Finite-dimensional preprojective algebras of finite ADE diagrams

The additive preprojective algebra of every orientation of a graph isomorphic to a standard Dynkin diagram is finite-dimensional over every field. The same holds for its signless presentation. For connected diagrams with at least three vertices, this is the quadratic-dual presentation used for zigzag algebras.

The type-specific calculations cover the simply-laced diagrams A, D and E. The diagram of every standard Dynkin type is isomorphic as a simple graph to one of these (TauCeti.DynkinType.exists_isSimplyLaced_diagramGraph_iso), and transporting along graph isomorphisms gives both finiteness theorems. Together with the non-Dynkin growth bound of TauCeti.RepresentationTheory.Quiver.Preprojective.InfiniteDimensional, this yields the complete finite/infinite-dimensionality dichotomy for connected finite simple graphs: both the signless preprojective algebra of DoubledQuiver G and the preprojective algebra of every orientation OrientedQuiver G o are finite-dimensional over k if and only if G is isomorphic to the diagram of a valid Dynkin type (equivalently, a valid simply-laced Dynkin type).

The signless results retain the caller's neighborhood Fintype instances as explicit parameters, so their conclusions refer to the quotient carrier formed with those enumerations.

Main results

• TauCeti.finiteDimensional_signlessPreprojectiveAlgebra_of_iso and TauCeti.finiteDimensional_preprojectiveAlgebra_of_iso: finiteness for any graph isomorphic to a standard Dynkin diagram.
• TauCeti.finiteDimensional_signlessPreprojectiveAlgebra_iff and TauCeti.finiteDimensional_preprojectiveAlgebra_orientedQuiver_iff: the finite/infinite dichotomy for connected finite simple graphs.

References

The A, D, and E calculations are supplied by the type-specific developments. The local presentation and finite-Dynkin finiteness theorem are discussed in Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne--Simpson problem, Section 1. The signless comparison follows Huerfano--Khovanov, A category for the adjoint representation, Section 3, https://arxiv.org/abs/math/0002060.

theorem TauCeti.finiteDimensional_signlessPreprojectiveAlgebra_of_iso (k : Type u_1) [Field k] {V : Type u_2} [Finite V] {G : SimpleGraph V} [(i : V) → Fintype ↑(G.neighborSet i)] {t : DynkinType} (e : G ≃g diagramGraph t.cartanMatrix) : FiniteDimensional k (signlessPreprojectiveAlgebra k (DoubledQuiver G))

The signless preprojective algebra of any graph isomorphic to a standard Dynkin diagram is finite-dimensional over every field.

theorem TauCeti.finiteDimensional_preprojectiveAlgebra_of_iso (k : Type u_1) [Field k] {V : Type u_2} [Finite V] {G : SimpleGraph V} {t : DynkinType} (e : G ≃g diagramGraph t.cartanMatrix) (o : DoubledQuiver.Orientation G) : FiniteDimensional k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o))

The additive preprojective algebra of every orientation of a graph isomorphic to a standard Dynkin diagram is finite-dimensional over every field.

theorem TauCeti.finiteDimensional_signlessPreprojectiveAlgebra_iff (k : Type u_1) [Field k] {V : Type u_2} [Finite V] {G : SimpleGraph V} [(i : V) → Fintype ↑(G.neighborSet i)] (hG : G.Connected) : FiniteDimensional k (signlessPreprojectiveAlgebra k (DoubledQuiver G)) ↔ ∃ (t : DynkinType), t.Valid ∧ Nonempty (G ≃g diagramGraph t.cartanMatrix)

The finite/infinite dichotomy for signless preprojective algebras: for a connected finite simple graph G, the signless preprojective algebra of DoubledQuiver G is finite-dimensional over k if and only if G is isomorphic to the diagram of a valid Dynkin type.

theorem TauCeti.finiteDimensional_preprojectiveAlgebra_orientedQuiver_iff (k : Type u_1) [Field k] {V : Type u_2} [Finite V] {G : SimpleGraph V} (hG : G.Connected) (o : DoubledQuiver.Orientation G) : FiniteDimensional k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o)) ↔ ∃ (t : DynkinType), t.Valid ∧ Nonempty (G ≃g diagramGraph t.cartanMatrix)

The finite/infinite dichotomy for preprojective algebras of oriented graphs: for a connected finite simple graph G and any orientation o of G, the preprojective algebra Π_k(OrientedQuiver G o) is finite-dimensional over k if and only if G is isomorphic to the diagram of a valid Dynkin type.