On deformed preprojective algebras

Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the Deligne-Simpson problem, integrable systems and noncommutative geometry. In this paper we offer three contributions to the study of such algebras: (1) the 2-Calabi-Yau property; (2) the unification of the reflection functors of Crawley-Boevey and Holland with reflection functors for the usual preprojective algebras; and (3) the classification of tilting ideals in 2-Calabi-Yau algebras, and especially in deformed preprojective algebras for extended Dynkin quivers.     

Representations of deformed preprojective algebras and quantum groups

Let (Γ,I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group Zd. In this paper, we list all indecomposable representations of (Γ,I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Πλ(Γ,I). It is shown that those representations given by extending indecomposable representations of (Γ,I) are all simple representations of Πλ(Γ,I). Therefore, it is concluded that all simple representa-tions of rest...     

Quiver varieties and Lusztig's algebra

We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig.     

Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem

We introduce a family of algebras which are multiplicative analogues of preprojective algebras, and their deformations, as introduced by M.P. Holland and the first author. We show that these algebras provide a natural setting for the ‘middle convolution’ operation introduced by N.M. Katz in his book ‘Rigid local systems’, and put in an algebraic setting by M. Dettweiler and S. Reiter, and H. Völklein. We prove a homological formula relating the dimensions of Hom and Ext spaces, study varieties of representations of multiplicative preprojective algebras, and use these results to study simple representations. We apply this work to the Deligne-Simpson problem, obtaining a sufficient (and conjecturally necessary) condition for the existence of an irreducible solution to the equation A₁A₂…A_k=1 with the A_i in prescribed conjugacy classes in GL_n(C).     

Connections for weighted projective lines

We introduce a notion of a connection on a coherent sheaf on a weighted projective line (in the sense of Geigle and Lenzing). Using a theorem of Hübner and Lenzing we show, under a mild hypothesis, that if one considers coherent sheaves equipped with such a connection, and one passes to the perpendicular category to a nonzero vector bundle without self-extensions, then the resulting category is equivalent to the category of representations of a deformed preprojective algebra.     

Quasitilted algebras admit a preprojective component

Quasitilted algebras are generalizations of tilted algebras. As a main result we show here that the Auslander-Reiten quiver of such an algebra has a preprojective component

     

The structure of AS-Gorenstein algebras

In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.     

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.     

Pre-Projective Parts of Tilting Quivers Over Certain Path Algebras

Happel and Unger defined a partial order on the set of basic tilting modules. We study the poset of basic preprojective tilting modules over path algebras of representation-infinite type. First we will give a criterion for Ext-vanishing for preprojective modules. With the using of this result, we will give combinatorial characterizations of the poset of basic preprojective tilting modules. Finally, we will see the structure of a preprojective part of tilting quivers.     

Reflection functors and symplectic reflection algebras for wreath products

We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of wreath product groups.     

Deformed preprojective algebras and symplectic reflection algebras for wreath products

We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding symplectic reflection algebra for wreath product.     

Pro-Species of Algebras I: Basic Properties

In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geiß et al. (2016). In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories.     

Irreducible Representations for the Baby Tits–Kantor–Koecher Algebra

The baby Tits–Kantor–Koecher (TKK) algebra constructed from the smallest (nonlattice) semilattice is related to the “smallest” extended affine Lie algebras other than the finite dimensional simple Lie algebras and the affine Kac–Moody algebras. In this article, we classify the finite dimensional irreducible representations for the baby TKK algebra. It turns out that such representations can be lifted from modules of direct sums of finitely many copies of the simple Lie algebra sp ₄(?).     

Minimal algebras of infinite representation type with preprojective component

Let A be a finite dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field k. Denote by e₁,...,e_n a complete set of primitive orthogonal idempotents in A and by A_i= A/Ae_iA. A is called a minimal algebra of infinite representation type provided A is itself of infinite representation type,whereas all A_i, 1≤i≤n,are of finite representation type. The main result gives the classification of the minimal algebras having a preprojective component in their Auslander-Reiten quiver. The classification is obtained by realizing that these algebras are essentially given by preprojective tilting modules over tame hereditary algebras.     

Derived dynkin extensions

The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri].

We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions.

Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category instead of combinatorial arguments.     

An algorithm for finding all preprojective components of the Auslander-Reiten quiver

The Auslander-Reiten quiver of a finite-dimensional associative algebra encodes information about the indecomposable finite-dimensional representations of and their homomorphisms. A component of the Auslander-Reiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the Auslander-Reiten translation. Preprojective components play an important role in the present research on algebras of finite and tame representation type. We present an algorithm which detects all preprojective components of a given algebra.

     

On simply laced Lie algebras and their minuscule representations

In this paper we adapt a known construction for the simply laced, semisimple Lie algebras (over Z), and thereby obtain a very simple construction for all minuscule representations of those Lie algebras (again over Z). We apply these results to give explicit formulas for tensors invariant under the exceptional algebras and . Received: November 3, 2000     

On a Result of Pointed Representations of finite Dimensional Simple Lie Algebras

There have been a great many of studies on the pointed representations of fi-nite-dimensional sanple Lie algebras.cf.[1][2]etc.In this paper we give a new proof of animpottant Lemma,and from this we derive our main result:Irreducible pointed modules of finite-dimesional simple Lie algebras are all Harish-Chandra modules.