On the quiver with relations of a quasitilted algebra and applications

In this paper we discuss, in terms of quiver with relations, su?cient and necessary conditions for an algebra to be a quasitilted algebra. We start with an algebra with global dimension at most two and we give a su?cient condition to be a quasitilted algebra. We show that this condition is not necessary. In the case of a strongly simply connected schurian algebra, we discuss necessary conditions, and combining both types of conditions, we are able to analyze if some given algebra is quasitilted. As an application we obtain the quiver with relations of all the tilted and cluster tilted algebras of Dynkin type E_p.     

Representation of a quiver with relations

Representations of a quiver that is a chain of commutative squares are studied. It is proved that the problem of classification of representations of such a quiver is a tame problem, and a structure of representations is given. This result is a generalization of several known results concerning some important matrix problems. Bibliography:3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 140–151.     

The Auslander-Reiten quiver of a modular group algebra revisited

This work has been supported by the DFG project “Darstellungstheorie” at the University of Essen.     

Bases of the quantum cluster algebra of the Kronecker quiver

We construct bar-invariant Z[q ±1/2 ]-bases of the quantum cluster algebra of Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the corresponding cluster algebra. As a byproduct, we prove positivity of the elements in these bases.     

Symmetric functions and the centre of the Ringel-Hall algebra of a cyclic quiver

We obtain explicit generators for the centre of the Ringel-Hall algebra of a cyclic quiver and define a canonical algebra monomorphism from Macdonald's ring of symmetric functions to the centre, which furthermore respects the comultiplication and the symmetric bilinear form. Dedicated to Claus Michael Ringel on the occasion of his 60th birthday     

A quiver presentation for Solomon's descent algebra

The descent algebra Σ(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of S, the set of simple reflections in W. From this construction we obtain some general information about the quiver of Σ(W) and an algorithm for the construction of a quiver presentation for the descent algebra Σ(W) of any given finite Coxeter group W.     

On the Hochschild (co)homology of a monomial algebra given by a cyclic quiver and two zero-relations

Let K be an algebraically closed field and Γ a cyclic quiver. Xu and Wang investigated the Hochschild (co)homology groups of KΓ∕I, where I is an ideal of KΓ generated by one path. In this paper, in the case that I is an ideal of KΓ generated by two paths, we give the module structure of the Hochschild (co)homology groups of KΓ∕I.     

On the algebra associated with a geometric lattice

Let L be a geometric lattice. Following P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. math. 56 (1980), 167–189, we associate with L a graded commutative algebra A(L). In this paper we introduce a new invariant ψ of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. The goal is to find examples of arrangements with non-isomorphic lattices but homotopy equivalent complements. The invariant introduced here effectively narrows the list of candidates. Nevertheless, we exhibit combinatorially inequivalent arrangements for which all known invariants, including ψ, coincide.     

A crossed product of the canonical anticommutation relations algebra in the Cuntz algebra

We show that the Cuntz algebra can be represented as a crossed product of the canonical anticommutation relations algebra by an endomorphism.     

On the cardinalities of Kronecker quiver Grassmannians

We deduce using the Ringel?CHall algebra approach explicit formulas for the cardinalities of some Grassmannians over a finite field associated to the Kronecker quiver. We realize in this way a quantification of the formulas obtained by Caldero and Zelevinsky for the Euler characteristics of these Grassmannians. Finally we present a recursive algorithm for computing the cardinality of every Kronecker quiver Grassmannian over a finite field.     

On a finite algebra with two operations

Acta Mathematica Hungarica -     

On the quiver Grassmannian in the acyclic case

Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.