Infinite-dimensional preprojective algebras

The additive preprojective algebra Π_k(Q) of a finite quiver is presented on the doubled quiver by the local relators ρ_v, one at each vertex, homogeneous of path length two and lying in the corner of v. The growth bound TauCeti.PathAlgebra.not_module_finite_quotient_span_range_of_two_mul_le_sum therefore applies to it. The arrows of the doubled quiver from i to j are the arrows of Q between i and j in either direction, so the condition on a nonzero nonnegative weight δ on the vertices reads

2 δ_i ≤ ∑_j (#(i ⟶ j) + #(j ⟶ i)) δ_j,

that is (2I - A) δ ≤ 0 for the adjacency matrix A of the underlying multigraph of Q, a loop counting twice. Under it Π_k(Q) is infinite-dimensional over every field k.

For an orientation of a finite simple graph G the condition is 2 δ_i ≤ ∑_{j ∼ i} δ_j. The marks of an affine simply-laced diagram satisfy it with equality, so the preprojective algebra of every orientation of a graphical affine diagram is infinite-dimensional. The affine diagrams of types D_n, E6, E7 and E8 are trees, so all their orientations are acyclic and the oriented-cycle argument of TauCeti.not_module_finite_preprojectiveAlgebra_of_length_pos does not reach them.

A weight exists as soon as the form of 2I - A is not positive definite: a symmetric matrix with nonpositive off-diagonal entries and a nonpositive value of its form has a nonzero nonnegative vector δ with (2I - A) δ ≤ 0 (Matrix.exists_nonneg_mulVec_nonpos_of_dotProduct_mulVec_nonpos). Twice the Tits form of Q is the form of 2I - A, so Π_k(Q) is infinite-dimensional whenever the Tits form is not positive definite. For a connected simple graph, a positive definite 2I - A makes the graph a Dynkin diagram by the classification of finite-type Cartan matrices, so the preprojective algebra of every orientation of a connected graph which is not a Dynkin diagram is infinite-dimensional.

Main results

• TauCeti.not_module_finite_preprojectiveAlgebra_of_two_mul_le_sum: a preprojective algebra whose underlying multigraph carries a nonzero nonnegative weight δ with (2I - A) δ ≤ 0 is infinite-dimensional.
• TauCeti.not_module_finite_signlessPreprojectiveAlgebra_of_two_mul_le_sum: the same for the signless preprojective algebra of a finite quiver with arrow reversal.
• TauCeti.not_module_finite_preprojectiveAlgebra_orientedQuiver_of_two_mul_le_sum and TauCeti.not_module_finite_signlessPreprojectiveAlgebra_doubledQuiver_of_two_mul_le_sum: the same for an orientation and for the doubled quiver of a finite simple graph, with the condition 2 δ_i ≤ ∑_{j ∼ i} δ_j.
• TauCeti.not_module_finite_preprojectiveAlgebra_of_not_posDef_titsForm: a preprojective algebra whose Tits form is not positive definite is infinite-dimensional.
• TauCeti.not_module_finite_preprojectiveAlgebra_orientedQuiver_of_not_posDef and TauCeti.not_module_finite_signlessPreprojectiveAlgebra_of_not_posDef: the same for a finite simple graph whose matrix 2I - A is not positive definite.
• TauCeti.not_module_finite_preprojectiveAlgebra_orientedQuiver_of_not_exists_dynkinType_iso and TauCeti.not_module_finite_signlessPreprojectiveAlgebra_of_not_exists_dynkinType_iso: the preprojective and signless preprojective algebras of a connected non-Dynkin graph are infinite-dimensional.
• TauCeti.AffineDynkinType.not_module_finite_preprojectiveAlgebra and TauCeti.AffineDynkinType.not_module_finite_signlessPreprojectiveAlgebra: the preprojective and signless preprojective algebras of a graphical affine simply-laced diagram are infinite-dimensional.

Implementation notes

The oriented quiver of a graph has Finite vertex and arrow types; as in TauCeti.RepresentationTheory.Quiver.Preprojective.GraphOrientation, the Fintype structures its preprojective algebra needs are supplied by Fintype.ofFinite.

References

• W. Crawley-Boevey, Quiver algebras, weighted projective lines, and the Deligne--Simpson problem, Section 1, for the preprojective algebra and its local relations.
• P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, for the infinite-dimensionality of the preprojective algebras of non-Dynkin quivers.
• V. Kac, Infinite dimensional Lie algebras, Chapter 4, for the marks of the affine diagrams and the separation of finite from non-finite types by nonnegative vectors.

theorem TauCeti.not_module_finite_preprojectiveAlgebra_of_two_mul_le_sum (k : Type w) {Q : Type u} [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {δ : Q → S} (hδ0 : 0 ≤ δ) (hδ : δ ≠ 0) (hδle : ∀ (i : Q), 2 * δ i ≤ ∑ j : Q, ↑(Fintype.card (i ⟶ j) + Fintype.card (j ⟶ i)) * δ j) : ¬Module.Finite k (preprojectiveAlgebra k Q)

A preprojective algebra whose underlying multigraph carries a suitable weight is infinite-dimensional. If a nonzero nonnegative weight δ on the vertices of Q satisfies 2 δ_i ≤ ∑_j (#(i ⟶ j) + #(j ⟶ i)) δ_j at every vertex, that is (2I - A) δ ≤ 0 for the adjacency matrix A of the underlying multigraph, then Π_k(Q) is not a finite-dimensional k-vector space.

theorem TauCeti.not_module_finite_signlessPreprojectiveAlgebra_of_two_mul_le_sum (k : Type w) {Q : Type u} [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] [Quiver.HasReverse Q] [(x : Q) → Fintype (Quiver.Star x)] {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {δ : Q → S} (hδ0 : 0 ≤ δ) (hδ : δ ≠ 0) (hδle : ∀ (i : Q), 2 * δ i ≤ ∑ j : Q, ↑(Fintype.card (i ⟶ j)) * δ j) : ¬Module.Finite k (signlessPreprojectiveAlgebra k Q)

A signless preprojective algebra carrying a suitable weight is infinite-dimensional. If a nonzero nonnegative weight δ on the vertices of Q satisfies 2 δ_i ≤ ∑_j #(i ⟶ j) δ_j at every vertex, then the signless preprojective algebra of Q is not a finite-dimensional k-vector space.

theorem TauCeti.not_module_finite_preprojectiveAlgebra_of_not_posDef_titsForm (k : Type w) {Q : Type u} [Field k] [Quiver Q] [Fintype Q] [(i j : Q) → Fintype (i ⟶ j)] (h : ¬(titsForm Q).PosDef) : ¬Module.Finite k (preprojectiveAlgebra k Q)

A preprojective algebra whose Tits form is not positive definite is infinite-dimensional. If the Tits form of the finite quiver Q is not positive definite, then Π_k(Q) is not a finite-dimensional k-vector space. In particular this holds for every quiver with a loop, since a loop at i makes the Tits form of the simple dimension vector at i nonpositive.

theorem TauCeti.not_module_finite_preprojectiveAlgebra_orientedQuiver_of_two_mul_le_sum {V : Type u} {G : SimpleGraph V} (k : Type w) [Field k] [Fintype V] [DecidableRel G.Adj] (o : DoubledQuiver.Orientation G) {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {δ : V → S} (hδ0 : 0 ≤ δ) (hδ : δ ≠ 0) (hδle : ∀ (i : V), 2 * δ i ≤ ∑ j ∈ G.neighborFinset i, δ j) : ¬Module.Finite k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o))

The preprojective algebra of an oriented graph carrying a suitable weight is infinite-dimensional. If a nonzero nonnegative weight δ on the vertices of a finite simple graph G satisfies 2 δ_i ≤ ∑_{j ∼ i} δ_j at every vertex, then the preprojective algebra of every orientation of G is not a finite-dimensional k-vector space.

theorem TauCeti.not_module_finite_signlessPreprojectiveAlgebra_doubledQuiver_of_two_mul_le_sum {V : Type u} {G : SimpleGraph V} (k : Type w) [Field k] [Finite V] [(i : V) → Fintype ↑(G.neighborSet i)] {S : Type u_1} [CommRing S] [LinearOrder S] [IsStrictOrderedRing S] {δ : V → S} (hδ0 : 0 ≤ δ) (hδ : δ ≠ 0) (hδle : ∀ (i : V), 2 * δ i ≤ ∑ j ∈ G.neighborFinset i, δ j) : ¬Module.Finite k (signlessPreprojectiveAlgebra k (DoubledQuiver G))

The signless preprojective algebra of a graph carrying a suitable weight is infinite-dimensional. If a nonzero nonnegative weight δ on the vertices of a finite simple graph G satisfies 2 δ_i ≤ ∑_{j ∼ i} δ_j at every vertex, then the signless preprojective algebra of its doubled quiver is not a finite-dimensional k-vector space.

theorem TauCeti.AffineDynkinType.not_module_finite_preprojectiveAlgebra (k : Type w) [Field k] {t : AffineDynkinType} (ht : t.Valid) (hg : t.IsGraphical) (o : DoubledQuiver.Orientation t.graph) : ¬Module.Finite k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver t.graph o))

The preprojective algebra of an affine simply-laced diagram is infinite-dimensional, for every orientation of a valid graphical affine diagram and over every field. The multiplicity-two diagram Ã₁, which is not a simple graph, is excluded by IsGraphical.

theorem TauCeti.AffineDynkinType.not_module_finite_signlessPreprojectiveAlgebra (k : Type w) [Field k] {t : AffineDynkinType} [hN : (i : Fin t.nodes) → Fintype ↑(t.graph.neighborSet i)] (ht : t.Valid) (hg : t.IsGraphical) : ¬Module.Finite k (signlessPreprojectiveAlgebra k (DoubledQuiver t.graph))

The signless preprojective algebra of an affine simply-laced diagram is infinite-dimensional, for every valid graphical affine diagram and over every field.

theorem TauCeti.not_module_finite_preprojectiveAlgebra_orientedQuiver_of_not_posDef {V : Type u} {G : SimpleGraph V} (k : Type w) [Field k] [Finite V] [DecidableEq V] [DecidableRel G.Adj] (o : DoubledQuiver.Orientation G) (h : ¬(G.graphCartanMatrix ℝ).PosDef) : ¬Module.Finite k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o))

The preprojective algebra of an oriented graph with non-positive-definite form is infinite-dimensional. If the generalized Cartan matrix 2I - A of a finite simple graph G is not positive definite over ℝ, then the preprojective algebra of every orientation of G is not a finite-dimensional k-vector space.

theorem TauCeti.not_module_finite_signlessPreprojectiveAlgebra_of_not_posDef {V : Type u} {G : SimpleGraph V} (k : Type w) [Field k] [Finite V] [DecidableEq V] [DecidableRel G.Adj] [hN : (i : V) → Fintype ↑(G.neighborSet i)] (h : ¬(G.graphCartanMatrix ℝ).PosDef) : ¬Module.Finite k (signlessPreprojectiveAlgebra k (DoubledQuiver G))

The signless preprojective algebra of a graph with non-positive-definite form is infinite-dimensional. If the generalized Cartan matrix 2I - A of a finite simple graph G is not positive definite over ℝ, then the signless preprojective algebra of its doubled quiver is not a finite-dimensional k-vector space.

theorem TauCeti.not_module_finite_preprojectiveAlgebra_orientedQuiver_of_not_exists_dynkinType_iso {V : Type u} {G : SimpleGraph V} (k : Type w) [Field k] [Finite V] (hG : G.Connected) (hD : ¬∃ (t : DynkinType), t.Valid ∧ Nonempty (G ≃g diagramGraph t.cartanMatrix)) (o : DoubledQuiver.Orientation G) : ¬Module.Finite k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o))

The preprojective algebra of a connected non-Dynkin graph is infinite-dimensional. If a connected finite simple graph G is not isomorphic to the diagram of any valid Dynkin type, then the preprojective algebra of every orientation of G is not a finite-dimensional k-vector space.

theorem TauCeti.not_module_finite_signlessPreprojectiveAlgebra_of_not_exists_dynkinType_iso {V : Type u} {G : SimpleGraph V} (k : Type w) [Field k] [Finite V] [(i : V) → Fintype ↑(G.neighborSet i)] (hG : G.Connected) (hD : ¬∃ (t : DynkinType), t.Valid ∧ Nonempty (G ≃g diagramGraph t.cartanMatrix)) : ¬Module.Finite k (signlessPreprojectiveAlgebra k (DoubledQuiver G))

The signless preprojective algebra of a connected non-Dynkin graph is infinite-dimensional. If a connected finite simple graph G is not isomorphic to the diagram of any valid Dynkin type, then the signless preprojective algebra of its doubled quiver is not a finite-dimensional k-vector space.