The three indecomposable representations of the A₂ quiver #
The A₂ quiver • → • is the generalized Kronecker quiver on a one-element arrow type. This file
classifies its finite-dimensional indecomposable representations: there are exactly three, the two
vertex simples S₁ = (k → 0) and S₂ = (0 → k) and the vertex projective
P₁ = kQ·e₁ = (k →^{id} k), of dimension vectors (1,0), (0,1) and (1,1) -- the three
positive roots of A₂. In particular the A₂ quiver has finite representation type, the
positive half of Gabriel's dichotomy at the smallest Dynkin diagram, mirroring
TauCeti.not_isFiniteRepType_kronecker on the other side of the boundary.
The proof needs no reflection functor and no Krull-Schmidt theory. Indecomposability of an object
in a category where idempotents split is exactly the triviality of its idempotent endomorphisms
(TauCeti.indecomposable_iff_idempotent_eq_zero_or_id), and an endomorphism of a representation of
the generalized Kronecker quiver is a pair of endomorphisms of the two vertex spaces intertwining
the action of every arrow. Feeding three linear projections into that dichotomy settles the
classification:
- the projection onto the kernel of the arrow, paired with
0at the target, forces the arrow to be injective once the target is nonzero; - the projection onto a complement of the range of the arrow, paired with
0at the source, forces it to be surjective once the source is nonzero; - with the arrow then invertible, conjugating an arbitrary idempotent of the source through it produces an intertwining pair, so the source -- and hence the target -- has no proper nonzero subspace and is a line.
Finite-dimensionality is therefore a conclusion, not a hypothesis: an indecomposable
representation of the A₂ quiver is automatically finite-dimensional.
The list is exact, not merely exhaustive: the three dimension vectors are distinct, so the three
representations are pairwise non-isomorphic, and the skeleton of the finite-dimensional
indecomposables is counted by Fin 3.
Main results #
TauCeti.eq_zero_or_eq_id_of_indecomposable_kronecker: an indecomposable representation of the generalized Kronecker quiver admits no nontrivial intertwining pair of idempotents.TauCeti.ker_map_arrowPath_eq_bot,TauCeti.range_map_arrowPath_eq_topandTauCeti.isIso_map_arrowPath: over theA₂quiver the arrow of an indecomposable representation is injective as soon as its target is nonzero and surjective as soon as its source is, hence an isomorphism when neither vertex space vanishes.TauCeti.finrank_src_eq_one_of_isZero_tgt,TauCeti.finrank_tgt_eq_one_of_isZero_src,TauCeti.finrank_src_eq_one_of_not_isZeroandTauCeti.finrank_tgt_eq_one_of_not_isZero: the nonzero vertex spaces of an indecomposable representation are lines.TauCeti.nonempty_iso_simpleRep_src_or_simpleRep_tgt_or_indecProjRep_of_indecomposable_kronecker: every indecomposable representation of theA₂quiver isS₁,S₂orP₁.TauCeti.not_nonempty_simpleRep_indecProjRep_iso: a vertex simple of theA₂quiver is notP₁, which withTauCeti.not_nonempty_simpleRep_isomakes the three pairwise non-isomorphic.TauCeti.indecomposableKroneckerEquiv: the explicit enumeration of these classes byFin 3.TauCeti.card_skeleton_indecomposable_kronecker: theA₂quiver has exactly three finite-dimensional indecomposable representations up to isomorphism.
The finite representation type read off that count is
TauCeti.isFiniteRepType_kronecker, in
TauCeti.RepresentationTheory.Quiver.Kronecker.FiniteRepType beside its negative counterpart.
Implementation notes #
The pair of a linear map at each vertex intertwining every arrow is turned into a morphism of
representations by TauCeti.kroneckerHom, in
TauCeti.RepresentationTheory.Quiver.Kronecker.Representation, and into an isomorphism by
TauCeti.kroneckerIso there: neither needs indecomposability, so neither lives here.
The classification is stated for an arrow type in Type, not in an arbitrary universe. That is
what puts the three comparison objects in the same universe as the representation: the vertex
simple TauCeti.simpleRep puts the base field itself at its vertex, while TauCeti.indecProjRep
puts the paths of the quiver into a Finsupp, and only for a Type-valued arrow type do the two
land in one universe. The idempotent dichotomy and the structure results below carry an arbitrary
arrow universe, and the [Unique A] hypothesis is spelled only where a single arrow is genuinely
used.
References #
This supplies the representation-level half of the "A₂ quiver" worked example of
TauCetiRoadmap/RepresentationTheory/QuiverRepresentations/README.md, whose path-algebra half is
TauCeti.RepresentationTheory.Quiver.Kronecker.UpperTriangular, together with the finite
representation type asked for in its Layer 5. See Assem--Simson--Skowroński, Elements of the
Representation Theory of Associative Algebras I, Ch. II and VII, and Schiffler, Quiver
Representations, Ch. 2.
An indecomposable representation of the generalized Kronecker quiver admits no nontrivial
pair of intertwining idempotents. The intertwining hypothesis is explicit: it is not determined
by the idempotency hypotheses, which name only p and q.
A vertex space of an indecomposable representation of the generalized Kronecker quiver whose
partner vanishes is a line. With the target zero, every idempotent of the source intertwines
every arrow trivially, so indecomposability leaves only 0 and the identity.
The mirror of TauCeti.finrank_src_eq_one_of_isZero_tgt: with the source zero, the target of
an indecomposable representation of the generalized Kronecker quiver is a line.
The arrow of an indecomposable representation of the A₂ quiver whose target vertex space is
nonzero is injective. The projection onto its kernel, paired with 0 at the target, is an
idempotent pair; the identity is excluded because the target does not vanish. Nothing is assumed at
the source, so this is stronger than the case of two nonzero vertex spaces.
The arrow of an indecomposable representation of the A₂ quiver whose source vertex space is
nonzero is surjective, the mirror of TauCeti.ker_map_arrowPath_eq_bot: the projection onto a
complement of its range, paired with 0 at the source, is an idempotent pair, and the identity is
excluded because the source does not vanish. Nothing is assumed at the target, so this too is
stronger than the case of two nonzero vertex spaces.
The arrow of an indecomposable representation of the A₂ quiver with both vertex spaces
nonzero is an isomorphism, by TauCeti.ker_map_arrowPath_eq_bot and
TauCeti.range_map_arrowPath_eq_top.
A nonzero source vertex space of an indecomposable representation of the A₂ quiver is a
line; for the target vertex space see TauCeti.finrank_tgt_eq_one_of_not_isZero. With the target
zero this is TauCeti.finrank_src_eq_one_of_isZero_tgt; otherwise the arrow is an isomorphism by
TauCeti.isIso_map_arrowPath, and conjugating an idempotent of the source through it produces an
intertwining idempotent pair.
A nonzero target vertex space of an indecomposable representation of the A₂ quiver is a
line, the companion of TauCeti.finrank_src_eq_one_of_not_isZero. With the source zero this is
TauCeti.finrank_tgt_eq_one_of_isZero_src; otherwise the arrow is an isomorphism, so it carries
the dimension of the source to the dimension of the target.
Every indecomposable representation of the A₂ quiver is one of the three: the two vertex
simples S₁, S₂ and the projective P₁ = kQ·e₁, of dimension vectors (1,0), (0,1) and
(1,1) -- the three positive roots of A₂. That the three are pairwise non-isomorphic, so that
this list is exact rather than merely exhaustive, is
TauCeti.not_nonempty_simpleRep_indecProjRep_iso together with
TauCeti.not_nonempty_simpleRep_iso; the resulting count is
TauCeti.card_skeleton_indecomposable_kronecker.
No finite-dimensionality is assumed: it is a conclusion. If the target vanishes, every idempotent
of the source intertwines the arrow, so the source is a line and the representation is S₁;
mirror-image if the source vanishes. Otherwise the arrow is injective and surjective, because the
projections onto its kernel and onto a complement of its range extend to idempotent endomorphisms,
so conjugation through it turns an idempotent of the source into an intertwining pair and both
vertex spaces are lines.
A vertex simple of the A₂ quiver is not the projective P₁: the vertex where Sᵢ
vanishes is one where P₁ is a line, P₁ having dimension vector (1,1). With
TauCeti.not_nonempty_simpleRep_iso this makes the three representations of the classification
TauCeti.nonempty_iso_simpleRep_src_or_simpleRep_tgt_or_indecProjRep_of_indecomposable_kronecker
pairwise non-isomorphic.
The A₂ quiver has exactly three finite-dimensional indecomposable representations up to
isomorphism: the classification
TauCeti.nonempty_iso_simpleRep_src_or_simpleRep_tgt_or_indecProjRep_of_indecomposable_kronecker
exhibits S₁, S₂ and P₁ as an exhaustive list, and their dimension vectors (1,0), (0,1) and
(1,1) keep them pairwise non-isomorphic, so the skeleton is counted by Fin 3.
The finite-dimensional indecomposable classes of the one-arrow quiver, listed as S₁,
S₂, P₁.
Equations
Instances For
The first indecomposable class is the source simple.
The second indecomposable class is the target simple.
The third indecomposable class is the source projective.