We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on affine the weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to a interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and tame type that arise from our study. We also prove that the path algebra of star shaped quivers are the Ext algebra of a suitable subcategory.
We develop the theory of 2-quivers and quiver 2-categories to run in parallel with the classical theory of quiver algebras. A quiver 2-category is always finitary, and, conversely, every finitary 2-category will be bi-equivalent with a quiver 2-category for a unique underlying reduced 2-quiver. As an application, we produce an example of a fiat 2-category, one of whose Duflo involutions is not self-adjoint.
Our published paper contains an incorrect statement of a result due to Artin and Zhang. This corrigendum gives the correct statement of their result and includes a new result that allows us to use their result to prove our main theorem. Thus the main theorem of our published paper is correct as stated but its proof must be modified.
Let $n \geq 2$. We introduce the notion of $n$-representations of quivers, and we explicitly provide concrete examples of $2$-representations of quivers. We establish the categories of $n$-representations and investigate kernels and cokernels in the categories of $n$-representations of quivers. Further, we construct them in terms of kernels and cokernels of morphisms in the usual categories of quiver representations. We show that every morphism in the categories of $n$-representations has a canonical decomposition. Most importantly, we prove that the categories of $n$-representations of quiver…
In this paper, we introduce multi-layer quiver and show how to construct an $(n+1)$-slice algebras of infinite type from an $n$-slice algebra of infinite type using the bound quivers. This leads to constructing $(n+1)$-slice algebras of infinite type as matrix algebra and as tensor algebra of an $n$-slice algebra and equivalences of their module categories as the module categories of diagram of some quiver of type $\widetilde{A}_{n+1}$.
We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.
We show that Derksen-Weyman-Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi-Yau triangulated categories. Our approach is related to that of Iyama-Reiten and Koszul dual to that of Kontsevich-Soibelman. It improves on previous work by Vitoria. In the appendix, the first-named author studies pseudo-compact derived categories of certain pseudo-compact dg algebras.
We introduce the notion of (twisted) quiver representations in abelian categories and study the category of such representations. We construct standard resolutions and coresolutions of quiver representations and study basic homological properties of the category of representations. These results are applied in the case of parabolic vector bundles and framed coherent sheaves.
Motivated by the definition of Duflo involution for fiat $2$-categories, we define certain analogues of Duflo involution for arbitrary finitary $2$-categories and show that such Duflo involutions exist for two classes of finitary $2$-categories associated with tree path algebras. Additionally, we describe the quiver for the algebra underlying the principal $2$-representation for these two classes of finitary $2$-categories.
We introduce a new method to construct a Grothendieck category from a given colored quiver. This is a variant of the construction used to prove that every partially ordered set arises as the atom spectrum of a Grothendieck category. Using the new method, we prove that for every finite partially ordered set, there exists a locally noetherian Grothendieck category such that every nonzero object contains a compressible subobject and its atom spectrum is isomorphic to the given partially ordered set.
We embed triangulated categories defined by quivers with potential arising from ideal triangulations of marked bordered surfaces into Fukaya categories of quasi-projective 3-folds associated to meromorphic quadratic differentials. Together with previous results, this yields non-trivial computations of spaces of stability conditions on Fukaya categories of symplectic 6-manifolds.
In this paper, we introduce generalized quiver varieties which include as special cases classical and cyclic quiver varieties. The geometry of generalized quiver varieties is governed by a finitely generated algebra P: the algebra P is self-injective if the quiver Q is of Dynkin type, and coincides with the preprojective algebra in the case of classical quiver varieties. We show that in the Dynkin case the strata of generalized quiver varieties are in bijection with the isomorphism classes of objects in proj P, and that their degeneration order coincides with the Jensen-Su-Zimmermann's degener…
In the first part of this paper, we prove the existence of torsion free covers in the category of representations of quivers, $(Q,R-Mod)$, for a wide class of quivers included in the class of the so-called source injective representation quivers provided that any direct sum of torsion free and injective $R$-modules is injective. In the second part, we prove the existence of $\mathscr{F}_{cw}$-covers and $\mathscr{F}_{cw}^{\perp}$-envelopes for any quiver $Q$ and any ring $R$ with unity, where $\mathscr{F}_{cw}$ is the class of all "componentwise" flat representations of $Q$.
We prove in full generality that the generalized quantum affine Schur-Weyl duality functor, introduced by Kang-Kashiwara-Kim, gives an equivalence between the category of finite-dimensional modules over a quiver Hecke algebra and a certain full subcategory of finite-dimensional modules over a quantum affine algebra which is a generalization of the Hernandez-Leclerc's category $\mathcal{C}_Q$. This was previously proved in untwisted $ADE$ types by Fujita using the geometry of quiver varieties, which is not applicable in general. Our proof is purely algebraic, and so can be extended uniformly to…
We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations. Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However,…
Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. We construct a triangulated category $\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When $\Cc_A$ is $\Hom$-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schr{ö}er and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluste…
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.
We study finite quasi-quantum groups in their quiver setting developed recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to …
We study representation theory of quantizations of Nakajima quiver varieties associated to bouquet quivers. We show that there are no finite dimensional representations of the quantizations $\overline{\mathcal{A}}_λ(n, \ell)$ if dim $V=n$ is greater than $1$ and so is the number of loops $\ell$. We find that there is a Hamiltonian torus action with finitely many fixed points in case $n\leq 3$, provide the dimensions of Hom-spaces between standard objects in category $\mathcal{O}$ and compute the multiplicities of simples in standards for $n=2$ in case of one-dimensional framing and generic one…
We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique to recover a cluster-tilting object from its graded quiver in the cluster category of coh $\mathbb{X}$.
We show that the $m$-cluster category of type $A_{n-1}$ is equivalent to a certain geometrically-defined category of diagonals of a regular $nm+2$-gon. This generalises a result of Caldero, Chapoton and Schiffler for $m=1$. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the $m$th power of a translation quiver and show how it can be used to realise the $m$-cluster category in terms of the cluster category.
Inspired by recent work of Hernandez-Leclerc and Leclerc-Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc-Plamondon have shown, the points of these varieties can be interpreted as representations of a category, which we call the (singular) Nakajima category S. We determine the quiver of S and the number of minimal relations between any two given vertices. We construct a delta-functor Phi taking each finite-dimensional representation of S to an object of the derived categor…
We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers and bilinear pairings as special cases. We prove a Krull-Schmidt Theorem in this general context, which results in a Krull-Schmidt Theorem for the special cases just mentioned. We also show that cancellation holds under milder assumptions. Using similar ideas we prove a version of Fitting's Lemma for natural transformations between functors.
We prove that the weight multiplicities of the integrable irreducible highest weight module over the Kac-Moody algebra associated to a quiver are equal to the root multiplicities of the Kac-Moody algebra associated to some enlarged quiver. To do this, we use the Kac conjecture for indivisible roots and a relation between the Poincare polynomials of quiver varieties and the Kac polynomials, counting the number of absolutely irreducible representations of the quiver over finite fields. As a corollary of this relation, we get an explicit formula for the Poincare polynomials of quiver varieties, w…
Consider a 2-Calabi--Yau triangulated category with a Bridgeland stability condition. We devise an effective procedure to reduce the phase spread of an object by applying spherical twists. Using this, we give new proofs of the following theorems for 2-Calabi--Yau categories associated to ADE quivers: (1) all spherical objects lie in a single orbit of the braid group, and (2) the space of Bridgeland stability conditions is connected.
We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain Iwanaga-Gorenstein algebras introduced in Part I. The proof relies on the realization of the positive part of the enveloping algebra of a simple Lie algebra of the same finite type as a convolution algebra of constructible functions on representation varieties of $H$, given in Part III. Along the way, we obtain a new result on the PBW basis of this convolution algeb…
We derive a geometric model for the module category $\operatorname{mod} \mathbf{k} Q$ of a Dynkin quiver $Q$ via the heart of a total stability condition on the bounded derived category of $\operatorname{mod} \mathbf{k} Q$. As an application, we prove Reineke's conjecture that there is a stability function on $\operatorname{mod} \mathbf{k} Q$ making any indecomposables stable.
The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang-Zelevinsky in finite type. In type $A_n^{\!(1)}$ and fin…
This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
In this paper, we study the space of stability conditions on a certain $N$-Calabi-Yau ($\text{CY}_N$) category associated to an $A_n$-quiver. Recently, Bridgeland and Smith constructed stability conditions on some $\text{CY}_3$ categories from meromorphic quadratic differentials with simple zeros. Generalizing their results to higher dimensional Calabi-Yau categories, we describe the space of stability conditions as the universal cover of the space of polynomials of degree $n+1$ with simple zeros. In particular, central charges of stability conditions on $\text{CY}_N$ categories are constructe…
Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretical interpretation of cluster mutation in case of acyclic cluster algebras of finite type.
EI-categories are a simultaneous generalisation of finite groups and finite quivers without oriented cycles. It is therefore a natural question to ask for a characterisation of finite representation type. For special classes of EI-categories a complete characterisation is obtained using quiver techniques. For EI-categories with two objects we present a necessary criterion for finite representation type. The complexity of this classification problem is illustrated by some examples.
Gillespie's Theorem gives a systematic way to construct model category structures on $\mathscr{C}( \mathscr{M} )$, the category of chain complexes over an abelian category $\mathscr{M}$. We can view $\mathscr{C}( \mathscr{M} )$ as the category of representations of the quiver $\cdots \rightarrow 2 \rightarrow 1 \rightarrow 0 \rightarrow -1 \rightarrow -2 \rightarrow \cdots$ with the relations that two consecutive arrows compose to $0$. This is a self-injective quiver with relations, and we generalise Gillespie's Theorem to other such quivers with relations. There is a large family of these, an…
Given a complete hereditary cotorsion pair $(\mathcal{A}, \mathcal{B})$ in an abelian category $\mathcal{C}$ satisfying certain conditions, we study the completeness of the induced cotorsion pairs $(Φ(\mathcal{A}), Φ(\mathcal{A})^{\perp})$ and $(^{\perp}Ψ(\mathcal{B}), Ψ(\mathcal{B}) )$ in the category $\mbox{Rep}(Q, \mathcal{C})$ of $\mathcal{C}$-valued representations of a given quiver $Q$. We show that if $Q$ is left rooted, then the cotorsion pair $(Φ(\mathcal{A}), Φ(\mathcal{A})^{\perp})$ is complete, and if $Q$ is right rooted, then the cotorsion pair $(^{\perp}Ψ(\mathcal{B}), Ψ(\mathcal…
We introduce generalizations of Kac-Moody 2-categories in which the quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced by the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka.
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using $d-$cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a $d-$compatibility degree $(-||-)$ on any pair of ``colored'' almost positive real Schur roots which generalizes previous definitions on the non-colored case, and call two such roots compatible provided the $d-$compatibility degree of them is zero. Associated to the root system $Φ$ corresponding to the va…
For a Dynkin quiver $Q$ (of type ADE), we consider a central completion of the convolution algebra of the equivariant K-group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc's monoidal category $\mathcal{C}_Q$ of modules over the quantum loop algebra $U_q(L\mathfrak{g})$ via Nakajima's homomorphism. As an application, we show that Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor gives an equivalence between the cate…
The spectral radius of matrix, also known as Frobenius-Perron dimension, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, find a way to calculate the Frobenius-Perron dimension of these algebras when they satisfy the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver …
As a generalization of acyclic 2-Calabi-Yau categories, we consider 2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study their cluster-tilting subcategories and the cluster combinatorics that they encode. We show that such categories have a cluster structure. Triangulated 2-Calabi-Yau categories with a directed cluster-tilting subcategory are closely related to representations of certain semi-hereditary categories, more specifically to representations of thread quivers. Thread quivers are a tool to classify and study certain semi-hereditary categories using both quiver…
We show that any simple transitive $2$-representation of the $2$-ca\-te\-go\-ry of projective endofunctors for the quiver algebra of $\Bbbk(\xymatrix{\bullet\ar[r]&\bullet})$ and for the quiver algebra of $\Bbbk(\xymatrix{\bullet\ar[r]\ar@/^/@{.}[rr]&\bullet\ar[r]&\bullet})$ is equivalent to a cell $2$-representation.
The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a geometric interpretation of the triangulated structure of $m$-cluster categories. Furthermore, we characterize all the connected components arising from a cluster category when taking the $m$-th power of its Auslander-Reiten quiver.
We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of $\underline{\mathsf{mod}}\,A$, that is, $\mathsf{D}_{\rm sg}(\mathsf{D}^{\rm b}(\mathsf{mod}\,A))\simeq \mathsf{D}^{\rm b}(\mathsf{mod}(\underline{\mathsf{mod}}\,A))$. This extends a result of Iyama-Oppermann for the path algebra $A$ of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras.
We first study the (canonical) orbit category of the bounded derived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where the quiver is of infinite Dynkin type. In particular, its Auslander-Reiten components are explicitly described. When the quiver is of type $\mathbb{A}_\infty$ or $\mathbb{A}_\infty^\infty$, we show that this orbit category is a cluster category, that is, its cluster-tilting subcategories form a cluster structure. When the quiver is of type $\mathbb{A}_\infty^\infty$, we shall give a geometrical descrip…
We develop a new approach to highest weight categories $\cal{C}$ with good (and cogood) posets of weights via pseudocompact algebras by introducing ascending (and descending) quasi-hereditary pseudocompact algebras. For $\cal{C}$ admitting a Chevalley duality, we define and investigate tilting modules and Ringel duals of the corresponding pseudocompact algebras. Finally, we illustrate all these concepts on an explicit example of the general linear supergroup $GL(1|1)$.
We show that for a finite-type Lie algebra $\mathfrak{g}$, the representation theory of quiver Hecke algebras provides a natural framework for the construction of Newton-Okounkov bodies associated to the quantum coordinate rings $\Aqnw$. When $\mathfrak{g}$ is simply-laced, we use Kang-Kashiwara-Kim-Oh's monoidal categorification to investigate the cluster theory of these bodies. In particular, our construction yields a simplex $\ds$ for every seed $\s$ of $\Aqnw$. We exhibit various properties of these simplices by characterizing their rational points, normal fans, and volumes. As an applicat…
We give a geometric realization of module categories of type $\tilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $\tilde{A}_n$. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.
We give a brief review of the cohomological Hall algebra CoHA $\mathcal{H}$ and the K-theoretical Hall algebra KHA $\mathcal{R}$ associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) $\mathcal{R}\to \hat{\mathcal{H}}^σ$ where $\hat{\mathcal{H}}^σ$ is a Zhang twist of the completion of $\mathcal{H}$. Moreover, we establish the equivalence of categories of ``locally finite'' graded modules $\hat{\mathcal{H}}^σ-{\rm Mod}_{lf}\simeq \mathcal{R}_{\mathbb Q}-{\rm Mod}_{lf}$. Examples of locally finite $\hat…
We study finitary 2-categories associated to dual projection functors for finite dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type A) we show that the monoid generated by dual projection functors is the Hecke-Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.
