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Under certain circumstances, the algebra admits a locally finite-dimensional grading of Gorenstein parameter $1$, in which case it is the $(n+1)$-preprojective algebra of its $n$-representation infinite degree $0$ piece, as defined by Herschend, Iyama and Oppermann. If the group $G$ is abelian, the $(n+1)$-preprojective algebra is said to be of type $\\~A$. 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Then, the skew-group algebra $R \\ast G$ is bimodule $(n+1)$-Calabi-Yau. Under certain circumstances, the algebra admits a locally finite-dimensional grading of Gorenstein parameter $1$, in which case it is the $(n+1)$-preprojective algebra of its $n$-representation infinite degree $0$ piece, as defined by Herschend, Iyama and Oppermann. If the group $G$ is abelian, the $(n+1)$-preprojective algebra is said to be of type $\\~A$. 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