scholarly journals The Nakayama automorphism of a self‐injective preprojective algebra

2019 ◽  
Vol 52 (1) ◽  
pp. 137-152 ◽  
Author(s):  
Joseph Grant
Keyword(s):  
Preprojective Algebra

scholarly journals THE COMBINATORICS OF TENSOR PRODUCTS OF HIGHER AUSLANDER ALGEBRAS OF TYPE A

2020 ◽  
pp. 1-21
Author(s):  
JORDAN MCMAHON ◽  
NICHOLAS J. WILLIAMS
Keyword(s):  
Tensor Product ◽  
Tensor Products ◽  
Type A ◽  
Coordinate Ring ◽  
Preprojective Algebra ◽  
Finite Algebras ◽  
Plücker Coordinates ◽  
Higher Dimensional ◽  
Dimensional Version

Abstract We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two d-representation-finite algebras has a d-precluster-tilting subcategory. Finally, we relate mutations of these collections to a form of tilting for these algebras.

scholarly journals Higher preprojective algebras, Koszul algebras, and superpotentials

2020 ◽  
Vol 156 (12) ◽  
pp. 2588-2627
Author(s):  
Joseph Grant ◽  
Osamu Iyama
Keyword(s):  
Trivial Extension ◽  
Koszul Algebras ◽  
Preprojective Algebra ◽  
Hereditary Algebra ◽  
Dual Pairs ◽  
Bimodule Resolution ◽  
Preprojective Algebras ◽  
Projective Resolutions ◽  
Trivial Extension Algebra ◽  
Extension Algebra

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.

scholarly journals Lattice structure of Weyl groups via representation theory of preprojective algebras

2018 ◽  
Vol 154 (6) ◽  
pp. 1269-1305 ◽  
Author(s):  
Osamu Iyama ◽  
Nathan Reading ◽  
Idun Reiten ◽  
Hugh Thomas
Keyword(s):  
Representation Theory ◽  
Weyl Group ◽  
Lattice Structure ◽  
Weak Order ◽  
Weyl Groups ◽  
Preprojective Algebra ◽  
Combinatorial Description ◽  
Irreducible Elements ◽  
Preprojective Algebras

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

scholarly journals Tilting and Cluster Tilting for Preprojective Algebras and Coxeter Groups

2017 ◽  
Vol 2019 (18) ◽  
pp. 5597-5634 ◽  
Author(s):  
Yuta Kimura
Keyword(s):  
Coxeter Groups ◽  
Cluster Category ◽  
Preprojective Algebra ◽  
Endomorphism Algebra ◽  
Factor Algebra ◽  
Stable Category ◽  
Preprojective Algebras ◽  
Tilting Object ◽  
Cluster Tilting ◽  
Silting Object

AbstractWe study the stable category of the graded Cohen–Macaulay modules of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\boldsymbol{w})$ of this category associated with each reduced expression $\boldsymbol{w}$ of $w$ and give a sufficient condition on $\boldsymbol{w}$ such that $M(\boldsymbol{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\boldsymbol{w})$. Moreover, we compare it with a triangle equivalence given by Amiot–Reiten–Todorov for a cluster category.

scholarly journals A functorial approach to monomorphism categories for species I

2021 ◽  
Author(s):  
Nan Gao ◽  
Julian Külshammer ◽  
Sondre Kvamme ◽  
Chrysostomos Psaroudakis
Keyword(s):  
General Result ◽  
Path Algebra ◽  
Preprojective Algebra ◽  
Adjoint Functors ◽  
Almost Split Sequences ◽  
Functorial Approach

We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalized species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit computations as in the case of Ringel and Schmidmeier.

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