On trivial extensions and higher preprojective algebras

Guo, Jin Yun

doi:10.1016/j.jalgebra.2019.11.022

Cited by 11 publications

(20 citation statements)

References 19 publications

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“…Note that a special case of Theorem C was independently obtained by Guo [Guo19,Theorem 5.3]. His result corresponds to the "if" part of our Theorem 5.2(c) under the assuption that Λ !…”

Section: Introductionmentioning

confidence: 74%

Higher preprojective algebras, Koszul algebras, and superpotentials

Grant¹,

Iyama

2020

Compositio Math.

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In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case, we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is $d$ -hereditary, all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a $d$ -hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.

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“…Note that a special case of Theorem C was independently obtained by Guo [Guo19,Theorem 5.3]. His result corresponds to the "if" part of our Theorem 5.2(c) under the assuption that Λ !…”

Section: Introductionmentioning

confidence: 74%

Higher preprojective algebras, Koszul algebras, and superpotentials

Grant¹,

Iyama

2020

Compositio Math.

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“…It is easy to see that maximal bound path of Q have the same length n (see also Lemma 6.1 of [10]). Let M be a set of linearly independent maximal bound paths.…”

Section: Application To Dual τ -Slice Algebrasmentioning

confidence: 97%

“…Denote ∆Λ = Λ ⋉ DΛ the trivial extension of Λ, the returning arrow quiver of Q is exact the bound quiver of ∆Λ = Λ ⋉ DΛ (Proposition 2.2 of [15], see also Proposition 3.1 and Lemma 3.2 of [10]).…”

Section: Application To Dual τ -Slice Algebrasmentioning

confidence: 99%

$n$-APR tilting and $τ$-mutations

Guo

Xiao

2019

Preprint

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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 ⊥ .

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“…By Lemma 6.1 of [12], Q is an n-properly graded quiver and maximal bound paths of Q have the same length n. Let M be a set of linearly independent maximal bound paths in Q. Define returning arrow quiver Q…”

Section: Application To Dual -Slice Algebrasmentioning

confidence: 99%

n-APR tilting and $$\tau $$-mutations

Guo

Xiao

2021

J Algebr Comb

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APR tilts for path algebra k Q 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 ⊥ .

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On trivial extensions and higher preprojective algebras

Cited by 11 publications

References 19 publications

Higher preprojective algebras, Koszul algebras, and superpotentials

Higher preprojective algebras, Koszul algebras, and superpotentials

$n$-APR tilting and $τ$-mutations

n-APR tilting and $$\tau $$-mutations

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