Dynkin Quivers Research Articles

Abstract representation theory of Dynkin quivers of type A

We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard groupoid to derived Picard groupoids over arbitrary fields.These results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators. As further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field. This includes Yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral Serre duality result.Finally, using abstract representation theory of linearly oriented An-quivers, we construct canonical higher triangulations in stable derivators and hence, a posteriori, in stable model categories and stable ∞-categories.

Explicit description of generic representations for quivers of type [formula omitted] or [formula omitted

We describe explicitly a generic representation for Dynkin quivers of type An or Dn for any dimension vector.

Orbifold equivalent potentials

To a graded finite-rank matrix factorisation of the difference of two homogeneous potentials one can assign two numerical invariants, the left and right quantum dimensions. The existence of such a matrix factorisation with non-zero quantum dimensions defines an equivalence relation between potentials, giving rise to non-obvious equivalences of categories.Restricted to ADE singularities, the resulting equivalence classes of potentials are those of type {Ad−1} for d odd, {Ad−1,Dd/2+1} for d even but not in {12,18,30}, and {A11,D7,E6}, {A17,D10,E7} and {A29,D16,E8}. This is the result expected from two-dimensional rational conformal field theory, and it directly leads to new descriptions of and relations between the associated (derived) categories of matrix factorisations and Dynkin quiver representations.

EXPLICIT CONSTRUCTION OF COMPANION BASES

AbstractA companion basis for a quiver Γ mutation equivalent to a simply-laced Dynkin quiver is a subset of the associated root system which is a$\mathbb{Z}$-basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of Γ. It is known in typeA(and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation typeAorD.

Lattice structure of torsion classes for path algebras

We consider module categories of path algebras of connected acyclic quivers. It is shown in this paper that the set of functorially finite torsion classes form a lattice if and only if the quiver is either Dynkin quiver of type A, D, E, or the quiver has exactly two vertices.

Symmetric quiver Hecke algebras and R -matrices of quantum affine algebras, II

Let $\g$ be an untwisted affine Kac-Moody algebra of type $A^{(1)}_n$ $(n \ge 1)$ or $D^{(1)}_n$ $(n \ge 4)$ and let $\g_0$ be the underlying finite-dimensional simple Lie subalgebra of $\g$. For each Dynkin quiver $Q$ of type $\g_0$, Hernandez and Leclerc (\cite{HL11}) introduced a tensor subcategory $\CC_Q$ of the category of finite-dimensional integrable $\uqpg$-modules and proved that the Grothendieck ring of $\CC_Q$ is isomorphic to $\C [N]$, the coordinate ring of the unipotent group $N$ associated with $\g_0$. We apply the generalized quantum affine Schur-Weyl duality introduced in \cite{KKK13} to construct an exact functor $\F$ from the category of finite-dimensional graded $R$-modules to the category $\CC_Q$, where $R$ denotes the symmetric quiver Hecke algebra associated to $\g_0$. We prove that the homomorphism induced by the functor $\F$ coincides with the homomorphism of Hernandez and Leclerc and show that the functor $\F$ sends the simple modules to the simple modules.

Stable categories of Cohen-Macaulay modules and cluster categories: Dedicated to Ragnar-Olaf Buchweitz on the occasion of his sixtieth birthday

By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is triangle equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e., the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity $R$ and the generalized (higher) cluster category of a finite dimensional algebra $\Lambda$. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of $R$ as well as the higher preprojective algebra of an extension of $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay $R$-modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.

Hall polynomials and the Gabriel–Roiter submodules of simple homogeneous modules

Let be an arbitrary field and be an acyclic quiver of tame type (that is, of type ). Consider the path algebra , the category of finite‐dimensional right modules , and the minimal positive imaginary root of , denoted by . In the first part of the paper, we deduce that the Gabriel–Roiter (GR) inclusions in preprojective indecomposables and homogeneous modules of dimension , as well as their GR measures are field independent (a similar result due to Ringel being true in general over Dynkin quivers). Using this result, we can prove in a more general setting a theorem by Bo Chen which states that the GR submodule of a homogeneous module of dimension is preprojective of defect and so the pair is a Kronecker pair. The generalization consists in considering the originally missing case and using arbitrary fields (instead of algebraically closed ones). Our proof is based on the idea of Ringel (used in the Dynkin quiver context) of comparing all possible Hall polynomials with the special form they take in case of a GR inclusion. For this purpose, we determine (with the help of a program written in GAP) a list of tame Hall polynomials which may have further interesting applications.

Grothendieck Classes of Quiver Cycles as Iterated Residues

In the case of Dynkin quivers we establish a formula for the Grothendieck class of a quiver cycle as the iterated residue of a certain rational function, for which we provide an explicit combinatorial construction. Moreover, we utilize a new definition of the double stable Grothendieck polynomials due to Rimanyi and Szenes in terms of iterated residues to exhibit how the computation of quiver coefficients can be reduced to computing coefficients in Laurent expansions of certain rational functions.

Stability conditions and quantum dilogarithm identities for Dynkin quivers

We study the fundamental groups of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(ΓNQ) of the Calabi–Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected. As an application, we show that the Donaldson–Thomas type invariant associated to Q can be calculated as a quantum dilogarithm function on its exchange graph. In the case of D(ΓNQ), we show that the faithfulness of the Seidel–Thomas braid group action (which is known for Q of type A or N=2) implies the simply connectedness of the space of stability conditions.

Hom-configurations in triangulated categories generated by spherical objects

Hom- and Riedtmann configurations were studied in the context of stable module categories of selfinjective algebras and a certain orbit category C1(Q) of the bounded derived category of a Dynkin quiver Q, which is highly reminiscent of the cluster category. The category C1(Q) is (−1)-Calabi–Yau. In this paper we consider triangulated categories generated by w-spherical objects, for w<0, and higher versions of the orbit category C1(An), denoted by C|w|(An). These categories have also negative Calabi–Yau dimension. We classify the (higher) Hom- and Riedtmann configurations for these categories, and link them with noncrossing partitions in the case w=−1. Along the way, we obtain a geometric model for C|w|(An).

Canonical basis, KLR algebras and parity sheaves

We give a construction of a basis of the positive part f of the Drinfeld–Jimbo quantum enveloping algebra associated with a Dynkin quiver in terms of parity sheaves over fields in any characteristic, and we categorify both the multiplication and the comultiplication of f via complexes of sheaves.

Graded quiver varieties, quantum cluster algebras and dual canonical basis

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier–Sato–Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra.In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez–Leclerc from the Kronecker and Dynkin quiver case to the acyclic case.The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization.

Hom-Orthogonal Partial Tilting Modules for Dynkin Quivers

We count the number of isomorphism classes of hom-orthogonal partial tilting modules over path algebras of Dynkin quiver of type 𝔸 n , 𝔻 n , 𝔼 n . This number is independent on the choice of an orientation of the arrows, and the number for 𝔸 n or 𝔻 n -type can be expressed as a special value of a hypergeometric function. As a consequence of our theorem, we obtain a minimum value of the number of basic relative invariants of corresponding regular prehomogeneous vector spaces.

Desingularizations of quiver Grassmannians via graded quiver varieties

Inspired by recent work of Cerulli, Feigin and Reineke on desingularizations of quiver Grassmannians of representations of Dynkin quivers, we obtain desingularizations in considerably more general situations and in particular for Grassmannians of modules over iterated tilted algebras of Dynkin type. Our desingularization map is constructed from Nakajima's desingularization map for graded quiver varieties.

Graded quiver varieties and derived categories

Abstract 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 δ-functor Φ taking each finite-dimensional representation of 𝒮 to an object of the derived category of Q. We show that the functor Φ establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Φ. If the underlying graph of Q is an ADE Dynkin diagram, the image is the whole derived category; otherwise, it is the category of `line bundles over the non-commutative curve given by Q'. We show that the degeneration order between strata corresponds to Jensen–Su–Zimmermann's degeneration order on objects of the derived category. Moreover, if Q is an ADE Dynkin quiver, the singular category 𝒮 is weakly Gorenstein of dimension 1 and its derived category of singularities is equivalent to the derived category of Q.

Homological approach to the Hernandez-Leclerc construction and quiver varieties

In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we prove that this algebra is isomorphic to an algebra constructed by Hernandez-Leclerc defined combinatorially and used to describe certain graded Nakajima quiver varieties. This approach is used to get an explicit realization of the orbit closures of representations of Dynkin quivers as affine quotients.

Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers

For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d(1), d(2),…, d(r)) of positive integers, we define a “torus-restricted” representation (GS, R d (Γ)) in natural way. Here we put GS = G1 × G2 × … ×Gr, where each Gi is either SL(d(i)) or GL(d(i)) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R d (Γ) has only finitely many GS-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (GS, R d (Γ)).

Nakajima Varieties and Repetitive Algebras

We realize certain graded Nakajima varieties of finite Dynkin type as orbit closures of repetitive algebras of Dynkin quivers. As an application, we show that the perverse sheaves introduced by Nakajima to describe irreducible characters of quantum loop algebras are isomorphic to the intersection cohomology sheaves of these orbit closures.

C-vectors and dimension vectors for cluster-finite quivers

Let (Q, W) be a quiver with a non-degenerate potential. We give a new description of the c-vectors of Q. We use it to show that, if Q is mutation equivalent to a Dynkin quiver, then the set of positive c-vectors of the cluster algebra associated to Q coincides with the set of dimension vectors of the indecomposable modules over the Jacobian algebra of (Q, W).