Acyclic Quiver Research Articles

Motivic Cluster Multiplication Formulas in 2-Calabi–Yau Categories

Abstract We introduce a notion of motivic cluster characters via virtual Poincaré polynomials, and prove a motivic version of multiplication formulas obtained by Chen–Xiao–Xu for weighted quantum cluster characters associated to a 2-Calabi–Yau triangulated category ${{\mathcal{C}}}$ with a cluster tilting object. Furthermore, a refined form of this formula is also given. When ${{\mathcal{C}}}$ is the cluster category of an acyclic quiver, our certain refined multiplication formula is a motivic version of the multiplication formula in SIGMA 19 (2023), 097-157.

Frobenius Morphisms and Stability Conditions

We generalize Deng–Du’s folding argument, for the bounded derived category \operatorname{\mathcal{D}}(Q) of an acyclic quiver Q , to the finite-dimensional derived category \operatorname{\mathcal{D}}(\Gamma Q) of the Ginzburg algebra \Gamma Q associated to Q . We show that the F -stable category of \operatorname{\mathcal{D}}(\Gamma Q) is equivalent to the finite-dimensional derived category \operatorname{\mathcal{D}}(\Gamma\mathbb{S}) of the Ginzburg algebra \Gamma\mathbb{S} associated to the species \mathbf{\mathbb{S}} , which is folded from Q . If (Q,\mathbf{\mathbb{S}}) is of Dynkin type, we prove that the space \operatorname{Stab}\operatorname{\mathcal{D}}(\mathbf{\mathbb{S}}) of the stability conditions on \operatorname{\mathcal{D}}(\mathbf{\mathbb{S}}) is canonically isomorphic to the space \operatorname{FStab}\operatorname{\mathcal{D}}(Q) of F -stable stability conditions on \operatorname{\mathcal{D}}(Q) . For the case of Ginzburg algebras, we also prove a similar isomorphism between principal components \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma\mathbb{S}) and \operatorname{FStab}^{\circ}\operatorname{\mathcal{D}}(\Gamma Q) . There are two applications. One is for the space \operatorname{NStab}\operatorname{\mathcal{D}}(\Gamma Q) of numerical stability conditions in \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma Q) . We show that \operatorname{NStab}\operatorname{\mathcal{D}}(\Gamma Q) consists of \operatorname{Br}Q/\operatorname{Br}\mathbf{\mathbb{S}} many connected components, each of which is isomorphic to \operatorname{Stab}^{\circ}\operatorname{\mathcal{D}}(\Gamma\mathbb{S}) , for (Q,\mathbf{\mathbb{S}}) is of type (A_{3}, B_{2}) or (D_{4}, G_{2}) . The other is that we relate the F -stable stability conditions to Gepner-type stability conditions.

A categorification of cluster algebras of type B and C through symmetric quivers

We express cluster variables of type Bn and Cn in terms of cluster variables of type An. Then we associate a cluster tilted bound symmetric quiver Q of type A2n−1 to any seed of a cluster algebra of type Bn and Cn. Under this correspondence, cluster variables of type Bn (resp. Cn) correspond to orthogonal (resp. symplectic) indecomposable representations of Q. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.

G-semisimple algebras

Let Λ be an Artin algebra and mod-(Gprj_-Λ) the category of finitely presented functors over the stable category Gprj_-Λ of finitely generated Gorenstein projective Λ-modules. This paper deals with those algebras Λ in which mod-(Gprj_-Λ) is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence (called representation equivalence in the terminology of Auslander), i.e. a full and dense functor that reflects isomorphisms, from the stable category of Gorenstein projective representations Gprj_(Q,Λ) of a finite acyclic quiver Q to the category of representations rep(Q,Gprj_-Λ) over Gprj_-Λ, provided Λ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra ΛQ of the G-semisimple algebra Λ is CM-finite if and only if Q is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within Gprj(An,Λ) of the linear quiver An over a G-semisimple algebra Λ. We also determine almost split sequences in Gprj(An,Λ) with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver Gprj(An,Λ).

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.

Representing polynomial of ST-CONNECTIVITY

We show that the coefficients of the representing polynomial of any monotone Boolean function are the values of the M\"obius function of an atomistic lattice related to this function. Using this we determine the representing polynomial of any Boolean function corresponding to a ST-CONNECTIVITY problem in acyclic quivers (directed acyclic multigraphs). Only monomials corresponding to unions of paths have non-zero coefficients which are $(-1)^D$ where $D$ is an easily computable function of the quiver corresponding to the monomial (it is the number of plane regions in the case of planar graphs). We determine that the number of monomials with non-zero coefficients for the two-dimensional $n \times n$ grid connectivity problem is $2^{\Omega(n^2)}$.

Skew group categories, algebras associated to Cartan matrices and folding of root lattices

For an action of a finite group on a finite EI quiver, we construct its ‘orbifold’ quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action. Specializing the result to a finite group action on a finite acyclic quiver, we prove that, under reasonable conditions, the skew group category of the path category is equivalent to a finite EI category of Cartan type. If the ground field is of characteristic $p$ and the acting group is a cyclic $p$ -group, we prove that the skew group algebra of the path algebra is Morita equivalent to the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc, and J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61–158]. We apply the Morita equivalence to construct a categorification of the folding projection between the root lattices with respect to a graph automorphism. In the Dynkin cases, the restriction of the categorification to indecomposable modules corresponds to the folding of positive roots.

On tau-tilting subcategories

Abstract The main theme of this paper is to study $\tau $ -tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $ -cotorsion torsion triples and investigate a bijection between the collection of $\tau $ -cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $ -tilting subcategories of $\mathscr {A}$ , generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$ . General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$ , where R is a unitary associative ring, we characterize all support $\tau $ -tilting (resp. all support $\tau ^-$ -tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support $\tau $ -tilting (resp. support $\tau ^{-}$ -tilting) subcategory of $\mathrm{Mod}\mbox {-}R$ . We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$ , where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $ -tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $ -tilting subcategories of $\mathscr {A}$ .

A deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers

A deterministic algorithm for Harder–Narasimhan filtrations for representations of acyclic quivers

Separated monic correspondence of cotorsion pairs and semi-Gorenstein-projective modules

Given a finite dimensional algebra A over a field k, and a finite acyclic quiver Q, let Λ=A⊗kkQ/I, where kQ is the path algebra of Q over k and I is a monomial ideal. This paper is devoted to studying the separated monic representations over the bounded quiver (Q,I) over a subcategory X of A-mod, denoted by smon(Q,I,X). We show that (X,Y) is a (complete) hereditary cotorsion pair in A-mod if and only if (smon(Q,I,X),rep(Q,I,Y)) is a (complete) hereditary cotorsion pair in Λ-mod. We also show that A is left weakly Gorenstein if and only if so is Λ. Provided that kQ/I is non-semisimple, the category ▪ of semi-Gorenstein-projective Λ-modules coincides with the category of separated monic representations ▪ if and only if A is left weakly Gorenstein.

On some Ringel-Hall polynomials associated to tame indecomposable modules

Let k be a finite field and Q an acyclic quiver of tame type (i.e. of type A˜m,D˜m,E˜6,E˜7,E˜8). Consider the path algebra kQ and the category of finite dimensional right modules mod-kQ. Using Schofield induction combined with some reduction tools, we determine all the Ringel-Hall polynomials associated to indecomposable modules of defect ranging from −2 to 2. In this way we obtain the full list of Ringel-Hall polynomials corresponding to indecomposables in the tame cases A˜m, D˜m and a partial list in the tame cases E˜6, E˜7, E˜8.

Harder-Narasimhan filtrations and zigzag persistence

We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type A˜ quivers, there is a precise relationship between the associated Harder-Narasimhan filtration and the barcode of the periodic zigzag persistence module obtained by unwinding the underlying quiver.

The Cluster Multiplication Theorem for Acyclic Quantum Cluster Algebras

Abstract Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can be viewed as the quantum version of the cluster multiplication theorem in the classical cluster algebra proved by Caldero–Keller for finite type, Hubery for affine type, and Xiao–Xu for acyclic quivers.

Q-stability conditions on Calabi–Yau-𝕏 categories

We introduce $q$ -stability conditions $(\sigma,s)$ on Calabi–Yau- $\mathbb {X}$ categories $\mathcal {D}_\mathbb {X}$ , where $\sigma$ is a stability condition on $\mathcal {D}_\mathbb {X}$ and $s$ a complex number. We prove the corresponding deformation theorem, that $\operatorname {QStab}_s\mathcal {D}_\mathbb {X}$ is a complex manifold of dimension $n$ for fixed $s$ , where $n$ is the rank of the Grothendieck group of $\mathcal {D}_\mathbb {X}$ over $\mathbb {Z}[q^{\pm 1}]$ . When $s=N$ is an integer, we show that $q$ -stability conditions can be identified with the stability conditions on $\mathcal {D}_N$ , provided the orbit category $\mathcal {D}_N=\mathcal {D}_\mathbb {X}/[\mathbb {X}-N]$ is well defined. To attack the questions on existence and deformation along the $s$ direction, we introduce the inducing method. Sufficient and necessary conditions are given, for a stability condition on an $\mathbb {X}$ -baric heart (that is, a usual triangulated category) of $\mathcal {D}_\mathbb {X}$ to induce $q$ -stability conditions on $\mathcal {D}_\mathbb {X}$ . As a consequence, we show that the space $\operatorname {QStab}^\oplus \mathcal {D}_\mathbb {X}$ of (induced) open $q$ -stability conditions is a complex manifold of dimension $n+1$ . Our motivating examples for $\mathcal {D}_\mathbb {X}$ are coming from (Keller's) Calabi–Yau- $\mathbb {X}$ completions of dg algebras. In the case of smooth projective varieties, the $\mathbb {C}^*$ -equivariant coherent sheaves on canonical bundles provide the Calabi–Yau- $\mathbb {X}$ categories. Another application is that we show perfect derived categories can be realized as cluster- $\mathbb {X}$ categories for acyclic quivers.

Categorical properties of generalized σ-derivations on modules

Let S be an algebra over a commutative ring R, and let σ be an automorphism on S. In this paper, we investigate the notion of generalized σ-derivations on modules, which is an extension of generalized derivations on modules introduced by Nakajima. Namely, we study homological properties of generalized σ-derivations. Also, we equip the category of functors that send S/R-modules to R-modules of generalized σ-derivations with a tensor product. We show this category is semi-monoidal. As an application, we characterize when a generalized derivation on a path algebra of an acyclic quiver is a generalized inner derivation.

Tame hereditary path algebras and amenability

AbstractIn this note we revisit the notion of amenable representation type introduced by Gábor Elek. We show that tame hereditary path algebras of quivers of extended Dynkin type over any field are of amenable type. This verifies a conjecture of Elek, which draws similarities to the tame‐wild dichotomy, for another class of tame algebras. We also show that path algebras of wild acyclic quivers over finite fields are not amenable.

Stability conditions and derived Hall algebras

In the present paper, we investigate the universal PBW basis and HN system for the derived Hall algebra of a triangulated category satisfying some finiteness conditions. Furthermore, we study the related stability conditions on the module category of the path algebra A of an acyclic quiver and on the bounded derived category of . Moreover, we investigate the relation between the Ringel-Hall algebra and the HN system of A defined by Reineke.

Edmonds' problem and the membership problem for orbit semigroups of quiver representations

A central problem in algebraic complexity, posed by J. Edmonds [16] , asks to decide if the span of a given l -tuple W = ( W 1 , … , W l ) of N × N complex matrices contains a non-singular matrix. In this paper, we provide a quiver invariant theoretic approach to this problem. Viewing W as a representation of the l -Kronecker quiver K l , Edmonds' problem can be rephrased as asking to decide if there exists a semi-invariant on the representation space ( C N × N ) l of weight ( 1 , − 1 ) that does not vanish at W . In other words, Edmonds' problem is asking to decide if the weight ( 1 , − 1 ) belongs to the orbit semigroup of W . Let Q be an arbitrary acyclic quiver and W a representation of Q . We study the membership problem for the orbit semi-group of W by focusing on the so-called W -saturated weights. We first show that for any given W -saturated weight σ , checking if σ belongs to the orbit semigroup of W can be done in deterministic polynomial time. Next, let ( Q , R ) be an acyclic bound quiver with bound quiver algebra A = K Q / 〈 R 〉 and assume that W satisfies the relations in R . We show that if A / Ann A ( W ) is a tame algebra then any weight σ in the weight semigroup of W is W -saturated. Our results provide a systematic way of producing families of tuples of matrices for which Edmonds' problem can be solved effectively.

Applications of spherical twist functors to Lie algebras associated to root categories of preprojective algebras

Let ΛQ be the preprojective algebra of a finite acyclic quiver Q of non-Dynkin type and Db(repnΛQ) be the bounded derived category of finite dimensional nilpotent ΛQ-modules. We define spherical twist functors over the root category RΛQ of Db(repnΛQ) and then realize the Weyl group associated to Q as certain subquotient of the automorphism group of the Ringel-Hall Lie algebra g(RΛQ) of RΛQ induced by spherical twist functors. We also present a concrete relation between certain Lie subalgebras of g(RΛQ) and g(RQ), where g(RQ) is the Ringel–Hall Lie algebra associated to the root category RQ of Q.

Monic modules and semi-Gorenstein-projective modules

The category gp ( Λ ) of Gorenstein-projective modules over tensor algebra Λ = A ⊗ k B can be described as the monomorphism category mon ( B , gp ( A ) ) of B over gp ( A ) . In particular, Gorenstein-projective Λ-modules are monic. In this paper, we find the similar relation between semi-Gorenstein-projective Λ-modules and A -modules, via monic modules, namely, mon ( B , A ⊥ ) = mon ( B , A ) ∩ Λ ⊥ . Using this, it is proved that if A is weakly Gorenstein, then Λ is weakly Gorenstein if and only each semi-Gorenstein-projective Λ-modules are monic; and that if B = k Q with Q a finite acyclic quiver, then Λ is weakly Gorenstein if and only if A is weakly Gorenstein. However, this relation itself does not answer the question whether there exist double semi-Gorenstein-projective Λ-modules which are not monic. Using the recent discovered examples of double semi-Gorenstein-projective A -modules which are not torsionless, we positively answer this question, by explicitly constructing a class of double semi-Gorenstein-projective T 2 ( A ) -modules with one parameter such that they are not monic, and hence not torsionless. The corresponding results are obtained also for the monic modules and semi-Gorenstein-projective modules over the triangular matrix algebras given by bimodules .