Dynkin Quivers Research Articles

Full Exceptional Collections and Stability Conditions for Dynkin Quivers

Abstract For a stability condition $\sigma $ on a triangulated category, Dimitrov–Katzarkov introduced the notion of a $\sigma $-exceptional collection. In this paper, we study full $\sigma $-exceptional collections in the derived category of an acyclic quiver. In particular, we prove that any stability condition $\sigma $ on the derived category of a Dynkin quiver admits a full $\sigma $-exceptional collection.

Classification results for n-hereditary monomial algebras

We classify n-hereditary monomial algebras in three natural contexts: First, we give a classification of the n-hereditary truncated path algebras. We show that they are exactly the n-representation-finite Nakayama algebras classified by Vaso. Next, we classify partially the n-hereditary quadratic monomial algebras. In the case n=2, we prove that there are only two examples, provided that the preprojective algebra is a planar quiver with potential. The first one is a Nakayama algebra and the second one is obtained by mutating kA3⊗kkA3, where A3 is the Dynkin quiver of type A with bipartite orientation. In the case n≥3, we show that the only n-representation finite algebras are the n-representation-finite Nakayama algebras with quadratic relations.

Geometric model for module categories of Dynkin quivers via hearts of total stability conditions

We derive a geometric model for the module category modkQ of a Dynkin quiver Q via the heart of a total stability condition on the bounded derived category of modkQ. As an application, we prove Reineke's conjecture that there is a stability function on modkQ making any indecomposables stable.

Namikawa-Weyl groups of affinizations of smooth Nakajima quiver varieties

We give a description of the Namikawa-Weyl group of affinizations of smooth Nakajima quiver varieties based on combinatorial data of the underlying quiver, and compute some explicit examples. This extends a result of McGerty and Nevins for quiver varieties associated to Dynkin quivers.

More on affine Dynkin quiver Yangians

We consider the quiver Yangians associated to general affine Dynkin diagrams. Although the quivers are generically not toric, the algebras have some similar structures. The odd reflections of the affine Dynkin diagrams should correspond to Seiberg duality of the quivers, and we investigate the relations of the dual quiver Yangians. We also mention the construction of the twisted quiver Yangians. It is conjectured that the truncations of the (twisted) quiver Yangians can give rise to certain \U0001d4b2-algebras. Incidentally, we give the screening currents of the \U0001d4b2-algebras in terms of the free field realization in the case of generalized conifolds. Moreover, we discuss the toroidal and elliptic algebras for any general quivers.

On the number of tilting modules over a class of Auslander algebras

Let [Formula: see text] be a radical square zero algebra of a Dynkin quiver and let [Formula: see text] be the Auslander algebra of [Formula: see text]. Then the number of tilting right [Formula: see text]-modules is [Formula: see text] if [Formula: see text] is of [Formula: see text] type for [Formula: see text]. Otherwise, the number of tilting right [Formula: see text]-modules is [Formula: see text] if [Formula: see text] is either of [Formula: see text] type for [Formula: see text] or of [Formula: see text] type for [Formula: see text].

Lattice theory of torsion classes: Beyond 𝜏-tilting theory

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set t o r s A \mathsf {tors} A of torsion classes over a finite-dimensional algebra A A . We show that t o r s A \mathsf {tors} A is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of t o r s A \mathsf {tors} A . In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that t o r s A \mathsf {tors} A is completely congruence uniform. When I I is a two-sided ideal of A A , t o r s ( A / I ) \mathsf {tors} (A/I) is a lattice quotient of t o r s A \mathsf {tors} A which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of t o r s A \mathsf {tors} A that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras Π \Pi , for which t o r s Π \mathsf {tors} \Pi is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between t o r s k Q \mathsf {tors} k Q and the Cambrian lattice when Q Q is a Dynkin quiver. We also prove that, in type A A , the algebraic quotients of t o r s Π \mathsf {tors} \Pi are exactly its Hasse-regular lattice quotients.

Approximation by interval-decomposables and interval resolutions of persistence modules

In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. In fact, for the latter equality, we obtained a general formula in the setting of finite-dimensional algebras and resolutions relative to a generator-cogenerator. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.

Singular quiver varieties over extended dynkin quivers

Abstract We classify the singularities in the unframed Nakajima quiver varieties associated with extended Dynkin quivers and the corresponding minimal imaginary root with a small restriction on the parameters and use this to construct a number of hyper-Kähler cobordisms between binary polyhedral spaces.

Multiplicative preprojective algebras of Dynkin quivers

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 the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest. In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etg\"{u}--Lekili arXiv:1502.07922 in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if $Q \neq A_1$, a departure from the additive preprojective algebra in characteristic 2 for $Q = D_{2n}$, $n \geq 2$ and $Q =E_{7}$, $E_{8}$.

Deformations of cluster mutations and invariant presymplectic forms

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4, we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions in the discrete sine-Gordon equation.

On total stability conditions for Dynkin quivers

AbstractWe show that for a Dynkin quiver of type with a specific orientation, the path algebra has no slope function of the form that defines a total stability condition. This gives a counterexample to a conjecture of Reineke.

Global dimension function on stability conditions and Gepner equations

We study the global dimension function \({\text {gldim}}:{\text {Aut}}\backslash {\text {Stab}}{\mathcal {D}}/\mathbb {C}\rightarrow \mathbb {R}_{\ge 0}\) on the quotient of the space of Bridgeland stability conditions on a triangulated category \({\mathcal {D}}\) as well as Toda’s Gepner equation \(\Phi (\sigma )=s\cdot \sigma \) for some \(\sigma \in {\text {Stab}}{\mathcal {D}}\) and \((\Phi ,s)\in {\text {Aut}}{\mathcal {D}}\times \mathbb {C}\). For the bounded derived category \({\mathcal {D}}^b(\textbf{k}Q)\) of a Dynkin quiver Q, we show that there is a unique minimal point \(\sigma _G\) of \({\text {gldim}}\) (up to the \(\mathbb {C}\)-action), with value \(1-2/h\). which is the solution of the Gepner equation \(\tau (\sigma )=(-2/h)\cdot \sigma \). Here \(\tau \) is the Auslander–Reiten functor and h is the Coxeter number. This solution \(\sigma _G\) was constructed by Kajiura–Saito–Takahashi. We also show that for an acyclic non-Dynkin quiver Q, the minimal value of \({\text {gldim}}\) is 1. Our philosophy is that the infimum of \({\text {gldim}}\) on \({\text {Stab}}{\mathcal {D}}\) is the global dimension for the triangulated category \({\mathcal {D}}\). We explain how this notion could shed light on the contractibility conjecture of the space of stability conditions.

On deformed preprojective algebras

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 preprojective algebras for extended Dynkin quivers.

Causal diamonds, cluster polytopes and scattering amplitudes

The “amplituhedron” for tree-level scattering amplitudes in the bi-adjoint ϕ3 theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1 + 1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic “spacetime” with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain “walk”, associated with any acyclic quiver, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The mathcal{A} n−3, mathcal{B} n−1/ mathcal{C} n−1 and mathcal{D} n polytopes are the amplituhedra for n-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope overline{mathcal{D}} n, which chops the mathcal{D} n polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.

Tangent Spaces of Orbit Closures for Representations of Dynkin Quivers of Type mathbb {D}

Let Bbbk be an algebraically closed field, Q a finite quiver, and denote by textup {rep}_{Q}^{mathbf {d}} the affine Bbbk -scheme of representations of Q with a fixed dimension vector d. Given a representation M of Q with dimension vector d, the set {mathcal {O}}_{M} of points in Bbbk isomorphic as representations to M is an orbit under an action on textup {rep}^{mathbf {d}}_{Q}Bbbk of a product of general linear groups. The orbit {mathcal {O}}_{M} and its Zariski closure overline {mathcal {O}}_{M}, considered as reduced subschemes of textup {rep}_{Q}^{{mathbf {d}}}, are contained in an affine scheme {mathcal {C}}_{M} defined by suitable rank conditions associated to M. For all Dynkin and extended Dynkin quivers, the sets of points of overline {{mathcal {O}}}_{M} and {mathcal {C}}_{M} coincide, or equivalently, overline {{mathcal {O}}}_{M} is the reduced scheme associated to {mathcal {C}}_{M}. Moreover, overline {mathcal {O}}_{M}={mathcal {C}}_{M} provided Q is a Dynkin quiver of type {mathbb {A}}, and this equality is a conjecture for the remaining Dynkin quivers (of type mathbb {D} and {mathbb {E}}). Let Q be a Dynkin quiver of type mathbb {D} and M a finite dimensional representation of Q. We show that the equality T_{N}overline {mathcal {O}}_{M}=T_{N}{mathcal {C}}_{M} of Zariski tangent spaces holds for any closed point N of overline {mathcal {O}}_{M}. As a consequence, we describe the tangent spaces to overline {mathcal {O}}_{M} in representation theoretic terms.

Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types

For a Dynkin quiver Q Q (of type A D E \mathrm {ADE} ), we consider a central completion of the convolution algebra of the equivariant K 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 C Q \mathcal {C}_{Q} of modules over the quantum loop algebra U q ( L g ) 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 category of finite-dimensional modules over the quiver Hecke algebra associated with Q Q and Hernandez-Leclerc’s category C Q \mathcal {C}_{Q} , assuming the simpleness of some poles of normalized R R -matrices for type E \mathrm {E} .

Microlocal characterization of Lusztig sheaves for affine quivers and 𝑔-loops quivers

We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we prove a similar result for a larger nilpotent variety and a larger class of perverse sheaves. We formulate conjectures concerning similar results for quivers with loops, for which we have to use the appropriate notion of nilpotent variety, due to Bozec, Schiffmann and Vasserot. We prove our conjecture for g g -loops quivers ( g ≥ 2 g\geq 2 ).

Exact structures and degeneration of Hall algebras

We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra H(E) of an exact category E is the Hall algebra of a smaller exact structure E′<E on the same additive category A. When E is admissible in the sense of Enomoto, for any E′<E satisfying suitable finiteness conditions, we prove that H(E′) is a degeneration of H(E) of this kind.In the additively finite case, all such degree functions form a simplicial cone whose face lattice reflects properties of the lattice of exact structures. For the categories of representations of Dynkin quivers, we recover degenerations of the negative part of the corresponding quantum group, as well as the associated polyhedral structure studied by Fourier, Reineke and the first author.Along the way, we give minor improvements to certain results of Enomoto and Brüstle-Langford-Hassoun-Roy concerning the classification of exact structures on an additive category. We prove that for each idempotent complete additive category A, there exists an abelian category whose lattice of Serre subcategories is isomorphic to the lattice of exact structures on A. We show that every Krull-Schmidt category admits a unique maximal admissible exact structure and that the lattice of smaller exact structures of an admissible exact structure is Boolean.

Calculating dimension of triangulated categories: Path algebras, their tensor powers and orbifold projective lines

This is a companion paper of [7], where different notions of dimension for triangulated categories are discussed. Here we compute dimensions for some examples of triangulated categories and thus illustrate and motivate material from [7]. Our examples include path algebras of finite ordered quivers, orbifold projective lines, some tensor powers of path algebras in Dynkin quivers of type A and categories, generated by an exceptional pair.