Quiver mutation

An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log canonical thresholds from algebraic geometry. Our analysis involves the computation of the singular loci of correlation hypersurfaces. Statistical applications include the strong-faithfulness assumption for the PC-algorithm, and the quantification of confounder bias in causal inference. A detailed analysis is presented for trees, bow-ties, tripartite graphs, and complete graphs.

An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log canonical thresholds from algebraic geometry. Our analysis involves the computation of the singular loci of correlation hypersurfaces. Statistical applications include the strong-faithfulness assumption for the PC-algorithm, and the quantification of confounder bias in causal inference. A detailed analysis is presented for trees, bow-ties, tripartite graphs, and complete graphs.

Let Cn be the group of conjugating automorphisms. We study the representation ρ of Cn, an extension of Lawrence-Krammer representation of the braid group Bn, defined by Valerij G. Bardakov. As Bardakov proved that the representation ρ is unfaithful for n ≥ 5, the cases n = 3, 4 remain open. In our work, we make attempts towards the faithfulness of ρ in the case n = 3.

We associate a quiver to a quasi-triangulation of a non-orientable marked surface and define a notion of quiver mutation that is compatible with quasi-cluster algebra mutation defined by Dupont and Palesi [DP15]. Moreover, we use our quiver to show the unistructurality of the quasi-cluster algebra arising from the Möbius strip.

We associate a quiver to a quasi-triangulation of a non-orientable marked surface and define a notion of quiver mutation that is compatible with quasi-cluster algebra mutation defined by Dupont and Palesi [DP15]. Moreover, we use our quiver to show the unistructurality of the quasi-cluster algebra arising from the Möbius strip.

In the present paper we introduce a notion of homotopy of two Volterra operators which is related to fixed points of such operators. It is establish a criterion when two Volterra operators are homotopic, as a consequence we obtain that the corresponding tournaments of that operators are the same. This, due to [4], gives us a possibility to know some information about the trajectory of homotopic Volterra operators. Moreover, it is shown that any Volterra q.s.o. given on a face has at least two homotopic extension to the whole simplex.

We study the projective dimension of finitely generated modules over cluster-tilted algebras End C (T) where T is a cluster-tilting object in a cluster category C. It is well-known that all End C (T)-modules are of the form Hom C (T, M) for some object M in C, and since End C (T) is Gorenstein of dimension 1, the projective dimension of Hom C (T, M) is either zero, one or infinity. We define in this article the ideal I M of End C (T [1]) given by all endomorphisms that factor through M , and show that the End C (T)-module Hom C (T, M) has infinite projective dimension precisely when I M is non-zero. Moreover, we apply the results above to characterize the location of modules of infinite projective dimension in the Auslander-Reiten quiver of cluster-tilted algebras of type A and D.

First we consider the solutions of the general "cubic" equation a_{1}x^{r1}a_{2}x^{r2}a_{3}x^{r3}=1 (with r1,r2,r3 in {1,-1}) in the symmetric group S_{n}. In certain cases this equation can be rewritten as aya^{-1}=y^{2} or as aya^{-1}=y^{-2}, where a in S_{n} depends on the a_{i}'s and the new unknown permutation y in S_{n} is a product of x (or x^{-1}) and one of the permutations a_{i}^{1} and a_{i}^{-1}. Using combinatorial arguments and some basic number theoretical facts, we obtain results about the solutions of the so-called power conjugate equation aya^{-1}=y^{e} in S_{n}, where e is an integer exponent.

We consider m-cluster tilted algebras arising from quivers of Euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite. For the caseÃ, using the geometric realization, we get a description of representation finite type in terms of (m + 2)-angulations. We establish which m-cluster tilted algebras arise at the same time from quivers of typeà and A. Finally, we characterize representation infinite m-cluster tilted algebras arising from a quiver of typeÃ, as m-relations extensions of some iterated tilted algebra of typeÃ. Dedicated to José Antonio de la Peña on the occasion of his 60th birthday

We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.

The effect of relaxation times is studied on plane waves propagating through semiconductor half-space medium by using the eigen value approach. The bounding surface of the half-space is subjected to a heat flux with an exponentially decaying pulse and taken to be traction free. Solution of the field variables are obtained in the form of series for a general semiconductor medium. For numerical values, Silicon is considered as a semiconducting material. The results are represented graphically to assess the influences of the thermal relaxations times on the plasma, thermal, and elastic waves.

To each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential, in such a way that whenever we apply a flip to a tagged triangulation, the Jacobian algebra of the QP associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author's previous work for ideal triangulations, are then applied to obtain information about cluster monomials in the cluster algebra associated to the given surface by Fomin-Shapiro-Thurston (with an arbitrary system of coefficients).

The effect of relaxation times is studied on plane waves propagating through semiconductor half-space medium by using the eigen value approach. The bounding surface of the half-space is subjected to a heat flux with an exponentially decaying pulse and taken to be traction free. Solution of the field variables are obtained in the form of series for a general semiconductor medium. For numerical values, Silicon is considered as a semiconducting material. The results are represented graphically to assess the influences of the thermal relaxations times on the plasma, thermal, and elastic waves.

In this paper, we show that the repetitive cluster category of type Dn, defined as the orbit category D b (modkDn)/(τ −1 [1]) p , is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon. This generalizes the construction of Schiffler, [23], which we recover when p = 1.

A simple method for transmitting quantum states within a quantum computer is via a quantum spin chain-that is, a path on n vertices. Unweighted paths are of limited use, and so a natural generalization is to consider weighted paths; this has been further generalized to allow for loops (potentials in the physics literature). We study the particularly important situation of perfect state transfer with respect to the corresponding adjacency matrix or Laplacian through the use of orthogonal polynomials. Low-dimensional examples are given in detail. Our main result is that PST with respect to the Laplacian matrix cannot occur for weighted paths on n ≥ 3 vertices nor can it occur for certain symmetric weighted trees. The methods used lead us to a conjecture directly linking the rationality of the weights of weighted paths on n > 3 vertices, with or without loops, with the capacity for PST between the end vertices with respect to the adjacency matrix.

Let H be a finite dimensional hereditary algebra over an algebraically closed field, and let CH be the corresponding cluster category. We give a description of the (standard) fundamental domain of CH in the bounded derived category D b (H), and of the cluster-tilting objects, in terms of the category mod Γ of finitely generated modules over a suitable tilted algebra Γ. Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra.

In this article we study modules over wild canonical algebras which correspond to extension bundles [9] over weighted projective lines. We prove that all modules attached to extension bundles can be established by matrices with coefficients related to the relations of the considered algebra. Moreover, we expand the concept of extension bundles over weighted projective lines with three weights to general weight type and establish similar results in this situation. Finally, we present a method to compute matrices for all modules attached to extension bundles using cokernels of maps between direct sums of line bundles.

Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the-suitably graded-triangle singularities f = x a + y b + z c of domestic type, that is, we assume that (a, b, c) are integers at least two satisfying 1/a + 1/b + 1/c > 1. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type (a, b, c). Equivalently, in a representation-theoretic context, we can work in the mesh category of Z∆ over k, where∆ is the extended Dynkin diagram corresponding to the Dynkin diagram ∆ = [a, b, c]. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the Z-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from {0, ±1}.

In this article we study modules over wild canonical algebras which correspond to extension bundles [9] over weighted projective lines. We prove that all modules attached to extension bundles can be established by matrices with coefficients related to the relations of the considered algebra. Moreover, we expand the concept of extension bundles over weighted projective lines with three weights to general weight type and establish similar results in this situation. Finally, we present a method to compute matrices for all modules attached to extension bundles using cokernels of maps between direct sums of line bundles.

Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the — suitably graded — triangle singularities f = x + y + z of domestic type, that is, we assume that (a, b, c) are integers at least two, satisfying 1/a + 1/b + 1/c > 1. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type (a, b, c). Equivalently, in a representation-theoretic context, we can work in the mesh category of Z∆̃ over k, where ∆̃ is the extended Dynkin diagram, corresponding to the Dynkin diagram ∆ = [a, b, c]. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the Z-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from {0,±1}.

Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the-suitably graded-triangle singularities f = x a + y b + z c of domestic type, that is, we assume that (a, b, c) are integers at least two, satisfying 1/a + 1/b + 1/c > 1. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type (a, b, c). Equivalently, in a representation-theoretic context, we can work in the mesh category of Z∆ over k, wherẽ ∆ is the extended Dynkin diagram, corresponding to the Dynkin diagram ∆ = [a, b, c]. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the Z-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from {0, ±1}.

Given a Jacobian algebra arising from the punctured disk, we show that all non-split extensions can be found using the tagged arcs and skein relations previously developed in cluster algebras theory. Our geometric interpretation can be used to find non-split extensions over other Jacobian algebras arising form surfaces with punctures. We show examples in type D and in a punctured surface.

We construct a tilting object for the stable category of vector bundles on a weighted projective line X of type (2, 2, 2, 2; λ), consisting of five rank two bundles and one rank three bundle, whose endomorphism algebra is a canonical algebra associated with X of type (2,2,2,2).

Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type (2, 2, 2, 2). More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.

Let A ∼ = kQ/I be a basic and connected finite dimension algebra over closed field k. In this note show that in case B = A[M ] is a tame one-point extension of a tame concealed algebra A by an indecomposable module M , then the trivial extension T (B) = B ∝ DB is tame if and only if the module M is regular.

The cluster morphism category of an hereditary algebra was introduced in [5] to show that the picture space of an hereditary algebra of finite representation type is a K(π, 1) for the associated picture group, thereby allowing for the computation of the homology of picture groups of finite type as carried out in [7] for the case of A n. In this paper we show that the cluster morphism category is a CAT (0)-category for hereditary algebras of finite or tame type with only small tubes. As a consequence, we get that the classifying space of the cluster morphism category is a locally CAT (0) space and, as a consequence of that, we get that this classifying space is a K(π, 1).

In [8] we constructed topological triangulated categories Cc as stable categories of certain topological Frobenius categories Fc. In this paper we show that these categories have a cluster structure for certain values of c including c = π. The continuous cluster categories are those Cc which have cluster structure. We study the basic structure of these cluster categories and we show that Cc is isomorphic to an orbit category Dr/F s of the continuous derived category Dr if c = rπ/s. In Cπ, a cluster is equivalent to a discrete lamination of the hyperbolic plane. We give the representation theoretic interpretation of these clusters and laminations.

In this paper, we characterize all the finite-dimensional algebras that are derived equivalent to an [Formula: see text]-cluster tilted algebras of type [Formula: see text]. This generalizes a result of Bobiński and Buan [The algebras derived equivalent to gentle cluster tilted algebras, J. Algebra Appl. 11(1) (2012), Article ID:1250012, 26 pp.].

Source-finite infinite quivers were introduced recently by Enochs, Estrada, and García Rozas. Their injective representations are characterized by local properties. Enochs et al. provide a partial characterization of source-finite trees, proving, e.g., that barren trees have this property. We give a complete characterization.

Let A ∼ = kQ/I be a basic and connected finite dimension algebra over closed field k. In this note show that in case B = A[M ] is a tame one-point extension of a tame concealed algebra A by an indecomposable module M , then the trivial extension T (B) = B ∝ DB is tame if and only if the module M is regular.

We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let Q be the Gabriel quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster category C X of a weighted projective line X. It is proved that there exists a quiver Q ′ in the mutation equivalence class Mut(Q) of Q such that Q ′ admits a maximal green sequence. Furthermore, there is a quiver in Mut(Q) which does not admit a maximal green sequence if and only if X is of wild type.

Let X be a weighted projective line of tubular type and coh X the category of coherent sheaves on X. The main purpose of this note is to show that the subgraph of the tilting graph consisting of all basic tilting bundles of coh X is connected. This yields an alternative proof for the connectedness of the tilting graph of coh X. Our approach leads to the investigation of the change of slopes of a tilting sheaf in coh X under (co-)APR mutations, which may be of independent interest.

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra [Formula: see text] such that [Formula: see text] has non-trivial open Frobenius structures.

In this paper, we show that the repetitive cluster category of type Dn, defined as the orbit category D b (modkDn)/(τ −1 [1]) p , is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon. This generalizes the construction of Schiffler, [23], which we recover when p = 1.

We are going to show that the sheafication of graded Koszul modules K Γ over Γn = K [x 0 , x 1 ...xn] form an important subcategory ∧ K Γ of the coherents sheaves on projective space, Coh(P n). One reason is that any coherent sheave over P n belongs to ∧ K Γ up to shift. More importantly, the category K Γ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Koszul modules over Γ and over the exterior algebra Λ. This is then used to develop a kind of relative Auslander-Reiten theory for the category Coh(P n), with respect to this theory, all but finitely many Auslander-Reiten components for Coh(P n) have the shape ZA∞. We also describe the remaining ones.

Let F \mathbb {F} be a finite field and ( Q , d ) (Q,\mathbf {d}) an acyclic valued quiver with associated exchange matrix B ~ \tilde {B} . We follow Hubery’s approach to prove our main conjecture from 2011: the quantum cluster character gives a bijection from the isoclasses of indecomposable rigid valued representations of Q Q to the set of non-initial quantum cluster variables for the quantum cluster algebra A | F | ( B ~ , Λ ) \mathcal {A}_{|\mathbb {F}|}(\tilde {B},\Lambda ) . As a corollary we find that for any rigid valued representation V V of Q Q , all Grassmannians of subrepresentations G r e V Gr_{\mathbf {e}}^V have counting polynomials.

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.

This present paper is devoted to the study of a class of Nakayama algebras Nn(r) given by the path algebra of the equioriented quiver An subject to the nilpotency degree r for each sequence of r consecutive arrows. We show that the Nakayama algebras Nn(r) for certain pairs (n, r) can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras Nn(r) of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classification result on Nakayama algebras of piecewise hereditary type, which have been done by Happel-Seidel before.

Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type (2, 2, 2, 2). More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.

Let A be a finitary hereditary abelian category and D(A) be its reduced Drinfeld double Hall algebra. By giving explicit formulas in D(A) for left and right mutations, we show that the subalgebras of D(A) generated by exceptional sequences are invariant under mutation equivalences. As an application, we obtain that if A is the category of finite dimensional modules over a finite dimensional hereditary algebra, or the category of coherent sheaves on a weighted projective line, the double composition algebra of A is generated by any complete exceptional sequence. Moreover, for the Lie algebra case, we also have paralleled results.

The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.

Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type (2, 2, 2, 2). More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.

Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type (2, 2, 2, 2). More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.

Any pair of consecutive B-smooth integers for a given smoothness bound B corresponds to a solution (x, y) of the equation x 2 − 2∆y 2 = 1 for a certain square-free, B-smooth integer ∆ and a B-smooth integer y. This paper describes algorithms to find such twin B-smooth integers that lie in a given interval by using the structure of solutions of the above Pell equation. The problem of finding such twin smooth integers is motivated by the quest for suitable parameters to efficiently instantiate recent isogeny-based cryptosystems. While the Pell equation structure of twin B-smooth integers has previously been used to describe and compute the full set of such pairs for very small values of B, increasing B to allow for cryptographically sized solutions makes this approach utterly infeasible. We start by revisiting the Pell solution structure of the set of twin smooth integers. Instead of using it to enumerate all twin smooth pairs, we focus on identifying only those that lie in a given interval. This restriction allows us to describe algorithms that navigate the vast set of Pell solutions in a more targeted way. Experiments run with these algorithms have provided examples of twin B-smooth pairs that are larger and have smaller smoothness bound B than previously reported pairs. Unfortunately, those examples do not yet provide better parameters for cryptography, but we hope that our methods can be generalized or used as subroutines in future work to achieve that goal.

We generalise the notion of cluster structures from the work of Buan-Iyama-Reiten-Scott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Hom-finite 2-Calabi-Yau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2-cycles in the quivers of their endomorphism rings. We apply this result to the cluster category of a tube, and show that this category forms a good model for the combinatorics of a type B cluster algebra.

Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional indecomposable objects in the cluster category inducing a correspondence between clusters and cluster-tilting objects. Fix a cluster-tilting object T and a corresponding initial cluster. By the Laurent phenomenon, every cluster variable can be written as a Laurent polynomial in the initial cluster. We give conditions on T equivalent to the fact that the denominator in the reduced form for every cluster variable in the cluster algebra has exponents given by the dimension vector of the corresponding module over the endomorphism algebra of T .

We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.

The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.