On $n$-translation algebras

Motivated by Iyama's higher representation theory, we introduce $n$-translation quivers and $n$-translation algebras. The classical $\mathbb Z Q$ construction of the translation quiver is generalized to construct an $(n+1)$-translation quiver from an $n$-translation quiver, using trivial extension and smash product. We prove that the quadratic dual of $n$-translation algebras have $(n-1)$-almost splitting sequences in the category of its projective modules. We also present a non-Koszul $1$-translation algebra whose trivial extension is $2$-translation algebra, thus also provides a class of examples of $(3,m-1)$-Koszul algebras (and also a class of $(m-1,3)$-Koszul algebras) for all $m \ge 2$.