An Explicit dg--Lie Model Governing Jordan Algebra Deformations

Abstract: Over a field of characteristic~$0$ we place the deformation theory of Jordan algebras on a concrete, computation--friendly differential graded Lie algebra. On the symmetric cochains $C^n(J)=\Sym^n(J^\vee)\otimes J$ we define an unshuffle--averaged insertion whose commutator is a graded Lie bracket. For the Jordan product $μ\in C^2(J)$ the differential $d_μ=[μ,\cdot]$ governs deformations. We prove that $[μ,μ]=0$ is equivalent to the Jordan identity (via polarization), hence $d_μ^2=0$, and record the standard Maurer--Cartan/gauge recursion together with low--degree coboundary formulas. A small two--dimensional unital example is worked out explicitly. The note is deliberately lean and self--contained.