Christof Geiss

Semicanonical bases and preprojective algebras

Abstract We study the multiplicative properties of the dual of Lusztigs semicanonical basis. The elements of this basis are naturally indexed by the irreducible components of Lusztigs nilpotent varieties, which can be interpreted as varieties of modules over preprojective algebras. We prove that the product of two dual semicanonical basis vectors ρ Z ′ and ρ Z ″ is again a dual semicanonical basis vector provided the closure of the direct sum of the corresponding two irreducible components Z ′ and Z ″ is again an irreducible component. It follows that the semicanonical basis and the canonical basis coincide if and only if we are in Dynkin type A n with n ⩽ 4 . Finally, we provide a detailed study of the varieties of modules over the preprojective algebra of type A 5 . We show that in this case the multiplicative properties of the dual semicanonical basis are controlled by the Ringel form of a certain tubular algebra of type ( 6 , 3 , 2 ) and by the corresponding elliptic root system of type E 8 ( 1 , 1 ) .

Semicanonical bases and preprojective algebras II: A multiplication formula

Let

Cluster Algebras of Finite Type and Positive Symmetrizable Matrices

n

Combinatorial derived invariants for gentle algebras

be a maximal nilpotent subalgebra of a complex symmetric Kac-Moody Lie algebra. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of nilpotent modules over a preprojective algebra of the same type as

Preprojective algebras and cluster algebras

n

n-ANGULATED CATEGORIES

. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinskys theory of cluster algebras. It was inspired by recent results of Caldero and Keller.

Quivers with relations for symmetrizable Cartan matrices I: Foundations

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However, the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.

Cluster algebras in algebraic lie theory

Abstract We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stable Auslander–Reiten quiver of its repetitive algebra. As a by-product we obtain that the number of arrows of the quiver of a gentle algebra is invariant under derived equivalence. Finally, our invariants separate the derived equivalence classes of gentle algebras with at most one cycle.

Verma modules and preprojective algebras

This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss properties of the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Corre- spondence. Finally, we describe matrix factorizations corresponding to indecomposable Cohen- Macaulay modules over the non-isolated singularities A1 and D1.For a finite dimensional algebra A of finite global dimension the bounded derived category of finite dimensional A-modules admits Auslander- Reiten triangles such that the Auslander-Reiten translation τ is an equivalence. On the level of the Grothendieck group τ induces the Coxeter transformation �A. More generally this extends to a homologically finite triangulated category T admitting Serre duality. In both cases the Coxeter polynomial, that is, the characteristic polynomial of the Coxeter transformation yields an important homological invariant of A or T. Spectral analysis is the study of this interplay, it often reveals unexpected links between apparently different subjects. This paper gives a summary on spectral techniques and studies the links to singularity theory. In particular, it offers a contribution to the categorifica- tion of the Milnor lattice through triangulated categories which are naturally attached to a weighted projective line.We review the definition of a Calabi-Yau triangulated category and survey examples coming from the representation theory of quivers and finite-dimensional algebras. Our main motivation comes from the links between quiver representations and Fomin-Zelevinsky’s cluster algebras. Mathematics Subject Classification (2000). Primary 18E30; Secondary 16D90, 18G10.The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincare duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed.We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce

Diagonalizable derivations of finite-dimensional algebras I

n