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

Abstract

We show that whenever \[ [\,\cdot,\cdot]_t = [\,\cdot,\cdot]_0 + t[\,\cdot,\cdot]_1,\qquad α_t = \mathrm{id} + tα_1 \] 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.

9 pages