On n-translation algebras

- We introduce n-translation quivers and n-translation algebras.
- We generalize the classical ZQ construction of the translation quiver to n-translation quiver.
- We find conditions for the quadratic dual of n-translation algebras has n-almost split sequences in the category of its projectives.

Motivated by Iyama's higher representation theory, we introduce n-translation quivers and n-translation algebras. The classical ZQ 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 find a sufficient and necessary condition for the quadratic dual of n-translation algebras to have n-almost split sequences in the category of its projective modules.