Injective modules over the Jacobson algebra $K<X,Y|XY=1>$

Abstract

For a field $K$, let $\mathcal{R}$ 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 $\mathcal{R}$-modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for $\mathcal{R}$. Our approach involves realizing $\mathcal{R}$ up to isomorphism as the Leavitt path $K$-algebra of an appropriate graph $\mathcal{T}$, which thereby allows us to utilize important machinery developed for that class of algebras.

16 pages