Higher zigzag algebras

Grant, Joseph

doi:10.48550/arxiv.1711.00794

Cited by 1 publication

(1 citation statement)

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“…We know by Theorem 5.2 and Corollary 5.4 that φ is always surjective. If s ≥ 3, then it is shown in [Gra,Section 3] that φ is an isomorphism.…”

Section: Homπ(−π)mentioning

confidence: 99%

Higher preprojective algebras, Koszul algebras, and superpotentials

Grant,

Iyama

2019

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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. Contents1. Introduction 1 2. Preliminaries 3 3. Description of higher preprojective algebras as higher Jacobi algebras 8 4. Resolutions of simple modules over higher preprojective algebras 16 5. Quadratic duals of higher preprojective algebras 23 References 28

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“…We know by Theorem 5.2 and Corollary 5.4 that φ is always surjective. If s ≥ 3, then it is shown in [Gra,Section 3] that φ is an isomorphism.…”

Section: Homπ(−π)mentioning

confidence: 99%