On $n$-translation algebras

Abstract

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 $m\ge 2$.

arXiv commentThe paper is revised, according to the referees' suggestions and comments. The definitions of $n$-translation quiver, admissibility are rewritten, and the results related to these definition are revised. The results concerning $n$-almost split sequence is revised. The Section 7 is removed and Section 6 is split into 3 sections. The mistake and typos pointed out are corrected