On n-translation algebras

2014/06/24 by Guo, Jin Yun
#FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1406.6136

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 ≥ 2.

Related

• \mathbb Z Q type constructions in higher representation theory
• τ-slice algebras of n-translation algebras and quasi n-Fano algebras
• Returning Arrows for Self-injective Algebras and Artin-Schelter Regular Algebras
• Categorification of quiver diagonalization and Koszul algebras
• n-quivers and A Universal Investigation of n-representations of Quivers