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arXiv · 2002.04593

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

Gene Abrams·Francesca Mantese·Alberto Tonolo
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Abstract

For a field $K$, let $\mathcal{R}$ denote the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle$. 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.

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Gene Abrams, Francesca Mantese, Alberto Tonolo. 2020-02-11. Injective modules over the Jacobson algebra $K< X,Y | XY = 1 >$. https://arxiv.org/abs/2002.04593

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