Quivers Of Type Research Articles

Mutation classes ofÃn-quivers and derived equivalence classification of cluster tilted algebras of typeÃn

We give an explicit description of the mutation classes of quivers of type \tilde{A}_n. Furthermore, we provide a complete classification of cluster tilted algebras of type \tilde{A}_n up to derived equivalence. We show that the bounded derived category of such an algebra depends on four combinatorial parameters of the corresponding quiver.

Semi-Invariants of Symmetric Quivers of Tame Type

A symmetric quiver (Q, σ) is a finite quiver without oriented cycles Q = (Q 0, Q 1) equipped with a contravariant involution σ on $Q_0\sqcup Q_1$ . The involution allows us to define a nondegenerate bilinear form $\langle -,-\rangle_V$ on a representation V of Q. We shall say that V is orthogonal if $\langle -,-\rangle_V$ is symmetric and symplectic if $\langle -,-\rangle_V$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if (Q, σ) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type c V and, when the matrix defining c V is skew-symmetric, by the Pfaffians pf V . To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector.

Counting the Number of Elements in the Mutation Classes of $\tilde A_n-$Quivers

In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type $\tilde A_n$ in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type $\tilde A_n$. As a by-product, this provides an alternative proof for the number of quivers mutation equivalent to a quiver of Dynkin type $D_n$ which was first determined by Buan and Torkildsen.

B-Functions associated with quivers of type A

We study the b-functions of relative invariants of the prehomogeneous vector spaces associated with quivers of type A. By applying the decomposition formula for b-functions, we determine explicitly the b-functions of one variable for each irreducible relative invariant. Moreover, we give a graphical algorithm to determine the b-functions of several variables.

On quantum cluster algebras of finite type

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined by D. Rupel. When Q is a quiver of finite type, we prove that the algebra (Q) generated by all cluster characters is exactly the quantum cluster algebra (Q).

The Gabriel–Roiter Measure for $\widetilde{\mathbb{A}}_{\boldsymbol n}$ II

Let Q be a tame quiver of type \(\widetilde{\mathbb{A}}_n\) and Rep(Q) the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. It will be shown that a regular string module has, up to isomorphism, at most two Gabriel–Roiter submodules. The quivers Q with sink-source orientations will be characterized as those, whose central parts do not contain preinjective modules. It will also be shown that there are only finitely many (central) Gabriel–Roiter measures admitting no direct predecessors. This fact will be generalized for all tame quivers.

The Dual Quiver of the Auslander–Reiten Quiver of Path Algebras

Let Q be a finite quiver of type An, n ≥ 1, Dn, n ≥ 4, E6, E7 and E8, σ ∈ Aut(Q), k be an algebraic closed field whose characteristic does not divide the order of σ. In this article, we prove that the dual quiver \(\widetilde{\Gamma_{Q}}\) of the Auslander–Reiten quiver ΓQ of kQ, the Auslander–Reiten quiver of \(kQ\#k\langle\sigma\rangle\), and the Auslander–Reiten quiver \(\Gamma_{\widetilde{Q}}\) of \(k\widetilde{Q}\), where \(\widetilde{Q}\) is the dual quiver of Q, are isomorphic.

Cluster Multiplication in Regular Components via Generalized Chebyshev Polynomials

We introduce a multivariate generalization of normalized Chebyshev polynomials of the second kind. We prove that these polynomials arise in the context of cluster characters associated to Dynkin quivers of type $ \mathbb{A} $ and representation-infinite quivers. This allows to obtain a simple combinatorial description of cluster algebras of type $ \mathbb{A} $ . We also provide explicit multiplication formulas for cluster characters associated to regular modules over the path algebra of any representation-infinite quiver.

Quiver Grassmannians associated with string modules

We provide a technique to compute the Euler characteristic of a class of projective varieties called quiver Grassmannians. This technique applies to quiver Grassmannians associated with "orientable string modules". As an application we explicitly compute the Euler characteristic of quiver Grassmannians associated with indecomposable preprojective, preinjective and regular homogeneous representations of an affine quiver of type $\tilde{A}_{p,1}$. For $p=1$, this approach provides another proof of a result due to P. Caldero and A. Zelevinsky in \cite{CZ}.

The Anticyclic Operad of Moulds

A new anticyclic operad Mould is introduced, on spaces of functions in several variables. It is proved that the Dendriform operad is an anticyclic suboperad of this operad. Many operations on the free Mould algebra on one generator are introduced and studied. Under some restrictions, a forgetful map from moulds to formal vector fields is then defined. A connection to the theory of tilting modules for quivers of type A is also described.

Generic variables in acyclic cluster algebras

Let Q be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra A(Q) . We prove that the set G(Q) of generic variables contains naturally the set M(Q) of cluster monomials in A(Q) and that these two sets coincide if and only if Q is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig’s dual semicanonical basis. This allows to compute explicitly the generic variables when Q is a quiver of affine type. When Q is the Kronecker quiver, the set G(Q) is a Z -basis of A(Q) and this basis is compared to Sherman–Zelevinsky and Caldero–Zelevinsky bases.

Quivers of finite mutation type and skew-symmetric matrices

Quivers of finite mutation type are certain directed graphs that first arised in Fomin–Zelevinsky’s theory of cluster algebras. It has been observed that these quivers are also closely related with different areas of mathematics. In fact, main examples of finite mutation type quivers are the quivers associated with triangulations of surfaces. In this paper, we study structural properties of finite mutation type quivers in relation with the corresponding skew-symmetric matrices. We obtain a characterization of finite mutation type quivers that are associated with triangulations of surfaces and give a new numerical invariant for their mutation classes.

The Mutation Class of D n Quivers

We give an explicit description of the mutation classes of quivers of type D, and thus obtain a complete description of type D cluster-tilted algebras.

An algorithm for the construction of exceptional modules over tubular canonical algebras

Let Λ be a tubular canonical algebra of quiver type. We describe an algorithm, which for numerical data computes all regular exceptional Λ -modules, or more generally all indecomposable modules in exceptional tubes. The input for the algorithm is a quadruple consisting of the slope, the number of the tube, the quasi-socle and the quasi-length, the output are explicit matrices for the module with the data above.

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 , E 6 , E 7 or E 8 ). We show that every sincere indecomposable Λ -module has at most three Gabriel–Roiter submodules.

On the zero set of semi-invariants for [formula omitted]-quivers

Let Q be a quiver of type D n , d a dimension vector for Q, and T a representative of the open orbit of the variety rep ( Q , d ) of d-dimensional representations of Q, under the product Gl ( d ) of the general linear groups at all vertices of Q. Let T = T 1 λ 1 ⊕ ⋯ ⊕ T r λ r be a decomposition of T into pairwise non-isomorphic indecomposable representations T i with multiplicities λ i . We show that it depends on the multiplicity of at most one such direct summand whether or not the set of common zeros of all non-constant semi-invariants for rep ( Q , d ) , with respect to the action of Gl ( d ) , is a set theoretical complete intersection.

Quantized Chebyshev polynomials and cluster characters with coefficients

We introduce quantized Chebyshev polynomials as deformations of generalized Chebyshev polynomials previously introduced by the author in the context of acyclic coefficient-free cluster algebras. We prove that these quantized polynomials arise in cluster algebras with principal coefficients associated to acyclic quivers of infinite representation types and equioriented Dynkin quivers of type $\mathbb A$. We also study their interactions with bases and especially canonically positive bases in affine cluster algebras.

The Number of Elements in the Mutation Class of a Quiver of Type $D_n$

We show that the number of quivers in the mutation class of a quiver of Dynkin type $D_n$ is given by $\sum_{d|n}\phi(n/d){2d\choose d}/(2n)$ for $n \geq 5$. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees.

Quantum algebras and quivers

Given a finite quiver (Q) without loops, we introduce a new class of quantum algebras D(Q) which are deformations of the enveloping algebra of a Lie algebra which is a central extension of \(\mathfrak{sl}_n (\Pi(Q))\) where \(\Pi(Q)\) is the preprojective algebra of (Q). When Q is an affine Dynkin quiver of type A, D or E, we can relate them to Γ-deformed double current algebras. We are able to construct functors between different categories of modules over D(Q). We also give some general results about \(\widehat{\mathfrak{sl}}_n(A)\), for a quadratic algebra A and about \(\widehat{\mathfrak{g}}({\mathbb{C}}[u,v])\), which we use to introduce deformed double current algebras associated to a simple Lie algebra \(\mathfrak{g}\).

On the Representation Ring of a Quiver

The Clebsch–Gordan problem for quiver representations is the problem of decomposing the tensor product of any two representations of a quiver, where the tensor product is defined point-wise and arrow-wise. We introduce the so-called characteristic representations and decompose their tensor product. This is then applied to solve the Clebsch–Gordan problem for quivers of type $\mathbb{A}_n$ and $\mathbb{D}_n$ . In both cases we also provide an explicit description of the representation ring.