In this paper, we study two-term tilting complexes for preprojective algebras of non-Dynkin type. We show that there exist two families of two-term tilting complexes, which are respectively parameterized by the elements of the corresponding Coxeter group. Moreover, we provide the complete classification in the case of affine type by showing that any two-term silting complex belongs one of them. For this purpose, we also discuss the Krull-Schmidt property for the homotopy category of finitely generated projective modules over a complete ring.
We prove that every deformed preprojective algebra of Dynkin type $\mathbb{E}_6$ is isomorphic to the preprojective algebra of Dynkin type $\mathbb{E}_6$.
We provide a complete classification of finite-dimensional self-injective algebras which are socle equivalent to preprojective algebras of generalized Dynkin type. In particular, we conclude that these algebras are deformed preprojective algebras of generalized Dynkin type (in the sense of [5, 12]), and hence are periodic algebras.
We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojective algebras of generalized Dynkin type are periodic. In particular, we obtain an algorithmic procedure showing that non-trivial deformed preprojective algebras of Dynkin types $\mathbb{E}_7$ and $\mathbb{E}_8$ exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types $\mathbb{E}_6$, $\mathbb{E}_7$ and $\mathbb{E}_8$ are periodic and we obtain an algorithm for a classification of such algebras, up to algebra isomorphism. We do it b…
In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we …
For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Geiß~ et al. (2016) introduced a generalized preprojective algebra $Π(C, D)$. We study tilting modules and support $τ$-tilting modules for the generalized preprojective algebra $Π(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $Π(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $τ$-tilting $Π(C,D)$-modules and the corresponding Weyl group $…
For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schröer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $Π$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We show that any basic support $τ$-tilting $Π$-module is locally free and gives a classification theorem of torsion-free classes in $\operatorname{\mathbf{rep}}Π$ as the generalization of the work of Mizuno [Math. Z. 277 (2014)].
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.
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…
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.
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 $\mathsf{mod}\,kQ$.
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 study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.
We give a combinatorial classification of cluster tilting subcategories and torsion pairs in Igusa--Todorov cluster categories of Dynkin type $A_{ \infty }$.
We compute the Hochschild Cohomology of a finite-dimensional preprojective algebra of generalized Dynkin type Ln over a field of characteristic different from 2 . In particular, we describe the ring structure of the Hochschild Cohomology ring under the Yoneda product by giving an explicit presentation by generators and relations.
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 study support $τ$-tilting modules over preprojective algebras of Dynkin type. We classify basic support $τ$-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and other important objects in representation theory. We also study $g$-matrices of support $τ$-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of $g$-matrices and prove that cones given by $g$-matrices co…
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.
This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 <= \ell <= d-1, we show a weakly d-cluster tilting subcategory T_{\ell} which has an indecomposable object with precisely \ell…
We apply tilting theory over preprojective algebras $Lambda$ to a study of moduli space of $Lambda$-modules. We define the categories of semistable modules and give an equivalence, so-called reflection functors, between them by using tilting modules over $Lambda$. Moreover we prove that the equivalence induces an isomorphism of algebraic varieties between moduli spaces. In particular, we study in the case when the moduli spaces related to the Kleinian singularity. We generalize a result of Crawley-Boevey which is known another proof of the McKay correspondence of Ito-Nakamura type.
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 show that $m$-APR tilting preserves $n$-representation infiniteness for $1\leq m\leq n$. Moreover, we show that these tilting modules provide different tilting modules for the corresponding higher preprojective algebras, which is $(n+1)$-CY algebras. We also study the interplay of the two kinds of tilting modules.
In this paper we characterize when a recollement of compactly generated triangulated categories admits a ladder of some height going either upwards or downwards. As an application, we show that the derived category of the preprojective algebra of Dynkin type $\mathbb{A}_n$ admits a periodic infinite ladder, where the one outer term in the recollement is the derived category of a differential graded algebra.
We compute the minimal model for Ginzburg algebras associated to acyclic quivers $Q$. In particular, we prove that there is a natural grading on the Ginzburg algebra making it formal and quasi-isomorphic to the preprojective algebra in non-Dynkin type, and in Dynkin type is quasi-isomorphic to a twisted polynomial algebra over the preprojective with a unique higher $A_\infty$-composition. To prove these results, we construct and study the minimal model of an $A_\infty$-envelope of the derived category $\mathcal{D}^b(Q)$ whose higher compositions encode the triangulated structure of $\mathcal{D…
We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of …
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…
We introduce (n+1)-preprojective algebras of algebras of global dimension n. We show that if an algebra is n-representation-finite then its (n+1)-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)-preprojective algebra is (n+1)-Calabi-Yau, and, more precisely, it is the (n+1)-Amiot cluster category of the stable n-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)-cluster tilting object. We show that even if the (n+1)-preprojective algebra is not self-injective, under certain assumptions …
The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper exhibits the number of support-tilting modules for any Dynkin algebra. Since the support-tilting modules for a Dynkin algebra correspond bijectively to the non-crossing partitions of the same type, the calculations presented here may also be considered as a categorification of results concerning non-crossing partitions.
Bialkowski, Erdmann and Skowronski classified those indecomposable self-injective algebras for which the Nakayama shift of every (non-projective) simple module is isomorphic to its third syzygy. It turned out that these are precisely the deformations, in a suitable sense, of preprojective algebras associated to the simply laced ADE Dynkin diagrams and of another graph L_n, which also occurs in the Happel-Preiser-Ringel classification of subadditive but not additive functions. In this paper we study these deformed preprojective algebras of type L via their Kuelshammer spaces, for which we give …
The aim of this paper is to study a poset isomorphism between two support $τ$-tilting posets. We take several algebraic information from combinatorial properties of support $τ$-tilting posets. As an application, we treat a certain class of basic algebras which contains preprojective algebras of type $A$, Nakayama algebras, and generalized Brauer tree algebras. We provide a necessary condition for that an algebra $Λ$ share the same support $τ$-tilting poset with a given algebra $Γ$ in this class. Furthermore, we see that this necessary condition is also a sufficient condition if $Γ$ is either a…
We consider the dual space of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of Dynkin or Euclidean quiver. We prove that this space contains an open dense subset isomorphic to the product of dual spaces of full linear groups and, perhaps, one more (explicitly described) space. The proof uses the technique of bimodule categories, deformations and representations of quivers.
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…
For a fixed finite dimensional algebra $A$, we study representation embeddings of the form $mod(B)\rightarrow mod(A)$. Such an embedding is called homological, if it induces an isomorphism on all Ext-groups and weakly homological, if only Ext$^1$ is preserved. In case $A$ is a preprojective algebra of Dynkin type, we give an explicit classification of all weakly homological and homological embeddings. Furthermore, we show that for self-injective algebras a classification of homological embeddings becomes accessible once these algebras fulfil the Tachikawa conjecture.
We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding symplectic reflection algebra for wreath product.
Following the article of C. M. Ringel we introduce preprojective algebras of a Dynkin quiver $Q$ starting from three definitions which, despite concerning completely different algebraic structures, turn out to be equivalent. Our main result is a new version of Ringel's proofs that applies a theorem by Happel and exploits the techniques of homological algebra. Moreover we show that the definition of the preprojective algebra given with the usual notion of commutator is equivalent to the definition with the "generalised" commutator.
Let $\mathcal{C}$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster tilting object. Under some constructibility assumptions on $\mathcal{C}$ which are satisfied for instance by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra, we prove a multiplication formula for the cluster character associated with any cluster tilting object. This formula generalizes those obtained by Caldero-Keller for representation finite path algebras and by Xiao-Xu for finite-dimensional path algebras. It is analogous to a formula obt…
We study higher cluster tilting objects in generalized higher cluster categories arising from dg algebras of higher Calabi-Yau dimension. Taking advantage of silting mutations of Aihara-Iyama, we obtain a class of $m$-cluster tilting objects in generalized $m$-cluster categories. For generalized $m$-cluster categories arising from strongly ($m+2$)-Calabi-Yau dg algebras, by using truncations of minimal cofibrant resolutions of simple modules, we prove that each almost complete $m$-cluster tilting $P$-object has exactly $m+1$ complements with periodicity property. This leads us to the conjectur…
Let $\la$ be a preprojective algebra of simply laced Dynkin type $Δ$. We study maximal rigid $\la$-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring $\C[N]$ of polynomial functions on a maximal unipotent subgroup $N$ of a complex Lie group of type $Δ$. As an application we obtain that all cluster monomials of $\C[N]$ belong to the dual semicanonical basis.
Let L be a preprojective algebra of Dynkin type, and let G be the corresponding complex semisimple simply connected algebraic group. We study rigid modules in subcategories sub(Q) for Q an injective L-module, and we introduce a mutation operation between complete rigid modules in sub(Q). This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.
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…
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.
Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the Deligne-Simpson problem, integrable systems and noncommutative geometry. In this paper we offer three contributions to the study of such algebras: (1) the 2-Calabi-Yau property; (2) the unification of the reflection functors of Crawley-Boevey and Holland with reflection functors for the usual preprojective algebras; and (3) the classification of tilting ideals in 2-Calabi-Yau algebras, and especially in deformed…
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…
In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $τ$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $\mathbb{A}_{1}, \mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type $\mathbb{A}_{1}$.
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 give a characterization of radical square zero bound quiver algebras $\mathbf{k} Q/\mathcal{J}^2$ that admit $n$-cluster tilting subcategories and $n\mathbb{Z}$-cluster tilting subcategories in terms of $Q$. We also show that if $Q$ is not of cyclically oriented extended Dynkin type $\tilde{A}$, then the poset of $n$-cluster tilting subcategories of $\mathbf{k} Q/\mathcal{J}^2$ with relation given by inclusion forms a lattice isomorphic to the opposite of the lattice of divisors of an integer which depends on $Q$.
We describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders. This completes the categorification of Geiss-Leclerc-Schröer by adding the missing coefficients. To achieve this, for an order $A$ and an idempotent $e \in A$, we introduce a subcategory $\operatorname{CM}\nolimits_e A$ of $\operatorname{CM}\nolimits A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $(\operatorname{CM}\nolimits_e A)…
Let $\mathbb{X}$ be a weighted projective line and $\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such that $τ^2 T\cong T$, where $τ$ is the Auslander-Reiten translation in $D^b(\operatorname{coh}\mathbb{X})$. As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we…
We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form T=R_U\oplus R_U /R where U is a union of tubes, and R_U denotes the universal localization of R at U in the sense of Schofield and Crawley-Boevey. Here R_U/R is a direct sum of the Prüfer modules corresponding to the tubes in U. Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class Gen L consists of all modules wit…
We study cluster tilting modules in mesh algebras of Dynkin type, providing a new proof for their existence. In all but one case, we show that these are precisely the maximal rigid modules, and that they are equivariant for a certain automorphism. We further study their mutation, providing an example of mutation in an abelian category which is not stably 2-Calabi-Yau, and explicitly describe the combinatorics.
Let $B$ be an one-point extension of a finite dimensional $k$-algebra $A$ by a simple $A$-module at a source point $i$. In this paper, we classify the $τ$-tilting modules over $B$. Moreover, it is shown that there are equations $$|\tilt B|=|\tilt A|+|\tilt A/\langle e_i\rangle|\quad \text{and}\quad |\stilt B|=2|\stilt A|+|\stilt A/\langle e_i\rangle|.$$ As a consequence, we can calculate the numbers of $τ$-tilting modules and support $τ$-tilting modules over linearly Dynkin type algebras whose square radical are zero.
