The representation type of the generalized Kronecker quiver #
This file settles the representation type of the generalized Kronecker quiver on both sides of
Gabriel's boundary: infinite as soon as there are two distinct arrows, finite for the A₂ quiver
• → • of a single arrow.
The negative half exhibits, over every field, an infinite family of pairwise non-isomorphic
finite-dimensional indecomposable representations of the generalized Kronecker quiver as soon as
there are two distinct arrows: TauCeti.kroneckerJordanRep puts the truncated polynomial algebra
k[X]/(Xⁿ⁺¹) at both vertices, lets one distinguished arrow act by multiplication by the class of
X and every other arrow by the identity, and is indecomposable because its endomorphism algebra
is the local ring k[X]/(Xⁿ⁺¹). The underlying construction of a representation from a pair of
vector spaces and a linear map along each arrow is TauCeti.kroneckerRep, in
TauCeti.RepresentationTheory.Quiver.Kronecker.Representation.
The Kronecker quiver • ⇉ • is the case of a two-element arrow type, and it is the boundary case
of Gabriel's theorem: connected and acyclic, but not of Dynkin type, its Tits form being the
positive semidefinite (a - b) ^ 2 of TauCeti.Quiver.Kronecker.titsForm_apply. The family below
supplies the representation-theoretic half of that boundary, namely that finite representation type
genuinely fails there.
The A₂ quiver • → • -- a one-element arrow type -- is of Dynkin type and does have finite
representation type, so the hypothesis that two distinct arrows exist is not an artefact: with a
single arrow there is no arrow beside the distinguished one, so nothing forces the two components
of an endomorphism to agree and the Jordan block below is not indecomposable -- the A₂
indecomposables have dimension vectors (1, 0), (0, 1) and (1, 1). That positive half,
TauCeti.isFiniteRepType_kronecker, is read off the count of those three isomorphism classes in
TauCeti.RepresentationTheory.Quiver.Kronecker.Indecomposable.
Main definitions #
TauCeti.kroneckerJordanRep: the nilpotent Jordan block of sizen + 1, spread over the two vertices.
Main results #
TauCeti.indecomposable_kroneckerJordanRep: the Jordan blocks are indecomposable, as soon as some arrow other than the distinguished one exists.TauCeti.nonempty_kroneckerJordanRep_iso_iff: two of them are isomorphic exactly when their sizes agree.TauCeti.not_isFiniteRepType_kronecker: over any field, a generalized Kronecker quiver with at least two arrows has infinite representation type.TauCeti.isFiniteRepType_kronecker: theA₂quiver, of a single arrow, has finite representation type.
Implementation notes #
TauCeti.kroneckerJordanRep carries @[expose], for the reason the loop-quiver file
TauCeti.RepresentationTheory.Quiver.OneLoop.FiniteRepType records, and
TauCeti.kroneckerRep does too: a functor built by CategoryTheory.Paths.lift reveals its value
on objects only through its definition. The obstruction is at the level of statements, not of
proofs, so no characteristic lemma can remove it: without @[expose], every statement below
reading an endomorphism of a Jordan block as a linear map on k[X]/(Xⁿ⁺¹) fails to elaborate.
Indecomposability runs through TauCeti.indecomposable_of_injective_of_isLocalRing rather than the
brick criterion: the endomorphism algebra of a Jordan block is k[X]/(Xⁿ⁺¹), which is not a field.
It is a local ring (TauCeti.isLocalRing_adjoinRoot_X_pow), and that is exactly enough: the
value of an endomorphism at 1 records it faithfully there, sending 0 to 0, the identity to
1 and squares to squares.
The two components of an endomorphism are forced to agree by naturality along an arrow acting as
the identity, which is why a second arrow is needed; the distinguished arrow then contributes the
one genuine relation, that the common component commutes with multiplication by the class of X,
and AdjoinRoot.eq_mulRight_of_root_mul turns that relation into the description of the
endomorphism algebra. Taking the vertex spaces to be a truncated polynomial algebra rather than a
line is what makes the result uniform in the base field: the one-dimensional representations
k ⇉ k, with the two arrows acting by 1 and by a scalar, are indecomposable and pairwise
non-isomorphic, but over a finite field there are only finitely many of them.
References #
This proves the ¬ IsFiniteRepType half of the “Kronecker quiver” worked example of
TauCetiRoadmap/RepresentationTheory/QuiverRepresentations/README.md, whose other half -- the
positive semidefinite Tits form (a - b) ^ 2 with radical (1, 1) -- is
TauCeti.Quiver.Kronecker.titsForm_apply and
TauCeti.Quiver.Kronecker.titsForm_eq_zero_iff_exists_smul. See Derksen--Weyman, An Introduction
to Quiver Representations, and Assem--Simson--Skowroński, Elements of the Representation Theory
of Associative Algebras I, Ch. II.
The Jordan blocks #
The Kronecker Jordan block of size n + 1: the truncated polynomial algebra k[X]/(Xⁿ⁺¹)
at both vertices of the generalized Kronecker quiver, with the arrow a₁ acting by multiplication
by the class of X and every other arrow by the identity.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The distinguished arrow acts on a Jordan block by multiplication by the class of X.
The action of the distinguished arrow, read on an element.
Every other arrow acts on a Jordan block by the identity.
The action of any other arrow, read on an element.
A Jordan block is finite-dimensional: both of its vertex spaces are k[X]/(Xⁿ⁺¹).
The dimension vector of a Jordan block is n + 1 at both vertices.
A Jordan block is nonzero: its vertex spaces are the nontrivial ring k[X]/(Xⁿ⁺¹).
A Kronecker Jordan block is indecomposable, as soon as some arrow other than the distinguished
one exists. An idempotent endomorphism is multiplication by an idempotent of k[X]/(Xⁿ⁺¹), and
that ring is local, so that idempotent is 0 or 1.
Jordan blocks of different sizes are non-isomorphic: their dimension vectors differ.
Two Kronecker Jordan blocks are isomorphic exactly when their sizes agree.
A generalized Kronecker quiver with at least two arrows has infinite representation type over
every field. The Jordan blocks TauCeti.kroneckerJordanRep are finite-dimensional,
indecomposable and pairwise non-isomorphic, so ℕ indexes an infinite family of them.
For a one-element arrow type this fails, and must: that is the A₂ quiver • → •, of Dynkin type,
which has exactly three indecomposables.
The A₂ quiver has finite representation type, the positive half of Gabriel's dichotomy
for the smallest Dynkin quiver: by TauCeti.card_skeleton_indecomposable_kronecker its
finite-dimensional indecomposables fall into exactly three isomorphism classes.
Contrast TauCeti.not_isFiniteRepType_kronecker above: as soon as a second arrow is added the
Kronecker quiver leaves Dynkin type and acquires infinitely many indecomposables.