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The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let ${\Lambda }_t$ be the Yoneda algebra of a reconstruction algebra of type ${𝐀}_1$ over a field $.Inthispaper,aminimalprojectivebimoduleresolutionof$t$isconstructed,andthe$-dimensions of all Hochschild homology and cohomology groups of ${\Lambda }_t$ are calculated explicitly.

Let $Q$ be a finite union of Dynkin quivers, $G\subseteq \mathrm{Aut}\left(𝕜Q\right)$ a finite abelian group, $\widehat{Q}$ the generalized McKay quiver of $(Q,G)$ and ${\Gamma }_Q$ the Auslander-Reiten quiver of $𝕜Q$. Then $G$ acts functorially on the quiver ${\Gamma }_Q$. We show that the Auslander-Reiten quiver of $𝕜\widehat{Q}$ coincides with the generalized McKay quiver of $({\Gamma }_Q,G)$.