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

Deren Luo; Jin Yun Guo; Xiaojian Lu

REVIEW 2 major objections 5 minor 24 references

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

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For n-slice algebras satisfying the $(n,n)$-condition, the higher preprojective and preinjective components are truncations of the stable $n$-translation quiver $\mathbb{Z}|_{n-1}Q^{op,\perp}$; strong Koszulness gives the derived-category…

desk verdict New and plausible conditional results on higher ZQ-type constructions, but the advertised scope outruns the hypotheses; worth a serious referee. read the letter →

arxiv 1908.06546 v3 pith:3MFYWYDA submitted 2019-08-19 math.RT math.RA

Jin Yun Guo , Xiaojian Lu , Deren Luo This is my paper · ORCID

classification math.RTmath.RA MSC 16G7016E3516S37

keywords higherrepresentationtheoryn-translationquivern-slicealgebraAuslander-Reitentau_n-closurenu_n-closurederivedcategoryKoszul

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Classical representation theory records the preprojective and preinjective components of a path algebra as truncations of a single infinite quiver, the translation quiver $\mathbb{Z}Q$. This paper proposes a higher-dimensional version: for an $n$-slice algebra $\Gamma$, the same role is played by the stable $n$-translation quiver $\mathbb{Z}|_{n-1}Q^{op,\perp}$, built from the bound quiver $Q^{op}$ of $\Gamma$ by a returning-arrow construction. The central claim is that, when $\Gamma$ satisfies the $(n,n)$-condition and the relevant closure is $\tau$-mature, the Auslander-Reiten quivers of the $\tau_n$-closures of the injectives and projectives are truncations of this quiver; under a stronger Koszul hypothesis the $\nu_n$-closure in the derived category has exactly this quiver as its Auslander-Reiten quiver. A sympathetic reader should care because this turns a whole family of higher homological categories into a single explicit combinatorial pattern, the way $\mathbb{Z}Q$ organizes the module category of a path algebra.

What carries the argument

The load-bearing object is the stable $n$-translation quiver $\mathbb{Z}|_{n-1}Q$, an infinite bound quiver built from an $n$-properly-graded quiver $Q$ by keeping the original arrows in each copy $\mathbb{Z}\times Q_1$ and adding a returning arrow $\beta_p\colon t(p)\to s(p)$ for each maximal bound path $p$ in a chosen basis, with relations forcing an $n$-translation $\tau(i,m)=(i,m-1)$. Its quadratic dual $\mathbb{Z}|_{n-1}Q^{op,\perp}$ is the quiver used in the theorems. The other essential piece is the $\tau$-hammock, a generalized mesh that lists, for each vertex $i$, the positions and multiplicities of the indecomposable projectives appearing in the Koszul complex ending at $\Gamma e_i$; hammocks are what produce the $n$-almost split sequences in the closure categories. Proposition 4.2 supplies the recognition criterion: if a Hom-finite Krull-Schmidt category carries a correspondence to such a quiver, respects the induced order, and sends sink and source sequences to those of the quiver category, then it is equivalent to the bound path category of a truncation. The paper also introduces $n$-slice quivers and algebras as quadratic duals of complete $\tau$-slices, so that $Q^{op,\perp}$ sits inside $\mathbb{Z}|_{n-1}Q^{op,\perp}$ as the zero slice.

What would settle it

Compute the Auslander-Reiten quiver of the $\nu_n$-closure for a concrete strongly Koszul acyclic $n$-slice algebra satisfying Conditions 6.2, for instance one of the $n$-cubic pyramid algebras of Section 5.1: if it contains a cycle, a non-locally-finite pattern, or a vertex whose arrow multiplicities differ from the corresponding vertex of $\mathbb{Z}|_{n-1}Q^{op,\perp}$, then Theorem 6.5 is false; separately, for a finite-$q$ example test $\tau$-maturity by checking whether some vertex $i$ in $Q(M_-)$ has both $\tau i$ and its dual-translation preimage in the truncation.

Watch

Extended reading notes

Core claim

The central discovery is that the higher analogues of the preprojective and preinjective components of an $n$-slice algebra $\Gamma$ --- the $\tau_n$-closures $M_+(\Gamma) = \operatorname{add}\{\tau_n^t D\Gamma \mid t \ge 0\}$ and $M_-(\Gamma) = \operatorname{add}\{\tau_n^{-t} \Gamma \mid t \ge 0\}$ --- are governed by one infinite quiver. Theorem 5.10 states that if $\Gamma$ satisfies the $(n,n)$-condition and these closures are $\tau$-mature, then their Auslander-Reiten quivers are truncations of the bound quiver $\mathbb{Z}|_{n-1}Q^{op,\perp}$. Theorem 6.5 states that for a strongly Koszul acyclic $n$-slice algebra satisfying one of the listed Conditions 6.2, the $\nu_n$-closure $U(\Gamma)$ in the derived category has Auslander-Reiten quiver exactly $\mathbb{Z}|_{n-1}Q^{op,\perp}$. For $n=1$ these statements recover the classical description of the Auslander-Reiten quiver of a path algebra as truncations of $\mathbb{Z}Q$. The arguments build an explicit equivalence sending $\tau_n^{-t}\Gamma e_u$ to $\Gamma e_{\tau^{-t}u}$, so the translation structure of $\mathbb{Z}|_{n-1}Q^{op,\perp}$ literally becomes the higher Auslander-Reiten translation of the closures.

Load-bearing premise

The theorems apply only when the relevant closure quivers are $\tau$-mature, and for finite $q$ the paper does not prove $\tau$-maturity for any $n$-slice algebra satisfying the $(n,n)$-condition; it also leaves open whether typical $n$-slice algebras satisfy the $(n,n)$-condition at all.

Editorial extensions

If this is right

For every $n$-slice algebra satisfying the $(n,n)$-condition whose closures are $\tau$-mature, the higher preprojective and preinjective components have Auslander-Reiten quivers that are truncations of $\mathbb{Z}|_{n-1}Q^{op,\perp}$, hence acyclic, locally finite, and explicitly computable from the bound quiver $Q^{op}$.
For strongly Koszul acyclic $n$-slice algebras satisfying Conditions 6.2, the $\nu_n$-closure in the derived category has Auslander-Reiten quiver equal to the whole infinite quiver $\mathbb{Z}|_{n-1}Q^{op,\perp}$, extending the classical description of the derived category of a path algebra.
In the $n$-representation-infinite case (Corollary 6.6), the closure $U(\Gamma)$ is equivalent to the bound path category on $\mathbb{Z}|_{n-1}Q^{op,\perp}$ and has $n$-almost split sequences; the paper leaves open when strongly Koszul $n$-slice algebras are $n$-representation-infinite.
The $n=1$ instance of the construction recovers the classical fact that the preprojective and preinjective components of a path algebra are truncations of $\mathbb{Z}Q$, with mesh relations supplied by the quadratic dual of the returning-arrow quiver.
Proposition 4.2 is a general recognition principle: any Hom-finite Krull-Schmidt category whose indecomposables correspond to a $\tau$-mature truncation, with matching sink and source sequences, is equivalent to the bound path category of that truncation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the open question about $\tau$-maturity has a positive answer for finite $q$, Theorem 5.10 would apply to the $n$-cubic pyramid algebras discussed in Section 5.1 and would yield a concrete family of higher representation-infinite algebras with fully transparent Auslander-Reiten quivers.
The equivalences $\Theta_\pm$ commute with the respective translations, which suggests the equivalence between $M_\pm(\Gamma)$ and the bound path category may lift to a stable equivalence or even a derived equivalence; the paper does not pursue this.
The recognition criterion in Proposition 4.2 is stated for arbitrary Hom-finite Krull-Schmidt categories, so it may be reusable beyond $n$-slice algebras, for instance to recognize categories arising in cluster theory as bound path categories of translation quivers.
A practical check on the smallest finite-$q$ example (the two-dimensional case of Section 3.2) would be to compute the hammocks and test $\tau$-maturity of $Q(M_-)$ directly; this would either produce the first non-trivial instance of the paper's theorem or expose the obstruction.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things first. This paper is a serious attempt to generalize the classical ZQ picture to higher Auslander-Reiten theory, and it proves genuinely new statements: under Iyama's (n,n)-condition plus τ-maturity, the AR quivers of τn-closures of an n-slice algebra are truncations of Z|n−1Qop,⊥, and under stronger Hypotheses 6.2 the νn-closure has AR quiver Z|n−1Qop,⊥. Second, the abstract oversells the reach. The theorems are conditional on τ-maturity, Iyama's condition, and Conditions 6.2, and the paper itself asks whether any n>1 algebra satisfies them. So the main claims are well-formed but not established in the advertised generality.

What is genuinely good: Proposition 4.2 is a useful recognition theorem for categories equivalent to bound path categories of truncations, and the inductive proof of Lemma 5.5 is detailed. The n=1 case recovers the classical preprojective/preinjective picture cleanly. The authors also flag the open questions honestly, and the heavy reliance on their earlier papers is not deceptive; those are prior results in the same program, used as lemmas.

Soft spots, in proportion. The biggest one is existential: no theorem here shows that the hypotheses are ever satisfied outside the classical n=1 case and the n-cubic pyramid algebras of [12]. τ-maturity is automatic when q=∞, but not in general; whether every n-slice algebra satisfies Iyama's (n,n)-condition is left open. Conditions 6.2 are stated without any discussion of how to check them. That is a real gap between what the abstract promises and what the theorems deliver. The proof of Lemma 5.5 also has a few compressed passages, especially around the exact sequences (16)–(20) and the passage from source sequence to sink sequence by [18, Prop. 3.3]; a referee should inspect that step carefully. Minor: the notation Q+, Q−, G± is easy to confuse, and the open questions should be moved from the remark after Theorem 5.10 into the introduction.

The stress-test note is mostly fair. The central claims are not false, but they are conditional, and the abstract should say so prominently.

Who is this for? Specialists in higher AR theory and Koszul duality. It deserves a serious referee, not a desk reject. I would send it to an expert and expect major revision: make the hypotheses explicit in the abstract, discuss examples, and expand the compressed proof of Lemma 5.5.

Referee Report

2 major / 5 minor

Summary. The paper introduces n-slice quivers and n-slice algebras, together with the stable n-translation quiver Z|_{n-1}Q, as higher-dimensional analogues of the classical translation quiver ZQ. Its main technical tool is Proposition 4.2, a recognition criterion for when a Hom-finite Krull-Schmidt category is equivalent to the bound path category of a convex full subquiver of a dual stable n-translation quiver. Using this criterion, the paper claims that for an n-slice algebra Γ satisfying Iyama's (n,n)-condition, the τ_n-closures M+(Γ) and M−(Γ) have Auslander-Reiten quivers that are truncations of Z|_{n-1}Q^{op,⊥}, provided the relevant subquivers are τ-mature (Theorem 5.10). Under additional Hom-isomorphism hypotheses, Conditions 6.2, it further claims that the ν_n-closure U(Γ) in the derived category has Auslander-Reiten quiver Z|_{n-1}Q^{op,⊥} (Theorem 6.5).

Significance. If the hypotheses can be verified in new cases, the paper would provide a uniform higher-representation-theoretic analogue of the classical description of preprojective and preinjective components and of derived-category components via ZQ. The definitions of n-slice algebras and the explicit comparison functors Θ± and Θ are original, and Proposition 4.2 is a potentially useful general criterion. The paper is also honest in flagging the open questions about τ-maturity and Iyama's condition after Theorem 5.10. However, the main theorems are substantially more conditional than the abstract and introduction suggest, and no n>1 example satisfying all hypotheses is actually exhibited in the paper.

major comments (2)

[Abstract, Theorem 5.10, Remark after Theorem 5.10, Theorem 6.5, Conditions 6.2] The central theorems are more conditional than the abstract and introduction advertise. Theorem 5.10 only applies to n-slice algebras that satisfy Iyama's (n,n)-condition and whose components M± are τ-mature, and Theorem 6.5 additionally requires one of the Hom-isomorphism Conditions 6.2. The Remark after Theorem 5.10 explicitly leaves open when τ-maturity holds for finite q and whether every n-slice algebra satisfies Iyama's condition, while Section 6 closes with the same kind of open question for n-representation-infinite algebras. The only fully verified example is the classical n=1 path algebra (Section 5.1); Section 3.2 shows that the Auslander algebra of a representation-directed algebra is a 2-slice algebra but does not verify Iyama's condition or τ-maturity. I ask the author to either prove the missing hypotheses in a nontrivial family, exhibit an n>1 example satisfying all hypotheses, or explicitly re-scope the abstract and introduction to the conditional statements.
[§4, Proposition 4.2, condition (iv) and proof around Eqs. (10)–(12)] Condition (iv) is stated as a one-way preservation statement: a sink/source sequence in C is sent to one in G(C). In the proof, however, the sink sequence (10) in G is used to conclude the existence of the sink sequence (11) in C ('So by condition (iv)...'), which is the converse direction. Since F is not yet known to be an equivalence on the full category C, this implication requires proof, or the hypothesis should be 'if and only if' with the converse verified. This is load-bearing because Proposition 4.2 is the engine behind Theorems 5.6, 5.10, and 6.4/6.5.

minor comments (5)

[Section 2, Corollary 2.3 and the definition of τ-maturity] The definition of τ-maturity is given as 'satisfies the condition in Corollary 2.3'; it would be clearer to state the condition explicitly, including the range of indices i for which the condition must hold.
[§4, proof of Proposition 4.2, displayed sequence (10)] In the paragraph before (10), the index is introduced as τ^{1−s}u, but the displayed sequence uses τ^{1−r}u; the exponent in (10) appears to be inconsistent with the preceding notation.
[Section 5, Theorem 5.6] The phrase 'M− have n-almost split sequence' should be 'M− has n-almost split sequences'; similar grammatical issues occur in Theorem 5.9 and Theorem 6.5 ('is satisﬁes').
[Section 6, final sentence] The closing question asks when a strongly Koszul n-slice algebra is 'n-representation-ﬁnite'; given the surrounding discussion of Corollary 6.6, this should presumably read 'n-representation-infinite'.
[References] Reference [5] spells the second author as 'Ginsberg'; the standard spelling is 'Ginzburg'. In reference [24], 'Lecture Note' should be 'Lecture Notes'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central theorems are conditional deductions, not reductions to their own hypotheses

full rationale

The paper's central claims (Theorems 5.10 and 6.5) are conditional mathematical theorems, not fitted predictions or definitions in disguise. The equivalence functors are introduced as vertex correspondences, e.g. Θ− : τ^{−t}_n Γe_u ↦ Γe_{τ^{−t}u}, but the crucial homomorphism property is not assumed; it is proved by induction in Lemma 5.5 from τ-maturity and n-rigidity, using the general recognition criterion Proposition 4.2. The extensive dependence on the author's earlier papers [9], [10], and [11] supplies the Z|n−1Q construction, Koszul complexes, and the identification of n-slices with complete τ-slices; those are previously published results with stated assumptions and do not include the target equivalence as an input. The hypotheses Iyama's (n,n)-condition, τ-maturity, and Conditions 6.2 are genuinely additional, and the paper explicitly flags their verification as open, for instance in the Remark after Theorem 5.10: 'It is also interesting to know if an n-slice algebra always satisfy Iyama's (n,n)-condition?'. This is an applicability gap, not circularity. No equation in the paper reduces a stated conclusion to its own input by construction, and the n=1 examples recover classical results rather than renaming them.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper's central claims rest on a large body of prior results, mostly the authors' own: the n-translation algebra machinery from [10], τ-slice and covering theory from [9], and trivial extension/preprojective computations from [11]. These are published, but not re-derived here, and they are invoked at critical junctures (Koszul complexes becoming n-almost split sequences, the identification of the quiver of the smash product, and the Loewy length bound). There are no fitted numerical parameters. The new concepts (n-slice algebra, τ-maturity) are definitions with worked examples, not entities with independent falsifiable handles.

assumptions (6)

domain assumption The results of [10]: n-translation algebras are (n+1,q)-Koszul for some q ≥ 2 or q = ∞, and the Koszul complexes (2) are n-almost split sequences under stated conditions.
Lemma 2.2, Proposition 2.4, and Proposition 4.2 rely on Lemma 7.1 and Theorem 7.2 of [10], the authors' own prior paper. These results are invoked as black boxes.
domain assumption The results of [9]: τ-slice algebras, coverings, and the identification of the repetitive algebra of a τ-slice algebra with the smash product (Propositions 3.4 and 3.5).
Proposition 3.5 cites Theorem 6.10 and Theorem 5.12 of [9] to conclude that Q is isomorphic to Z|n−1Q. This is an unverified input in the current paper.
domain assumption The results of [11]: for an n-properly-graded algebra, the (n+1)-preprojective algebra equals the quadratic dual of the twisted trivial extension, and the bounds on Loewy length used in Lemma 5.1.
Section 3 and Lemma 5.1 cite Proposition 6.3 and 6.5 of [11] to conclude that Λ has Loewy length at most n+1 and that Γ is Koszul of global dimension n.
domain assumption Iyama's higher Auslander-Reiten theory results: Theorem 1.5 of [18], Proposition 2.3 and 2.5 of [15], Proposition 2.5 of [20], Lemma 3.3 of [20], used to establish n-rigidity and the order property for Orlov categories.
The proof of Lemma 5.8 and Lemma 5.5 use these as black boxes. These are standard results in the field, but they are load-bearing for the n-rigidity of M±.
standard math Koszul duality results from Beilinson-Ginzburg-Soergel [5] used to conclude that Γ has global dimension n (Lemma 5.1).
Theorem 2.10.2 and 2.6.1 of [5] are cited to pass from Koszulity of Λ to the global dimension of its quadratic dual Γ.
domain assumption The algebra Γ is acyclic, so the path category coincides with the complete path category, and Hom-spaces are finite dimensional (as used throughout Propositions 2.6 and 4.2).
The paper assumes Q is acyclic and locally finite throughout; this is stated in Section 2 and used in Proposition 4.1. This is a domain hypothesis of the theory, satisfied for the intended examples.

invented entities (2)

n-slice quiver and n-slice algebra
purpose: New classes of objects for which the Z|n−1Q description applies; the quadratic dual of an n-properly-graded quiver.
The paper introduces n-slices (Section 3) as the quadratic duals of n-properly-graded quivers, with n-slice algebras defined by requiring the trivial extension of the dual to be an n-translation algebra. This is a definition, not an empirical entity; independent evidence is not applicable. Examples are given for n=1 and n=2, but no new external phenomena are predicted.
τn-closure and νn-closure categories M± and U
purpose: The categories whose Auslander-Reiten quivers are claimed to be truncations of Z|n−1Qop,⊥.
M± and U are defined in Section 5 and 6 as additive closures of iterated τn and νn translates. They are standard constructions in higher Auslander-Reiten theory, not invented entities with independent empirical handles.

how reviews work

0 comments

Cite this review

Pith. "Pith review of $\mathbb Z Q$ type constructions in higher representation theory." pith.science (2026). https://pith.science/paper/3MFYWYDA

@misc{pith2026190806546,
  author       = {Pith},
  title        = {Pith review of: $\mathbb Z Q$ type constructions in higher representation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MFYWYDA}},
  note         = {Machine review of arXiv:1908.06546}
}

abstract

Let $Q$ be an acyclic quiver, it is classical that certain truncations of the translation quiver $\mathbb Z Q$ appear in the Auslander-Reiten quiver of the path algebra $kQ$. The stable $n$-translation quiver $\mathbb Z|_{n-1} Q$ is introduced as a generalization of the $\mathbb Z Q$ construction in studying higher representation theory of algebras for an acyclic bound quiver $Q$. In this paper, we find conditions for a Hom-finite Krull-Schmidt $k$-category to be realized as the bound path category of a convex full subquiver of an stable $n$-translation quiver.We show that for $n$-slice algebra $\Gamma$, which is an $n$-hereditary algebra whose $(n+1)$-preprojective algebra is $(q+1,n+1)$-Koszul, with bound quiver $Q^{op}$, its $n$-preprojective and $n$-preinjective components in the module category and truncations of the stable $n$-translation quiver $\mathbb Z|_{n-1} Q^{op}$. We also use $\mathbb Z|_{n-1} Q^{op}$ to describe the $\nu_n$-closure of $\Gamma$ in the derived category.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[12]

Y., Luo, D.: On n-cubic Pyramid Algebras

Guo, J. Y., Luo, D.: On n-cubic Pyramid Algebras. Algebr . Represent. Theory 19, 991– 1016(2016)

work page 2016
[1]

N., Riche, S., Koszul duality and semisimplici ty of Frobenius

Achar, P. N., Riche, S., Koszul duality and semisimplici ty of Frobenius. Ann. Inst. Fourier 63(4), 1511–1612(2013)

work page 2013
[2]

36, Cambridge Univ

Auslander, M., Reiten, I., and Smalø S.: Representation theory of artin algebras , Cambridge Studies in Advanced Math., vol. 36, Cambridge Univ. Press, ( 1995)

work page 1995
[3]

I., Kapranov, M

Bondal, A. I., Kapranov, M. M.: Representable functors, Serre functors, and reconstructions, Math. USSR-Izv. 35(3), 519–541(1990)

work page 1990
[4]

Brenner, S., Butler, M. C. R., King, A. D.: Periodic algeb ras which are almost Koszul, Algebr. Represent. Theory 5(4), 331–367(2002)

work page 2002
[5]

of AMS 9(2), 473–527(1996)

Beilinson, A., Ginsberg, V, Soergel, W.: Koszul duality patterns in Representation theory, J. of AMS 9(2), 473–527(1996)

work page 1996
[6]

Buan, A., Iyama, O., Reiten, I., Smith, D.: Mutation of cl uster-tilting objects and potentials, Amer. J. Math. 133(4), 835–887(2011)

work page 2011
[7]

A., Platzeck, M

Fern´ andez, E. A., Platzeck, M. I.: Presentations of Tri vial Extensions of Finite Dimensional Algebras and a Theorem of Sheila Brenner, J. Algebra 249, 326 –344(2002)

work page 2002

Show all 24 references

[8]

Y.: Translation algebras and applications, J

Guo, J. Y.: Translation algebras and applications, J. Al gebra, 255 1–21(2002)

2002
[9]

Y.: Coverings and Truncations of Graded Self-inj ective Algebras, J

Guo, J. Y.: Coverings and Truncations of Graded Self-inj ective Algebras, J. Algebra 355, 9–34(2012)

2012
[10]

Y.: On n-translation algebras

Guo, J. Y.: On n-translation algebras. J. Algebra 453, 400–428(2016)

2016
[11]

Y.: On Trivial extensions and higher preproject ive algebras, J

Guo, J. Y.: On Trivial extensions and higher preproject ive algebras, J. Algebra 547, 379– 397(2020)

2020
[13]

Y., W u, Q.: Loewy matrix, Koszul cone and applica tions, Comm

Guo, J. Y., W u, Q.: Loewy matrix, Koszul cone and applica tions, Comm. Algebra 28, 925– 941(2000)

2000
[14]

London Math

Happel, D.: Triangulated categories in the representation theory of ﬁn ite-dimensional alge- bras. London Math. Soc. Lecture Note Ser., vol. 119. Cambridge Un iversity Press, Cambridge, (1988)

1988
[15]

Herschend, M., Iyama, O.,Minamoto, H., Oppermann, S.: n-Representation inﬁnite algebras, Adv. Math. 252(3), 292–342(2014)

2014
[16]

Herschend, M., Iyama, O.,Minamoto, H., Oppermann, S.: Representation theory of Geigle- Lenzing complete intersections, preprint arXiv:1409.066 8
[17]

Iyama, O.: τ-Categories III: Auslander Orders and Auslander-Reiten Qu ivers. Algebr. Rep- resent. Theory 8, 601–619(2005)

2005
[18]

Iyama, O.: Higher-dimensional Auslander-Reiten theo ry on maximal orthogonal subcate- gories, Adv. Math. 210(1) 22–50(2007)

2007
[19]

Iyama, O.: Auslander correspondence, Adv. Math. 210(1 ), 51–82(2007)

2007
[20]

Iyama, O.: Cluster tilting for higher Auslander algebr as, Adv. Math. 226(1), 1–61(2011)

2011
[21]

Oppermann, S.: n-Representation-ﬁnite alge bras and n-APR tilting, Trans

Iyama, O. Oppermann, S.: n-Representation-ﬁnite alge bras and n-APR tilting, Trans. Amer. Math. Soc. 363 (12), 6575–6614(2012),

2012
[22]

Oppermann, S.: Stable categories of higher pr eprojective algebras, Adv

Iyama, O. Oppermann, S.: Stable categories of higher pr eprojective algebras, Adv. Math. 244, 23–68(2013)

2013
[23]

Mori, I.: Structures of AS-regular algebr as, Adv

Minamoto, H. Mori, I.: Structures of AS-regular algebr as, Adv. Math. 226, 4061–4095(2011)

2011
[24]

M.: Tame algebra and integral quadratic forms

Ringel, C. M.: Tame algebra and integral quadratic forms . Lecture Note in Mathematics no.1099 Springer Verlag Berlin (1984) Jin Yun Guo, LCSM( Ministry of Education), School of Mathemat ics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China

1984

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [12]

Y., Luo, D.: On n-cubic Pyramid Algebras

Guo, J. Y., Luo, D.: On n-cubic Pyramid Algebras. Algebr . Represent. Theory 19, 991– 1016(2016)

work page 2016

[2] [1]

N., Riche, S., Koszul duality and semisimplici ty of Frobenius

Achar, P. N., Riche, S., Koszul duality and semisimplici ty of Frobenius. Ann. Inst. Fourier 63(4), 1511–1612(2013)

work page 2013

[3] [2]

36, Cambridge Univ

Auslander, M., Reiten, I., and Smalø S.: Representation theory of artin algebras , Cambridge Studies in Advanced Math., vol. 36, Cambridge Univ. Press, ( 1995)

work page 1995

[4] [3]

I., Kapranov, M

Bondal, A. I., Kapranov, M. M.: Representable functors, Serre functors, and reconstructions, Math. USSR-Izv. 35(3), 519–541(1990)

work page 1990

[5] [4]

Brenner, S., Butler, M. C. R., King, A. D.: Periodic algeb ras which are almost Koszul, Algebr. Represent. Theory 5(4), 331–367(2002)

work page 2002

[6] [5]

of AMS 9(2), 473–527(1996)

Beilinson, A., Ginsberg, V, Soergel, W.: Koszul duality patterns in Representation theory, J. of AMS 9(2), 473–527(1996)

work page 1996

[7] [6]

Buan, A., Iyama, O., Reiten, I., Smith, D.: Mutation of cl uster-tilting objects and potentials, Amer. J. Math. 133(4), 835–887(2011)

work page 2011

[8] [7]

A., Platzeck, M

Fern´ andez, E. A., Platzeck, M. I.: Presentations of Tri vial Extensions of Finite Dimensional Algebras and a Theorem of Sheila Brenner, J. Algebra 249, 326 –344(2002)

work page 2002