Injective modules over the Jacobson algebra $Klangle X, Y | XY=1rangle $

For a field K, let R denote the Jacobson algebra K⟨X,Y | XY = 1⟩. We give an explicit construction of the injective envelope of each of the (infinitely many) simple left R-modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for R. Our approach involves realizing R up to isomorphism as the Leavitt path K-algebra of an appropriate graph T , which thereby allows us to utilize important machinery developed for that class of algebras.