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 Jordan centralizer maps on trivial extension algebras

2020 ◽  
Vol 53 (1) ◽  
pp. 58-66
Author(s):  
Mohammad Ali Bahmani ◽  
Fateme Ghomanjani ◽  
Stanford Shateyi
Keyword(s):  
Trivial Extension ◽  
Triangular Algebra ◽  
Trivial Extension Algebra ◽  
Extension Algebra

AbstractThe structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.

scholarly journals Lie Derivations on Trivial Extension Algebras

2017 ◽  
Vol 31 (1) ◽  
pp. 141-153 ◽  
Author(s):  
Amir Hosein Mokhtari ◽  
Fahimeh Moafian ◽  
Hamid Reza Ebrahimi Vishki
Keyword(s):  
Trivial Extension ◽  
Lie Derivation ◽  
Trivial Extension Algebra ◽  
Triangular Algebras ◽  
Extension Algebra ◽  
Lie Derivations

Abstract In this paper we provide some conditions under which a Lie derivation on a trivial extension algebra is proper, that is, it can be expressed as a sum of a derivation and a center valued map vanishing at commutators. We then apply our results for triangular algebras. Some illuminating examples are also included.

Hochschild cohomology, finiteness conditions and a generalization of d-Koszul algebras

2021 ◽  
pp. 2250147
Author(s):  
Ruaa Jawad ◽  
Nicole Snashall
Keyword(s):  
Hochschild Cohomology ◽  
Finiteness Condition ◽  
Koszul Algebras ◽  
Finite Dimensional Algebra ◽  
Koszul Algebra ◽  
Dimensional Algebra ◽  
Finiteness Conditions ◽  
Finite Dimensional ◽  
Projective Resolutions

Given a finite-dimensional algebra [Formula: see text] and [Formula: see text], we construct a new algebra [Formula: see text], called the stretched algebra, and relate the homological properties of [Formula: see text] and [Formula: see text]. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that [Formula: see text] has (Fg) if and only if [Formula: see text] has (Fg). We also consider projective resolutions and apply our results in the case where [Formula: see text] is a [Formula: see text]-Koszul algebra for some [Formula: see text].

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.

COMPLEXITY OF TRIVIAL EXTENSIONS OF ITERATED TILTED ALGEBRAS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250067 ◽  
Author(s):  
MARJU PURIN
Keyword(s):  
Representation Type ◽  
Trivial Extension ◽  
Closed Field ◽  
Hereditary Algebra ◽  
Algebraically Closed Field ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras

We study the complexity of a family of finite-dimensional self-injective k-algebras where k is an algebraically closed field. More precisely, let T be the trivial extension of an iterated tilted algebra of type H. We prove that modules over the trivial extension T all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H.

scholarly journals Verma Modules and Preprojective Algebras

2006 ◽  
Vol 182 ◽  
pp. 241-258 ◽  
Author(s):  
Christof Geiss ◽  
Bernard Leclerc ◽  
Jan Schröer
Keyword(s):  
Lie Algebra ◽  
Geometric Construction ◽  
Verma Modules ◽  
Preprojective Algebra ◽  
Finite Dimensional ◽  
Preprojective Algebras ◽  
Constructible Functions

AbstractWe give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra g in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra Λ.

scholarly journals Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

2020 ◽  
Author(s):  
Piotr M. Hajac ◽  
Tomasz Maszczyk
Keyword(s):  
Cyclic Homology ◽  
Quantum Deformation ◽  
Homotopy Formula ◽  
Standard Quantum ◽  
Unital Algebra ◽  
Hopf Fibrations ◽  
Chain Homotopy ◽  
Cartan Model ◽  
Extension Algebra ◽  
Hochschild Complex

AbstractViewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

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