Infinitesimal deformations of $\mathfrak{sl}_2$ with a twisted Jacobi identity

TL;DR

This paper proves that infinitesimal Hom-Lie deformations of rak{sl}_2 over a dual number ring satisfy the classical Jacobi identity, confirming a conjecture from 2010.

Contribution

It demonstrates that such deformations inherently satisfy the classical Jacobi identity, resolving a longstanding conjecture in Hom-Lie algebra theory.

Findings

Deformed brackets satisfy classical Jacobi identity over rak{K}[t]

Confirms the conjecture of Makhlouf and Silvestrov (2010)

Establishes a link between Hom-Lie deformations and classical Lie algebra structures

Abstract

We show that whenever \[ [\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad \alpha_t = \mathrm{id} + t\alpha_1 \] define an infinitesimal Hom--Lie deformation of $\mathfrak{sl}_2(\mathbb K)$ over $\mathbb K[t]/(t^2)$ and $(\mathfrak{sl}_2(\mathbb K),[\,\cdot,\cdot]_0,\alpha_1)$ is a Hom--Lie algebra, then the deformed bracket $[\,\cdot,\cdot]_t$ satisfies the ordinary Jacobi identity over $\mathbb K[t]$. This solves a conjecture of Makhlouf and Silvestrov from 2010.