Dynkin Type Research Articles

Frieze patterns over integers and other subsets of the complex numbers

We study (tame) frieze patterns over subsets of the complex numbers, with particular emphasis on the corresponding quiddity cycles. We provide new general transformations for quiddity cycles of frieze patterns. As one application, we present a combinatorial model for obtaining the quiddity cycles of all tame frieze patterns over the integers (with zero entries allowed),generalising the classic Conway-Coxeter theory. This model is thus also a model for the set of specializations of cluster algebras of Dynkin type A in which all cluster variables are integers. Moreover, we address the question of whether for a given height there are only finitely many non-zero frieze patterns over a given subset R of the complex numbers. Under certain conditions on R , we show upper bounds for the absolute values of entries in the quiddity cycles. As a consequence, we obtain that if R is a discrete subset of the complex numbers then for every height there are only finitely many non-zero frieze patterns over R . Using this, we disprove a conjecture of Fontaine, by showing that for a complex d -th root of unity \zeta_d there are only finitely many non-zero frieze patterns for a given height over R=\mathbb{Z}[\zeta_d] if and only if d\in \{1,2,3,4,6\} .

Deformed mesh algebras of Dynkin type 𝔽4

This paper belongs to the series of articles devoted to the classification of deformed mesh algebras of Dynkin types [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We prove here that every deformed mesh algebra of type [Formula: see text] is isomorphic to the canonical mesh algebra of type [Formula: see text].

Deformed preprojective algebras of Dynkin type

We prove that every deformed preprojective algebra of Dynkin type is isomorphic to the preprojective algebra of Dynkin type .

Derived Picard Groups of Preprojective Algebras of Dynkin Type

Abstract In this paper, we study two-sided tilting complexes of preprojective algebras of Dynkin type. We construct the most fundamental class of two-sided tilting complexes, which has a group structure by derived tensor products and induces a group of auto-equivalences of the derived category. We show that the group structure of the two-sided tilting complexes is isomorphic to the braid group of the corresponding folded graph. Moreover, we show that these two-sided tilting complexes induce tilting mutation and any tilting complex is given as the derived tensor products of them. Using these results, we determine the derived Picard group of preprojective algebras for type $A$ and $D$.

Simple transitive 2-representations of left cell 2-subcategories of projective functors for star algebras

In this article, we study simple transitive 2-representations of certain 2-subcategories of the 2-category of projective functors over a star algebra. We show that in the simplest case, which is associated to the Dynkin type A2, simple transitive 2-representations are classified by cell 2-representations. In the general case we conjecture that there exist many simple transitive 2-representations which are not cell 2-representations and provide some evidence for our conjecture.

Socle deformed preprojective algebras of generalized Dynkin type

We provide a complete classification of finite-dimensional self-injective algebras which are socle equivalent to preprojective algebras of generalized Dynkin type. In particular, we conclude that these algebras are deformed preprojective algebras of generalized Dynkin type (in the sense of [5, 12]), and hence are periodic algebras.

Oriented Cohomology Sheaves on Double Moment Graphs

In the present paper we extend the theory of sheaves on moment graphs due to Braden–MacPherson and Fiebig to the context of an arbitrary oriented equivariant cohomology \mathtt h (e.g. to algebraic cobordism). We introduce and investigate structure \mathtt h -sheaves on double moment graphs to describe equivariant oriented cohomology of products of flag varieties. We show that in the case of a total flag variety X of Dynkin type A the space of global sections of the double structure \mathtt h -sheaf also describes the endomorphism ring of the equivariant \mathtt h -motive of X .

Generalized quiver varieties and triangulated categories

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 $${{\mathcal {P}}}$$ : the algebra $${{\mathcal {P}}}$$ 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 $${\text {proj}}\,{{\mathcal {P}}}$$ , and that their degeneration order coincides with the Jensen–Su–Zimmermann’s degeneration order on the triangulated category $${\text {proj}}\,{{\mathcal {P}}}$$ . Furthermore, we prove that classical quiver varieties of type $$A_n$$ can be realized as moduli spaces of representations of an algebra $${{\mathcal {S}}}$$ .

Incidence graphs and non-negative integral quadratic forms

We present directed multigraphs (quivers) and their walks as combinatorial models for non-negative semi-unit integral quadratic forms of Dynkin type A and their associated roots and radical vectors. This point of view leads to an easy description of those forms which are positive and weakly positive (properties of interest in Lie theory and the representation theory of algebras), and derives into a categorification of such unitary forms.

Tilting Modules and Support τ-Tilting Modules over Preprojective Algebras Associated with Symmetrizable Cartan Matrices

For any given symmetrizable Cartan matrix C with a symmetrizer D, Geis et al. (2016) introduced a generalized preprojective algebra Π(C, D). We study tilting modules and support τ-tilting modules for the generalized preprojective algebra Π(C, D) and show that there is a bijection between the set of all cofinite tilting ideals of Π(C, D) and the corresponding Weyl group W(C) provided that C has no component of Dynkin type. When C is of Dynkin type, we also establish a bijection between the set of all basic support τ-tilting Π(C, D)-modules and the corresponding Weyl group W(C). These results generalize the classification results of Buan et al. (Compos. Math. 145(4), 1035–1079 2009) and Mizuno (Math. Zeit. 277(3), 665–690 2014) over classical preprojective algebras.

Graphical characterization of positive definite non symmetric quasi-Cartan matrices

It is known that each positive definite quasi-Cartan matrix A is Z-equivalent to a Cartan matrix AΔ called Dynkin type of A, the matrix AΔ is uniquely determined up to conjugation by permutation matrices. However, in most of the cases, it is not possible to determine the Dynkin type of a given connected quasi-Cartan matrix by simple inspection. In this paper, we give a graph theoretical characterization of non-symmetric connected quasi-Cartan matrices. For this purpose, a special assemblage of blocks is introduced. This result complements the approach proposed by Barot (1999, 2001), for An, Dn and Em with m=6,7,8.

Polytopal realizations of finite type g-vector fans

This paper shows the polytopality of any finite type g-vector fan, acyclic or not. In fact, for any finite Dynkin type Γ, we construct a universal associahedron Assoun(Γ) with the property that any g-vector fan of type Γ is the normal fan of a suitable projection of Assoun(Γ).

Strongness of companion bases for cluster-tilted algebras of finite type

For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e. gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.

Cubic Algorithm to Compute the Dynkin Type of a Positive Definite Quasi-Cartan Matrix

Inflations algorithm is a procedure that appears implicitly in Ovsienko’s classical proof for the classification of positive definite integral quadratic forms. The best known upper asymptotic bound for its time complexity is an exponential one. In this paper we show that this bound can be tightened to O(n6) for the naive implementation. Also, we propose a new approach to show how to decide whether an admissible quasi-Cartan matrix is positive definite and compute the Dynkin type in just O(n3) operations taking an integer matrix as input.

Ladders of Compactly Generated Triangulated Categories and Preprojective Algebras

In this paper we characterize when a recollement of compactly generated triangulated categories admits a ladder of some height going either upwards or downwards. As an application, we show that the derived category of the preprojective algebra of Dynkin type $$\mathbb {A}_n$$ admits a periodic infinite ladder, where the one outer term in the recollement is the derived category of a differential graded algebra.

Cluster structures in 2-Calabi–Yau triangulated categories of Dynkin type with maximal rigid objects

In this paper, we consider two kinds of 2-Calabi–Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called An,t = D b (KA(2t+1)(n+1)−3)/τt(n+1)−1[1], where n, t ≥ 1, and Dn,t = D b (KD2t(n+1))/τ(n+1)φ n , where n, t ≥ 1, and φ is induced by an automorphism of D2t(n+1) of order 2. Except the categories An,1, they all contain non-zero maximal rigid objects which are not cluster tilting. An,1 contain cluster tilting objects. We define the cluster complex of An,t (resp. Dn,t) by using the geometric description of cluster categories of type A (resp. type D). We show that there is an isomorphism from the cluster complex of An,t (resp. Dn,t) to the cluster complex of root system of type B n . In particular, the maximal rigid objects are isomorphic to clusters. This yields a result proved recently by Buan–Palu–Reiten: Let $${R_{{A_{n,t}}}}$$ , resp. $${R_{{D_{n,t}}}}$$ , be the full subcategory of An,t, resp. Dn,t, generated by the rigid objects. Then $${R_{{A_{n,t}}}} \simeq {R_{{A_{n,1}}}}$$ and $${R_{{D_{n,t}}}} \simeq {R_{{A_{n,1}}}}$$ as additive categories, for all t ≥ 1.

Mutation of friezes

We study mutations of Conway–Coxeter friezes which are compatible with mutations of cluster-tilting objects in the associated cluster category of Dynkin type A. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. We observe how the frieze can be divided into four distinct regions, relative to the entry at which we want to mutate, where any two entries in the same region obey the same mutation rule. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type A in the sense of Caldero and Chapoton.

Classifying tilting complexes over preprojective algebras of Dynkin type

We study tilting complexes over preprojective algebras of Dynkin type. We classify all tilting complexes by giving a bijection between tilting complexes and the braid group of the corresponding folded graph. In particular, we determine the derived equivalence class of the algebra. For the results, we develop the theory of silting-discrete triangulated categories and give a criterion of silting-discreteness.

Partition identities and quiver representations

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy’s Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke’s identity in the case of quivers \({\mathcal {Q}}\) of Dynkin type A. Our identity is stated in terms of the lacing diagrams of S. Abeasis–A. Del Fra, which parameterize orbits of the representation space of \({\mathcal {Q}}\) for a fixed dimension vector.

Intersection Local Times, Loop Soups and Permanental Wick Powers

Several stochastic processes related to transient Levy processes with potential densities u(x,y) = u(y x), that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures V endowed with a metric d. Su- cient conditions are obtained for the continuity of these processes on (V,d). The processes include n-fold self-intersection local times of tran- sient Levy processes and permanental chaoses, which are 'loop soup n- fold self-intersection local times' constructed from the loop soup of the Levy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of n-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional proper- ties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.