Higher Auslander-Reiten sequences and $t$-structures

Abstract

Let $R$ be an artin algebra and $\mathcal{C}$ an additive subcategory of $\mathrm{mod}(R)$. We construct a $t$-structure on the homotopy category $K^{-}(\mathcal{C})$ whose heart ${\mathcal{H}}_{\mathcal{C}}$ is a natural domain for higher Auslander-Reiten (AR) theory. The abelian categories ${\mathcal{H}}_{\mathrm{mod}(R)}$ (which is the natural domain for classical AR theory) and ${\mathcal{H}}_{\mathcal{C}}$ interact via various functors. If $\mathcal{C}$ is functorially finite then ${\mathcal{H}}_{\mathcal{C}}$ is a quotient category of ${\mathcal{H}}_{\mathrm{mod}(R)}$. We illustrate the theory with two examples: Iyama developed a higher AR theory when $\mathcal{C}$ is a maximal $n$-orthogonal subcategory, see \cite{I}. In this case we show that the simple objects of ${\mathcal{H}}_{\mathcal{C}}$ correspond to Iyama's higher AR sequences and derive his higher AR duality from the existence of a Serre functor on the derived category $D^b({\mathcal{H}}_{\mathcal{C}})$. The category $\mathcal{O}$ of a complex semi-simple Lie algebra $\mathfrak{g}$ fits into higher AR theory by considering $R$ to be the coinvariant algebra of the Weyl group of $\mathfrak{g}$.

26 pages, accepted for publication in Journal of Algebra 2016

Cited by in corpus (4)

• d-Representation-finite self-injective algebras
• Auslander-Gorenstein algebras and precluster tilting
• Continuous Quivers of Type A (II) The Auslander-Reiten Space
• Derived counterparts of fusion categories of quantum groups