$n$-APR tilting and $τ$-mutations

Abstract

APR tilts for path algebra kQ can be realized as the mutation of the quiver Q in ZQ with respect to the translation. In this paper, we show that we have similar results for the quadratic dual of truncations of n-translation algebras, that is, under certain condition, the n-APR tilts of such algebras are realized as τ -mutations.For the dual τ -slice algebras with bound quiver Q ⊥ , we show that their iterated n-APR tilts are realized by the iterated τ -mutations in Z| n−1 Q ⊥ .

On $n$-slice Algebras and Related Algebras

“…The quiver Q(G) is not unique by the definition, but they can be related by a sequences of τ -mutations. Thus n-slice algebra Γ(G) associated to the McKay quiver of G is not unique, and such algebras are related by a sequence of n-APR tilts (see [18]). Let Γ N = Λ N,!,op be the quadratic dual of Λ N .…”

“…But they all can be obtained from any one of them by a sequence of τ -mutations by [12]. By [18], the corresponding 2-slice algebras are obtained by a sequence of 2-APR tilts from the one described in Propositions 5.…”

“…The n-slice algebra is introduced and studied in [15] as the quadratic dual of an n-properly-graded algebra whose trivial extension is an n-translation algebra (see Section 2 for the details). The n-slice algebra was called dual τslice algebras in [14,18], for studying algebras related to the higher representation theory introduced and studied by Iyama and his coauthors [21,22,20]. The nslice algebras bear some interesting features of path algebras in higher dimensional representation theory setting: the τ n -closures, higher dimensional counterparts of preprojective and preinjective components, are truncations of Z| n−1 Q, a higher version of ZQ [15], and the n-APR tilts and colts can be realized by τ -mutations in Z| n−1 Q ⊥,op [18], which is obtained by removing a source (resp.…”