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- W1541773251 abstract "Let $R$ be an artin algebra and $mathcal{C}$ an additive subcategory of $operatorname{mod}(R)$. We construct a $t$-structure on the homotopy category $operatorname{K}^{-}(mathcal{C})$ whose heart $mathcal{H}_{mathcal{C}}$ is a natural domain for higher Auslander-Reiten (AR) theory. The abelian categories $mathcal{H}_{operatorname{mod}(R)}$ (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 $mathcal{H}_{operatorname{mod}(R)}$. 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 $operatorname{D}^b(mathcal{H}_{mathcal{C}})$. 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 $mathfrak{g}$." @default.
- W1541773251 created "2016-06-24" @default.
- W1541773251 creator A5059989914 @default.
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- W1541773251 date "2013-12-16" @default.
- W1541773251 modified "2023-09-26" @default.
- W1541773251 title "Higher Auslander-Reiten sequences and $t$-structures" @default.
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