Christof Geiß

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.

Cluster structures on quantum coordinate rings

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

Auslander algebras and initial seeds for cluster algebras

{\fancyscript{C}_{w}}

The representation type of Jacobian algebras

of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities that can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.

Extension-orthogonal components of preprojective varieties

Let Q be a finite quiver without oriented cycles, and let

Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras

\Lambda

Quivers with relations for symmetrizable Cartan matrices V: Caldero–Chapoton formulas

be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a unipotent cell N^w of the Kac-Moody group with Lie algebra g. In previous work we proved that the coordinate ring \C[N^w] of N^w is a cluster algebra in a natural way. A central role is played by generating functions \vphi_X of Euler characteristics of certain varieties of partial composition series of X, where X runs through all modules in a Frobenius subcategory C_w of the category of nilpotent