$\mathbb Z Q$ type constructions in higher representation theory

Abstract

Let Q be an acyclic quiver, it is classical that certain truncations of the translation quiver ZQ appear in the Auslander-Reiten quiver of the path algebra kQ. We introduce the n-translation quiver Z| n−1 Q as a generalization of the ZQ construction in our recent study on n-translation algebras. In this paper, we show that for certain algebra Γ of global dimension n, the quiver Z| n−1 Q can be used to describe the τ n -closure of DΓ and τ −1 n -closure of Γ in its module category and the νn-closure of Γ in the derived category.

Multi-layer quivers and higher slice algebras

“…We remark that in this case, Theorem 5.2 describes the (n + 1)-almost split sequences in the (n + 1)-preprojective and (n + 1)-preinjective components of Γ, by Theorem 5.6 of [17].…”

“…certain module categories [27,28,16,17]. Quivers are used extensively in studying n-slice algebras, especially for generalizing results of hereditary algebras [17,19].…”

On $n$-slice Algebras and Related Algebras

“…Path algebras are very important in representation theory and related mathematical fields, it is interesting to study their counterparts in higher representation theory [21,20]. 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 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. sink) and adding a corresponding sink (resp.…”

“…McKay quiver for a finite subgroup of a general linear group is introduced in [27], it connecting representation theory, singularity theory and many other mathematical fields. McKay quiver are also very interesting in studying higher representation theory of algebras [20,15]. Over the algebraically closed field of characteristic zero, the preprojective algebras of path algebras of tame type are Morita equivalent to the skew group algebras of finite subgroups of SL(2, C) over the polynomial algebra of two variables [29,8].…”