Derived Equivalence Classification of the Cluster-Tilted Algebras of Dynkin Type E

We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E ₆, E ₇ and E ₈ which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of “good” mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.     

Adjoint Action of Automorphism Groups on Radical Endomorphisms,Generic Equivalence and Dynkin Quivers

Let Q be a connected quiver with no oriented cycles, k the field of complex numbers and P a projective representation of Q. We study the adjoint action of the automorphism group Aut _kQ P on the space of radical endomorphisms radEnd _kQ P. Using generic equivalence, we show that the quiver Q has the property that there exists a dense open Aut _kQ P-orbit in radEnd _kQ P, for all projective representations P, if and only if Q is a Dynkin quiver. This gives a new characterisation of Dynkin quivers.     

Mutation of Torsion Pairs in Cluster Categories of Dynkin Type D

In cluster categories, mutation of torsion pairs provides a generalisation for the mutation of cluster tilting subcategories, which models the combinatorial structure of cluster algebras. In this paper we present a geometric model for mutation of torsion pairs in the cluster category \(\mathcal {C}_{D_{n}}\) of Dynkin type D _n . Using a combinatorial model introduced by Fomin and Zelevinsky in [7], subcategories in \(\mathcal {C}_{D_{n}}\) correspond to rotationally invariant collections of arcs in a regular 2n-gon, called diagrams of Dynkin type D _n . Torsion pairs in \(\mathcal {C}_{D_{n}}\) have been classified by Holm, Jørgensen and Rubey in [10] and correspond to diagrams of Dynkin type D _n satisfying a certain combinatorial condition, called Ptolemy diagrams of Dynkin type D _n . We define mutation of a diagram \(\mathcal {X}\) of Dynkin type D _n with respect to a compatible diagram \(\mathcal {D}\) of Dynkin type D _n consisting of pairwise non-crossing arcs. Such a diagram \(\mathcal {D}\) partitions the regular 2n-gon into cells and mutation of \(\mathcal {X}\) with respect to \(\mathcal {D}\) can be thought of as a rotation of each of these cells. We show that mutation of Ptolemy diagrams of Dynkin type D _n corresponds to mutation of the corresponding torsion pairs in the cluster category of Dynkin type D _n .     

Exceptional Sequences over Path Algebras of Type A n and Non-Crossing Spanning Trees

Exceptional sequences are fundamental to investigate the derived categories of finite dimensional algebras. The aim of this note is to classify all the complete exceptional sequences over the path algebra of a Dynkin quiver of type A _n in terms of non-crossing spanning trees.     

Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals

We describe the limit distribution of V- and U-statistics in a new fashion. In the case of V-statistics the limit variable is a multiple stochastic integral with respect to an abstract Brownian bridge G_Q. This extends the pioneer work of Filippova (1961) [8]. In the case of U-statistics we obtain a linear combination of G_Q-integrals with coefficients stemming from Hermite Polynomials. This is an alternative representation of the limit distribution as given by Dynkin and Mandelbaum (1983) [7] or Rubin and Vitale (1980) [13]. It is in total accordance with their results for product kernels.     

The number of Gabriel-Roiter submodules

Let k be an algebraically closed field and Λ a tilted algebra of a path algebra of a Dynkin quiver (of type A_n, D_n, E₆, E₇ or E₈). We show that every sincere indecomposable Λ-module has at most three Gabriel-Roiter submodules.     

Hecke-Kiselman monoids of small cardinality

In this paper, we give a characterization of digraphs Q, |Q|≤4 such that the associated Hecke-Kiselman monoids H _Q are finite. In general, a necessary condition for H _Q to be a finite monoid is that Q is acyclic and its Coxeter components are Dynkin diagrams. We show, by constructing examples, that such conditions are not sufficient.     

Stable equivalence classes of self-injective algebras of tree class <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A classification up to stable equivalence of representation-finite self-injective algebras with associated Dynkin diagram D _n is suggested. In each stable equivalence class, a representative is chosen, and the representatives are partitioned into five families of algebras, which are described in terms of quivers with relations.     

Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is basic connected with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ? _A as the automorphism of the Grothendieck group K ₀(A) induced by the Auslander-Reiten translation τ in the derived category Der(modA) of the module category modA of finite dimensional left A-modules. We say that A is an algebra of cyclotomic type if the characteristic polynomial χ _A of ? _A is a product of cyclotomic polynomials. There are many examples of algebras of cyclotomic type in the representaton theory literature: hereditary algebras of Dynkin and extended Dynkin types, canonical algebras, some supercanonical and extended canonical algebras. Among other results, we show that: (a) algebras satisfying the fractional Calabi-Yau property have periodic Coxeter transformation and are, therefore, of cyclotomic type, and (b) algebras whose homological form h _A is non-negative are of cyclotomic type. For an algebra A of cyclotomic type we describe the shape of the Auslander-Reiten components of Der(modA).     

Normality of orbit closures for Dynkin quivers¶of type ?n

It is known that the orbit closures for the representations of the equioriented Dynkin quivers ? _n are normal and Cohen–Macaulay varieties with rational singularities. In the paper we prove the same for the Dynkin quivers ? _n with arbitrary orientation. Received: 25 October 2000 / Revised version: 28 February 2001     

Absolute continuity results for superdiffusions with applications to differential equations

Let X=(X_D,P_μ) be a superdiffusion in a domain $\text{E?}\text{R}{}^{\mathrm{d}}$. We introduce a germ σ-algebra $\text{F}{}_{\mathrm{E?}}$ at the boundary of E and we prove that, on this σ-algebra, P_μ1 is absolutely continuous with respect to P_μ2 if μ₁ and μ₂ are concentrated on compact subsets of E. In combination with previous results of Dynkin, Kuznetsov and Mselati, this leads to a complete classification of positive solutions of equation Δu=u^α in a bounded domain E of class C⁴ for the case 1<α?2. To cite this article: E.B. Dynkin, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

BGP-reflection functors and cluster combinatorics

We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the “truncated simple reflections” on the set of almost positive roots Φ_≥−1 associated with a finite dimensional semi-simple Lie algebra. Combining this with the tilting theory in cluster categories developed in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054], we give a unified interpretation via quiver representations for the generalized associahedra associated with the root systems of all Dynkin types (simply laced or non-simply laced). This confirms the Conjecture 9.1 in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054] for all Dynkin types.     

On automorphisms of finite simple Chevalley groups

The paper deals with the splitting properties of the automorphism groups of finite Chevalley groups. Using the action of symmetries of a Dynkin diagram on the corresponding Weyl group, a sufficient condition is developed for the existence of a complement for the inner automorphism group in the automor ph-ism group of a finite Chevalley group. The condition is verified for Chevalley groups of the classical types (viz., the group of type A_l, B_l, C_l,and D_l) as well as the exceptional groups of type E₆ and E₇, under suitable restrictions on the base fields.     

Representation-finite trivial extension algebras

Let A be a finite-dimensional basic connected associative algebra over an algebraically closed field, and $\text{T(A)=A ? DA}$ its trivial extension by its minimal injective cogenerator. We prove that T(A) is representation-finite of Cartan class Δ if and only if A is an iterated tilted algebra of Dynkin class Δ. The proof also yields a construction procedure for iterated tilted algebras of Dynkin type.     

Linear quivers, generic extensions and Kashiwara operators

In the present paper, we introduce the generic extension graph G of a Dynkin or cyclic quiver Q and then compare this graph with the crystal graph C for the quantized enveloping algebra associated to Q via two maps ℘_Q, _Q : Ω → Λ_Q induced by generic extensions and Kashiwara operators, respectively, where Λ_Q is the set of isoclasses of nilpotent representations of Q, and Ω is the set of all words on the alphabet I, the vertex set of Q. We prove that, if Q is a (finite or infinite) linear quiver, then the intersection of the fibres ℘_Q⁻¹ (λ) and KQ⁻¹ (λ) is non-empty for every λ ∈ Λ _Q. We will also show that this non-emptyness property fails for cyclic quivers.     

On the uniqueness of promotion operators on tensor products of type <Emphasis Type="Italic">A</Emphasis> crystals

The affine Dynkin diagram of type A _n ⁽¹⁾ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type A _n crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type A _n crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara’s conjecture that all ‘good’ affine crystals are tensor products of Kirillov-Reshetikhin crystals.     

AR quivers, exceptional sequences and algorithms in derived Hall algebras

Consider the canonical isomorphism between the positive part U ⁺ of the quantum group U _q (g) and the Hall algebra H(Λ), where the semisimple Lie algebra g and the finite-dimensional hereditary algebra Λ share a Dynkin diagram. Chen and Xiao have given two algorithms to decompose the root vectors into linear combinations of monomials of Chevalley generators of U ⁺, respectively induced by the braid group action on the exceptional sequences of Λ-modules and the structure of the Auslander-Reiten quiver of Λ. In this paper, we obtain the corresponding algorithms for the derived Hall algebra DH(Λ), which was introduced by Toën. We show that both algorithms are applicable to the lattice algebra and Heisenberg double in the sense of Kapranov. All the new recursive formulae have the same flavor with the quantum Serre relations.