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TauCeti.RepresentationTheory.Quiver.Zigzag.Socle

The socle and the top of a zigzag algebra #

For a finite simple graph without isolated vertices over a field, the Jacobson radical of the zigzag relation quotient is its positive-length part, its square is the span of the volume classes, and its cube vanishes. This file reads the two ends of that filtration as module invariants: the socle of the algebra, viewed as a left module over itself, is the span of the volume classes, and the top — the quotient by the radical — is the product of one copy of the ground field for each vertex. Both have dimension |V|.

The socle is computed as the square of the radical. One containment is a nilpotence argument: a simple submodule m is killed by the radical, since otherwise J • m = m forces m = J³ • m = 0, and an element killed on the left by every arrow has no vertex and no arrow coordinate. The other containment needs the volume classes themselves to be simple, and that is where the multiplication table enters: a left ideal absorbs an idempotent, an arrow and a volume class in only three ways, and each of them sends the volume class at a vertex back into its own k-line. So that line is a left ideal of dimension one, hence a simple module.

The top is the first isomorphism theorem applied to the vertex-coefficient homomorphism, whose kernel is the radical. It exhibits the |V| simple modules of a zigzag algebra as the characters reading a single vertex coordinate.

Main declarations #

Main results #

References #

See Huerfano--Khovanov, A category for the adjoint representation, Section 3, and Ehrig--Tubbenhauer, Algebraic properties of zigzag algebras, Section 2.

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theorem TauCeti.mul_zigzagVolume_mem_span_singleton {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] (z : nonisolatedZigzagQuotient k G) (i : V) :
z * zigzagVolume k G i ∈ k ∙ zigzagVolume k G i

Left multiplication by any element carries a volume class into its own k-line: the multiplication table sends it to itself, up to a scalar, or to zero.

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theorem TauCeti.restrictScalars_span_singleton_zigzagVolume {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] (i : V) :
Submodule.restrictScalars k (nonisolatedZigzagQuotient k G ∙ zigzagVolume k G i) = k ∙ zigzagVolume k G i

The left ideal generated by a volume class is nothing but its k-line.

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theorem TauCeti.mem_zigzagVolumeSpan_of_forall_zigzagMk_ofArrow_mul_eq_zero {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {x : nonisolatedZigzagQuotient k G} (hx : ∀ (d : G.Dart), (zigzagMk k G) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯)) * x = 0) :
x ∈ zigzagVolumeSpan k G

An element killed on the left by every arrow is a combination of volume classes.

The socle #

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theorem TauCeti.isSimpleModule_span_singleton_zigzagVolume {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (i : V) (hi : ∃ (j : V), G.Adj i j) :
IsSimpleModule (nonisolatedZigzagQuotient k G) ↥(nonisolatedZigzagQuotient k G ∙ zigzagVolume k G i)

The line spanned by a volume class is a simple left module. It is a one-dimensional k-subspace by TauCeti.restrictScalars_span_singleton_zigzagVolume, and a left ideal by construction, so it has no proper nonzero submodule.

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theorem TauCeti.zigzagVolume_mem_socle {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (i : V) (hi : ∃ (j : V), G.Adj i j) :
zigzagVolume k G i ∈ socle (nonisolatedZigzagQuotient k G) (nonisolatedZigzagQuotient k G)

Every volume class lies in the socle.

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theorem TauCeti.socle_nonisolatedZigzagQuotient_eq_jacobson_sq {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) :
socle (nonisolatedZigzagQuotient k G) (nonisolatedZigzagQuotient k G) = Ring.jacobson (nonisolatedZigzagQuotient k G) ^ 2

The socle of a zigzag algebra is the square of its Jacobson radical. Equivalently, by TauCeti.restrictScalars_jacobson_sq_nonisolatedZigzagQuotient_eq_zigzagVolumeSpan, it is the span of the volume classes.

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theorem TauCeti.restrictScalars_socle_nonisolatedZigzagQuotient_eq_zigzagVolumeSpan {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) :
Submodule.restrictScalars k (socle (nonisolatedZigzagQuotient k G) (nonisolatedZigzagQuotient k G)) = zigzagVolumeSpan k G

The socle of a zigzag algebra is the span of its volume classes.

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theorem TauCeti.finrank_restrictScalars_socle_nonisolatedZigzagQuotient {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] [Fintype V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) :
Module.finrank k ↥(Submodule.restrictScalars k (socle (nonisolatedZigzagQuotient k G) (nonisolatedZigzagQuotient k G))) = Fintype.card V

The socle of a zigzag algebra has one dimension for each vertex.

The top #

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noncomputable def TauCeti.zigzagTopEquiv {V : Type u} (k : Type w) [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) :
(nonisolatedZigzagQuotient k G ⧸ Ring.jacobson (nonisolatedZigzagQuotient k G)) ≃ₐ[k] DoubledQuiver G → k

The top of a zigzag algebra. The quotient by the Jacobson radical is the product of one copy of the ground field for each vertex, the vertex coefficients of an element being its coordinates.

Instances For
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    @[simp]
    theorem TauCeti.zigzagTopEquiv_mk {V : Type u} (k : Type w) [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (x : nonisolatedZigzagQuotient k G) :
    (zigzagTopEquiv k hns) ((Ideal.Quotient.mk (Ring.jacobson (nonisolatedZigzagQuotient k G))) x) = (zigzagTrivialCoeff k G) x
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    theorem TauCeti.isSemisimpleRing_quotient_jacobson_nonisolatedZigzagQuotient {V : Type u} (k : Type w) [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) :
    IsSemisimpleRing (nonisolatedZigzagQuotient k G ⧸ Ring.jacobson (nonisolatedZigzagQuotient k G))

    The top of a zigzag algebra is semisimple.

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    theorem TauCeti.finrank_quotient_jacobson_nonisolatedZigzagQuotient {V : Type u} (k : Type w) [Field k] {G : SimpleGraph V} [Finite V] [Fintype V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) :
    Module.finrank k (nonisolatedZigzagQuotient k G ⧸ Ring.jacobson (nonisolatedZigzagQuotient k G)) = Fintype.card V

    The top of a zigzag algebra has one dimension for each vertex, matching the socle.