This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)-admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shortest (+)-admissible sequence annihilating the module. A (+)-admissible sequence is the shortest sequence annihilating some preprojective module if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. These statements have the following application that strengthens known results of Howlett and Fomin-Zelevinsky. For any fixed Coxeter element of the Weyl group associated to an indecomposable symmetric generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words.
This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the gr…
Certain results on representations of quivers have analogs in the structure theory of general Coxeter groups. A fixed Coxeter element turns the Coxeter graph into an acyclic quiver, allowing for the definition of a preprojective root. A positive root is an analog of an indecomposable representation of the quiver. The Coxeter group is finite if and only if every positive root is preprojective, which is analogous to the well-known result that a quiver is of finite representation type if and only if every indecomposable representation is preprojective. Combinatorics of orientation-admissible word…
We study minimal degenerations between preprojective modules over wild quivers. Asymptotic properties of such degenerations are studied, with respect to codimension and numbers of indecomposable direct summands. We provide families of minimal disjoint degenerations of arbitrary codimension for almost all wild quivers and show that no such examples exist in the remaining cases.
In this paper, we compute the product structure of the Hochschild cohomology of preprojective algebras of quivers of type D and E over a field of characteristic zero.
This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver. The results apply to the study of redu…
The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that.
We calculate the additive and multiplicative structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types T. We also compute the calculus structure which is formed by the Hochschild homology/cohomology pair.
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the r…
We calculate the additive structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types ADE.
Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassm…
Let $Q$ be a tree-type quiver, $\mathbf{k} Q$ its path algebra, and $λ$ a nonzero element in the field $\mathbf{k}$. We construct irreducible morphisms in the Auslander-Reiten quiver of the transjective component of the bounded derived category of $\mathbf{k} Q$ that satisfy what we call the $λ$-relations. When $λ=1$, the relations are known as mesh relations. When $λ=-1$, they are known as commutativity relations. Using this technique together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree-type quiver, several descriptions of its p…
Let A be the central extension of the preprojective algebra of an ADE quiver introduced by P. Etingof and E. Rains in math/0503393. The paper math/0606403 computes the structure of the zeroth Hochschild (co)homology of A. We generalize the results of math/0606403 by calculating the additive structure of all the Hochschild homology and cohomology groups of A and the cyclic homology of A, and to describe the universal deformation of A. Namely, we show that the (co)homology is periodic with period 4, and compute the first four (co)homology groups in each case.
This paper aims at a geometric realization of the Yangian of non-simply laced type in terms of quiver with potentials. For every quiver with symmetrizer, there is an extended quiver with superpotential, whose Jacobian algebra is the generalized preprojective algebra of Geiß, Leclerc, and Schröer arXiv:1410.1403. We study the cohomological Hall algebra of Kontsevich and Soibelman associated to this quiver with potential. In particular, we prove a dimensional reduction result, and provide a shuffle formula of this cohomological Hall algebra. In the case when the quiver with symmetrizer comes fro…
In this paper, we study a preprojective algebra for quivers decorated with $k$-algebras and bimodules, which generalizes work of Gabriel for ordinary quivers, work of Dlab and Ringel for $k$-species, and recent work of de Thanhoffer de Völcsey and Presotto, which has recently appeared from a different perspective in work of Külshammer. As for undecorated quivers, we show that its moduli space of representations recovers the Hamiltonian reduction of the cotangent bundle over the space of representations of the decorated quiver. These algebras yield degenerations of ordinary preprojective algebr…
Let $Q$ be a wild $n$-Kronecker quiver, i.e., a quiver with two vertices, labeled by 1 and 2, and $n\geq 3$ arrows from 2 to 1. The indecomposable regular modules with preprojective Gabriel-Roiter submodules, in particular, those $τ^{-i}X$ with $\udim X=(1,c)$ for $i\geq 0$ and some $1\leq c\leq n-1$ will be studied. It will be shown that for each $i\geq 0$ the irreducible monomorphisms starting with $τ^{-i}X$ give rise to a sequence of Gabriel-Roiter inclusions, and moreover, the Gabriel-Roiter measures of those produce a sequence of direct successors. In particular, there are infinitely many…
Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that…
The aim of this paper is to study the space of stability conditions on the bounded derived category of nilpotent modules over the preprojective algebra associated with a quiver without loops. We describe this space as a covering space of some open set determined by the root system of the Kac-Moody Lie algebra associated with the quiver.
For a commutative ring $R$ and an ADE Dynkin quiver $Q$, we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter $q=1$, is isomorphic to the (additive) preprojective algebra as $R$-algebras if and only if the bad primes for $Q$ (2 in type D, 2 and 3 for $Q = E_{6}$, $E_{7}$ and 2, 3 and 5 for $Q= E_{8}$) are invertible in $R$. We construct an explicit isomorphism over $\mathbb{Z}[1/2]$ in type D, over $\mathbb{Z}[1/2, 1/3]$ for $Q = E_{6}$, $E_{7}$ and over $\mathbb{Z}[1/2, 1/3, 1/5]$ for $Q=E_{8}$. Conversely, if some bad prime is not invertible in $R…
We compute explicitly up to Morita-equivalence the skew group algebra of a finite group acting on the path algebra of a quiver and the skew group algebra of a finite group acting on a preprojective algebra. These results generalize previous results of Reiten and Riedtmann for a cyclic group acting on the path algebra of a quiver and of Reiten and Van den Bergh for a finite subgroup of $\SL(\C X \oplus \C Y)$ acting on $\C[X, Y]$.
For any valued quiver, by using BGP-reflection functors, an injection from the set of preprojective objects in the cluster category to the set of cluster variables of the corresponding cluster algebra is given, the images are called preprojective cluster variables. It is proved that all preprojective cluster variables have denominators $u^{\underline{dim}M}$ in their irreducible fractions of integral polynomials, where $M$ is the corresponding preprojective module or preinjective module. If the quiver is of Dynkin type, we generalize the denominator theorem in [FZ2] to any seed, and also gener…
We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cell…
Suppose that $Q$ is a finite quiver and $G\subseteq \Aut(Q)$ is a finite group, $k$ is an algebraic closed field whose characteristic does not divide the order of $G$. For any algebra $Λ=kQ/{\mathcal {I}}$, $\mathcal {I}$ is an arbitrary ideal of path algebra $kQ$, we give all the indecomposable $ΛG$-modules from indecomposable $Λ$-modules when $G$ is abelian. In particular, we apply this result to the deformed preprojective algebra $Π_{Q}^λ$, and get a reflection functor for the module category of $Π_{Q}^λG$. Furthermore, we construct a new quiver $Q_{G}$ and prove that $Π_{Q}^λG$ is Morita e…
The properties of the preprojective algebra are very di fferent whether the associated quiver is of Dynkin type or not. However in both cases, one can construct from it a triangulated category of Calabi-Yau dimension 2. In this note we explain the generalizations of this fact in the context of higher preprojective algebra, and we give some homological properties that characterize preprojective algebras.
We introduce for each quiver $Q$ and each algebraic oriented cohomology theory $A$, the cohomological Hall algebra (CoHA) of $Q$, as the $A$-homology of the moduli of representations of the preprojective algebra of $Q$. This generalizes the $K$-theoretic Hall algebra of commuting varieties defined by Schiffmann-Vasserot. When $A$ is the Morava $K$-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups. We construct an action of the preprojective CoHA on the $A$-homology of Nakajima quiver varieties. We comp…
Let $Q$ be a finite type quiver i.e. ADE Dynkin quiver. Denote by $Λ$ its preprojective algebra. It is known that there are finitely many indecomposable $Λ$-modules if and only if $Q$ is of type $A_1,A_2,A_3,A_4$. In this paper, extending Lusztig's construction of $U\frak{n}_+$, we study an algebra generated by these indecomposable submodules. It turns out that it forms the universal enveloping algebra of some nilpotent Lie algebra inside the function algebra on Lusztig's nilpotent scheme. The defining relations of the corresponding nilpotent Lie algebra for type $A_1, A_2,A_3,A_4$ are given h…
Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \mathbb{Z} Q.
The work of the first author on the moment map for representations of quivers included a classification of the possible dimension vectors of simple modules for deformed preprojective algebras. That classification was later used to solve an additive analogue of the Deligne-Simpson problem. The last step in the proof of the classification involved some general position arguments; here we give a new approach which avoids such arguments.
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 r…
We describe a method for an explicit determination of indecomposable preprojective and preinjective representations for extended Dynkin quivers by vector spaces and matrices. This method uses tilting theory and the explicit knowledge of indecomposable modules over the corresponding canonical algebra of domestic type.
We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they may be used to describe the components of Lusztig quiver varieties.
The goal of this paper is to prove that if Q is a connected non-Dynkin quiver then the preprojective algebra of Q over any field k is Koszul, and has Hilbert series 1/(1-Ct+t^2), where C is the adjacency matrix of the double of Q. (This result, in somewhat less general formulations, was previously obtained by Martinez-Villa and Malkin-Ostrik-Vybornov). We also prove a similar result for the partial preprojective algebra of any connected quiver Q, associated to a subset J of the set I of vertices of Q (by definition, this is the quotient of the path algebra of the double by the preprojective al…
For a finite acyclic quiver $Q$ and the corresponding preprojective algebra $Π$, we study the factor algebra $Π_w$ associated with a element $w$ in the Coxeter group introduced by Buan-Iyama-Reiten-Scott. The algebra $Π_w$ has a natural $\mathbb{Z}$-grading, and we prove that $\underline{\mathsf{Sub}}^{\mathbb{Z}}Π_w$ has a tilting object $M$. Moreover, we show that the endomorphism algebra of $M$ is isomorphic to the stable Auslander algebra of a certain torsion free class of $\mathsf{mod}\,kQ$.
We study the stable category 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 $M(\bf{w})$ of this category associated with each reduced expression $\bf{w}$ of $w$ and give a sufficient condition on $\bf{w}$ such that $M(\bf{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\bf{w})$. Moreover, we compare it with a triangle equivalence given by Amiot-Reiten-Todorov for a cluster category.
We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig.
Let Γbe a Dynkin diagram of type A,D,E and let R denote the corresponding root system. In this paper we give a categorical construction of R from Γ. Instead of choosing an orientation of Γand studying representations of the associated quiver, we study representations of a canonical quiver \Gammahat associated to Γ. This construction is very closely related to the preprojective algebra of Γ. In particular, the construction gives a certain periodicity result about the preprojective algebra.
Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers.
We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new $p$-torsion classes in degrees 2p^l, l >= 1, We relate these classes by p-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. An important tool is a generalization of the Diamond Lemma to modules over commutative rings, which we give in the ap…
We give a class of finite subgroups $G<SL(n, k)$ for which the skew-group algebra $k[x_1,\ldots, x_n]\#G$ does not admit the grading structure of a higher preprojective algebra. Namely, we prove that if a finite group $G<SL(n, k)$ is conjugate to a finite subgroup of $SL(n_1, k)\times SL(n_2, k)$, for some $n_1, n_2\geq 1$, then the skew-group algebra $k[x_1,\ldots,x_n]\#G$ is not Morita equivalent to a higher preprojective algebra. This is related to the preprojective algebra structure on the tensor product of two Koszul bimodule Calabi-Yau algebras. We prove that such an algebra cannot be en…
Let $Q$ be a finite quiver of Dynkin type and $Λ=Λ_Q$ be the preprojective algebra of $Q$ over an algebraically closed field $k$. Let $\mathcal {T}_Λ$ be the mutation graph of maximal rigid $Λ$ modules. Geiss, Leclerc and Schr$\ddot{\rm o}$er conjectured that $\mathcal {T}_Λ$ is connected, see [C.Geiss, B.Leclerc, J.Schröer, Rigid modules over preprojective algebras, Invent.Math., 165(2006), 589-632]. In this paper, we prove that this conjecture is true when $Λ$ is of representation finite type or tame type. Moreover, we also prove that $\mathcal {T}_Λ$ is isomorphic to the tilting graph of ${…
We compare two cohomological Hall algebras (CoHA). The first one is the preprojective CoHA introduced by the authors in arXiv:1407.7994 associated to each quiver Q, and each algebraic oriented cohomology theory A. It is defined as the A-homology of the moduli of representations of the preprojective algebra of Q, generalizing the K-theoretic Hall algebra of commuting varieties of Schiffmann-Vasserot. The other one is the critical CoHA defined by Kontsevich-Soibelman associated to each quiver with potentials. It is defined using the equivariant cohomology with compact support with coefficients i…
Let $Q$ be an acyclic quiver and $Λ$ be the complete preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and studied in \cite{Bua2} a finite dimensional algebra $Λ_w=Λ/I_w$. In this paper we look at filtrations of $Λ_w$ associated to any reduced expression $\mathbf{w}$ of $w$. We are especially interested in the case where the word $\mathbf{w}$ is $c$-sortable, where $c$ is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting $kQ…
We apply the notion of hyperfinite families of modules to the wild path algebras of generalised Kronecker quivers $kΘ(d)$. While the preprojective and postinjective component are hyperfinite, we show the existence of a family of non-hyperfinite modules in the regular component for some $d$. Making use of dimension expanders to achieve this, our construction is more explicit than previous results. From this it follows that no finitely controlled wild algebra is of amenable representation type.
We provide a technique to compute the Euler characteristic of a class of projective varieties called quiver Grassmannians. This technique applies to quiver Grassmannians associated with "orientable string modules". As an application we explicitly compute the Euler characteristic of quiver Grassmannians associated with indecomposable preprojective, preinjective and regular homogeneous representations of an affine quiver of type $\tilde{A}_{p,1}$. For $p=1$, this approach provides another proof of a result due to P. Caldero and A. Zelevinsky in \cite{CZ}.
We give an interpretation of the map $π^c$ defined by Reading, which is a map from the elements of a Coxeter group to the $c$-sortable elements, in terms of the representation theory of preprojective algebras. Moreover, we study a close relationship between $c$-sortable elements and torsion pairs, and give an explicit description of the cofinite torsion classes in the context of the Coxeter group. As a consequence, we give a proof of some conjectures proposed by Oppermann, Reiten, and the second author.
