5‑dimensional Heisenberg Lie algebra $\mathfrak{h}_2$ over $\mathbb{R}$ or $\mathbb{C}$. It computes the second adjoint cohomology $H^2(\mathfrak{h}_2,\mathfrak{h}_2)$, finding $20$ independent infinitesimal deformations, and explicitly exhibits eight cocycle families $\phi_i$ that generate these deformations. Jacobi identities are checked to determine which infinitesimals extend to real deformations; the authors show that $18$ of the deformations lift to real deformations while $2$ are purely infinitesimal. All resulting algebras are solvable, with two nilpotent cases identified up to isomorphism, and the real deformation picture aligns with the complex one via a standard real form analysis. The work provides a complete, explicit deformation moduli for $\mathfrak{h}_2$ and contributes to the understanding of the moduli space of 5‑dimensional Lie algebras.
Abstract
In this note we explicitly give all the equivalent classes of deformations of the 5-dimensional Heisenberg Lie algebra $\mathfrak{h}_2$ over complex or real number fields. We show that there are altogether 20 infinitesimal deformations (families), 18 of them being extendable to real deformations and 2 of them are only infinitesimal.