Jan Schröer

Rigid modules over preprojective algebras

Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis.

Kac–Moody groups and cluster algebras

Abstract Let Q be a finite quiver without oriented cycles, let Λ be the associated preprojective algebra, let g be the associated Kac–Moody Lie algebra with Weyl group W, and let n be the positive part of g . For each Weyl group element w, a subcategory C w of mod ( Λ ) was introduced by Buan, Iyama, Reiten and Scott. It is known that C w is a Frobenius category and that its stable category C w is a Calabi–Yau category of dimension two. We show that C w yields a cluster algebra structure on the coordinate ring C [ N ( w ) ] of the unipotent group N ( w ) : = N ∩ ( w − 1 N − w ) . Here N is the pro-unipotent pro-group with Lie algebra the completion n ˆ of n . One can identify C [ N ( w ) ] with a subalgebra of U ( n ) gr ⁎ , the graded dual of the universal enveloping algebra U ( n ) of n . Let S ⁎ be the dual of Lusztigʼs semicanonical basis S of U ( n ) . We show that all cluster monomials of C [ N ( w ) ] belong to S ⁎ , and that S ⁎ ∩ C [ N ( w ) ] is a C -basis of C [ N ( w ) ] . Moreover, we show that the cluster algebra obtained from C [ N ( w ) ] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C [ N w ] of regular functions on the unipotent cell N w of the Kac–Moody group with Lie algebra g . We obtain a corresponding dual semicanonical basis of C [ N w ] . As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.

Radical embeddings and representation dimension

Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.

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 structures on quantum coordinate rings

n

Generic bases for cluster algebras and the Chamber Ansatz

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

Auslander algebras and initial seeds for cluster algebras

n

The representation type of Jacobian algebras

. 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.

Preprojective algebras and cluster algebras

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac–Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory