Quiver Representations and ADE

Why quivers? Quiver representations Gabriel’s theorem But really, why quivers?

Quiver representations and ADE

Sira Gratz

Sira Gratz Quiver representations “Deﬁnition” A representation of a quiver associates to every vertex a vector space, and to every arrow a compatible linear map.

Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? Quiver

Deﬁnition A quiver is a directed graph, where loops and multiple edges between two vertices are allowed.

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? Quiver

Deﬁnition A quiver is a directed graph, where loops and multiple edges between two vertices are allowed.

“Deﬁnition” A representation of a quiver associates to every vertex a vector space, and to every arrow a compatible linear map.

Sira Gratz Quiver representations Let A be a square matrix. When does there exist an invertible matrix S and a diagonal matrix D such that

SAS−1 = D?

Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? A question from linear algebra

Question: When is a square matrix diagonalisable?

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? A question from linear algebra

Question: When is a square matrix diagonalisable? Let A be a square matrix. When does there exist an invertible matrix S and a diagonal matrix D such that

SAS−1 = D?

Sira Gratz Quiver representations SAS−1 = D ⇔ SA = DS

n A K

S S

n n D K D K

Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? The same question, in a diagram

n A K

Sira Gratz Quiver representations n A K

n D K

Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? The same question, in a diagram

SAS−1 = D ⇔ SA = DS

n A K

n D K

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? The same question, in a diagram

SAS−1 = D ⇔ SA = DS

n n A K A K

S S

n n D K D K

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? The same question, in a diagram

study representations of A n K the loop:

n D K •

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? Another question from linear algebra

Question: Let (A, B) and (A0, B0) be pairs of matrices, all of the same dimension. When do there exist invertible matrices S, T such that

SAT −1 = A0; SBT −1 = B0?

Sira Gratz Quiver representations A A

m n m n K K K K B B S T S T A0 A0

m n m n K K K K B0 B0

A A

m n m n K K K K B B S T S T A0 A0

m n m n K K K K B0 B0

Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? The same question, in a diagram

m n K K B

m n K K B0

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? The same question, in a diagram

A A

m n m n K K K K B B S T S T A0 A0

m n m n K K K K B0 B0

A A

m n m n K K K K B B S T S T A0 A0

m n m n K K K K B0 B0

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? The same question, in a diagram

m n study representations of K K the 2-Kronecker quiver: B S T A

m n K K • • B

Sira Gratz Quiver representations More precisely, a quiver consists of the following data:

a set of vertices Q0;

a set of arrows Q1;

a map s : Q1 → Q0 that maps an arrow to its source;

a map t : Q1 → Q0 that maps an arrow to its target.

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Quivers

Deﬁnition A quiver is a directed graph, where loops and multiple edges between two vertices are allowed.

Sira Gratz Quiver representations Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Quivers

Deﬁnition A quiver is a directed graph, where loops and multiple edges between two vertices are allowed. More precisely, a quiver consists of the following data:

a set of vertices Q0;

a set of arrows Q1;

a map s : Q1 → Q0 that maps an arrow to its source;

a map t : Q1 → Q0 that maps an arrow to its target.

Sira Gratz Quiver representations We have

Q0 = {1, 2, 3, 4}; Q1 = {α, β, γ, δ, ε, ζ} s(α) = 1, t(α) = 2; s(ζ) = t(ζ) = 1, etc.

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Example

ε δ γ α ζ 1 2 4

We have

Q0 = {1, 2, 3, 4}; Q1 = {α, β, γ, δ, ε, ζ} s(α) = 1, t(α) = 2; s(ζ) = t(ζ) = 1, etc.

Sira Gratz Quiver representations Throughout, we will assume that both Q0 and Q1 are ﬁnite.

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Quivers

Deﬁnition A quiver is a directed graph. More precisely, a quiver consists of the following data:

a set of vertices Q0;

a set of arrows Q1;

a map s : Q1 → Q0 that maps an arrow to its source;

a map t : Q1 → Q0 that maps an arrow to its target.

Deﬁnition A quiver is a directed graph. More precisely, a quiver consists of the following data:

a set of vertices Q0;

a set of arrows Q1;

a map s : Q1 → Q0 that maps an arrow to its source;

a map t : Q1 → Q0 that maps an arrow to its target.

Throughout, we will assume that both Q0 and Q1 are ﬁnite.

Sira Gratz Quiver representations Deﬁnition

Let Q be a quiver. A representation (Vi , Mα)i∈Q0,α∈Q1 of Q is a collection of vector spaces Vi of vector spaces over K, indexed by Q0, along with a collection Mα of linear maps, indexed by Q1, such that for all α ∈ Q1 we have

Mα : Vs(α) → Vt(α).

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Quiver representations

Throughout we work over an algebraically closed ﬁeld K.

Throughout we work over an algebraically closed ﬁeld K. Deﬁnition

Mα : Vs(α) → Vt(α).

Throughout we work over an algebraically closed field K. Definition Let Q be a quiver. A (finite dimensional) representation

(Vi , Mα)i∈Q0,α∈Q1 of Q is a collection of (ﬁnite dimensional) vector spaces Vi of vector spaces over K, indexed by Q0, along with a collection Mα of linear maps, indexed by Q1, such that for all α ∈ Q1 we have Mα : Vs(α) → Vt(α).

1 1 3 5 1 0 1 1 2 100 0 0 C C C 2 1

Sira Gratz Quiver representations Answer What do you mean by “diﬀerent”?

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Finding representations

Question Can we ﬁnd all diﬀerent representations of a given quiver?

Answer What do you mean by “diﬀerent”?

Deﬁnition

Let V = (Vi , Mα)i∈Q0,α∈Q1 and W = (Wi , Nα)i∈Q0,α∈Q1 be representations of a quiver Q.A morphism of quiver representations ϕ: V → W is a collection of linear maps

ϕ = (ϕi : Vi → Wi )i∈Q0 , such that for each α ∈ Q1 the diagram

Mα Vs(α) / Vt(α)

ϕs(α) ϕt(α)

Nα Ws(α) / Wt(α)

commutes.

Sira Gratz Quiver representations 1 0 3 1 1 2 0 0 C C C 2 5 1

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Identity morphism

1 0 3 1 1 2 0 0 C C C 2 5 1

Sira Gratz Quiver representations −1 0 2 0 5 C −2 1 1 1 1 4 2 2 3 C

−2 1 −1 0 2 5 1 4 −2 1 = = 1 1 0 5 −1 5 2 3 1 1

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Example

−1 0 2 0 5 C

Sira Gratz Quiver representations −1 0 2 0 5 C −2 1 1 1 1 4 2 2 3 C

−2 1 −1 0 2 5 1 4 −2 1 = = 1 1 0 5 −1 5 2 3 1 1

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Example

−1 0 2 0 5 C −2 1 1 1 1 4 2 2 3 C

−2 1 −1 0 2 5 1 4 −2 1 = = 1 1 0 5 −1 5 2 3 1 1

Sira Gratz Quiver representations 1 2

2 C C 2 1 3 2 1 3 1 1

2 C C 4 1

1 1 2 1 = 4 = 1 1 2 3 2 1 2 1 = 7 = 4 1 3 3

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Example

1 2

2 C C 2 3

Sira Gratz Quiver representations 1 2

2 C C 2 1 3 2 1 3 1 1

2 C C 4 1

1 1 2 1 = 4 = 1 1 2 3 2 1 2 1 = 7 = 4 1 3 3

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Example

1 2

2 C C 2 1 3 2 1 3 1 1

2 C C 4 1

1 2

2 C C 2 1 3 2 1 3 1 1

2 C C 1 1 2 1 = 4 = 1 1 4 1 2 3 2 1 2 1 = 7 = 4 1 3 3

Sira Gratz Quiver representations Objects: ﬁnite dimensional representations of Q over K Maps: morphisms of quiver representations

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? repKQ

We obtain the category repKQ of ﬁnite dimensional quiver representations of Q over K:

Sira Gratz Quiver representations Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? repKQ

We obtain the category repKQ of ﬁnite dimensional quiver representations of Q over K: Objects: ﬁnite dimensional representations of Q over K Maps: morphisms of quiver representations

Deﬁnition

Let V and W be quiver representations in repK(Q). A morphism ϕ: V → W of quiver representations is an isomorphism of quiver representations if there exists a morphism of quiver representations ϕ−1 : W → V such that

−1 −1 ϕ ◦ ϕ = idV ; ϕ ◦ ϕ = idW .

Sira Gratz Quiver representations −1 0 2 0 5 C

−2 1−1 −1 1 = 1 1 1 3 1 2 1 4 2 2 3 C

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Example

−1 0 2 0 5 C −2 1 1 1 1 4 2 2 3 C

−1 0 −1 0 2 2 0 5 C 0 5 C

−2 1 −2 1−1 −1 1 = 1 1 1 1 1 3 1 2 1 4 1 4 2 2 2 3 C 2 3 C

Deﬁnition

A quiver representation V in repK(Q) is called indecomposable if ∼ V = V1 ⊕ V2

implies V1 = 0 or V2 = 0.

Sira Gratz Quiver representations • •

• •

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Acyclic quivers

Deﬁnition A quiver is called acyclic, if it does not have any oriented cycles.

• •

Deﬁnition A quiver is called acyclic, if it does not have any oriented cycles.

• •

Sira Gratz Quiver representations Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers?

Theorem

Let Q be a acyclic quiver. The category repK(Q) is Krull-Schmidt, that is, we can write every representation of Q as a sum of indecomposable representations in a unique way (up to isomorphism and permutation of summands).

Sira Gratz Quiver representations 1 0 1 1 1 0 = 1 2 −1 2 0 1 2 −1 1 0 2 2 0 = 2 −1 4 0 2 C C 2 0

Why quivers? Deﬁnitions and Examples Quiver representations A category of quiver representations Gabriel’s theorem Indecomposables But really, why quivers? Example

1 2

2 C C 2 4

1 2

2 C C 2 1 0 1 1 1 0 = 1 4 2 −1 2 0 1 2 −1 1 0 2 2 0 = 2 −1 4 0 2 C C 2 0

1 2 3 0

2 ∼ C C = C C ⊕ 0 C 2 6 0 1 1 1 4 3 2 −1 0

2 C C 6 0

1 2 1 0

2 ∼ C C = C C ⊕ 0 C 2 2 0 1 1 1 4 3 2 −1 0

2 C C 6 0

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Rephrasing our question

Question Given a quiver Q, can we describe all indecomposable quiver representations of Q up to isomorphism?

Answer Sometimes, and it depends what you mean by “describe”.

Sira Gratz Quiver representations From now on, we assume all quivers to be acyclic and connected.

Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? The simplest case

Deﬁnition A quiver Q is of ﬁnite representation type if, up to isomorphism,

there are only ﬁnitely many indecomposable objects in repK(Q).

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? The simplest case

Deﬁnition A quiver Q is of ﬁnite representation type if, up to isomorphism,

there are only ﬁnitely many indecomposable objects in repK(Q). From now on, we assume all quivers to be acyclic and connected.

Sira Gratz Quiver representations If Q is an orientation of the ADE diagram ∆, then the number of isomorphism classes of non-trivial indecomposable objects in

repK(Q) is equal to the number of positive roots in the root system of ∆.

Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Gabriel’s theorem

Gabriel’s Theorem An (acyclic, connected) quiver is of ﬁnite representation type if and only if it is an orientation of an ADE diagram.

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Gabriel’s theorem

Gabriel’s Theorem An (acyclic, connected) quiver is of ﬁnite representation type if and only if it is an orientation of an ADE diagram. If Q is an orientation of the ADE diagram ∆, then the number of isomorphism classes of non-trivial indecomposable objects in

repK(Q) is equal to the number of positive roots in the root system of ∆.

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Yesterday’s numerology

Type # roots # positive roots # simple roots 2 n2+n An n + n 2 n Dn 2n(n − 1) n(n − 1) n E6 72 36 6 E7 126 63 7 E8 240 120 8

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Today’s numerology

Type # roots # indecomposable reps # simple reps 2 n2+n An n + n 2 n Dn 2n(n − 1) n(n − 1) n E6 72 36 6 E7 126 63 7 E8 240 120 8

Sira Gratz Quiver representations The number of positive roots is 12, so up to isomorphism we have

12 indecomposable representations in repK(Q).

Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Example

Consider the following orientation of D4: •

• •

•

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Example

Consider the following orientation of D4: •

• •

•

The number of positive roots is 12, so up to isomorphism we have

12 indecomposable representations in repK(Q).

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers?

Indecomposable representations of D4

0 0 K 0

0 K K 0 0 0 0 0

0 0 0 K

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers?

Indecomposable representations of D4

0 K 0

K K 0 K 0 K

0 0 K

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers?

Indecomposable representations of D4

K K 0

0 K K K K K

K 0 K

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers?

Indecomposable representations of D4

K K 1 1 1 0 2 K K K K 0 1

K K

Sira Gratz Quiver representations We expect inﬁnitely many isomorphism classes of indecomposable representations.

Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Non-example

The 2-Kronecker quiver

• •

is a connected acyclic quiver which is not an orientation of an ADE diagram.

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Non-example

The 2-Kronecker quiver

• •

is a connected acyclic quiver which is not an orientation of an ADE diagram. We expect inﬁnitely many isomorphism classes of indecomposable representations.

Sira Gratz Quiver representations 1 0 1n n  0 1 0 0 0  n n n K K n+1 K  ..  K  0 0 1 . 0   . .  . . . . .  .. ..  ......  0 1n λ 1 0    .  . . .   0 λ 1 .. 0   ...... 0 1      1n . . . . .  . . . .  ......  ...... 0 0   . . .  ......  λ 1  n n n n+1 . . . .  K K K K ...... λ 1n 0 for all λ ∈ K 1n

Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Representations of the 2-Kronecker quiver

For each n ∈ Z>0 we get the following pairwise non-isomorphic indecomposable representations:

Sira Gratz Quiver representations 1n 0  0 1 0 0 0  n n+1 K  ..  K  0 0 1 . 0  . . . . .   ......  0 1n   . . .   ...... 0 1    1n ...... 0 0

n n n n+1 K K K K

1n 0 1n

Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Representations of the 2-Kronecker quiver

For each n ∈ Z>0 we get the following pairwise non-isomorphic indecomposable representations:

n n K K  . .  λ 1 .. .. 0  .   ..   0 λ 1 0  . . . . .   ......    . . .   ...... λ 1  . . . .  ...... λ

for all λ ∈ K

Sira Gratz Quiver representations 1n 0

n+1 n K K 0 1n

1n 0

n n+1 K K 0 1n

Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Representations of the 2-Kronecker quiver

For each n ∈ Z>0 we get the following pairwise non-isomorphic indecomposable representations:

1n  0 1 0 0 0  n n K K  ..   0 0 1 . 0   . .  . . . . .  .. ..  ......  λ 1 0    .  . . .   0 λ 1 .. 0   ...... 0 1      . . . . .  . . . .  ......  ...... 0   ......   . . . λ 1  n n . . . .  K K ...... λ 1n for all λ ∈ K

Sira Gratz Quiver representations 1n 0

n n+1 K K 0 1n

Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Representations of the 2-Kronecker quiver

For each n ∈ Z>0 we get the following pairwise non-isomorphic indecomposable representations: 1 0 1n n  0 1 0 0 0  n n n K K n+1 K  ..  K  0 0 1 . 0   . .  . . . . .  .. ..  ......  0 1n λ 1 0    .  . . .   0 λ 1 .. 0   ...... 0 1      . . . . .  . . . .  ......  ...... 0   ......   . . . λ 1  n n . . . .  K K ...... λ 1n for all λ ∈ K

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Representations of the 2-Kronecker quiver

For each n ∈ Z>0 we get the following pairwise non-isomorphic indecomposable representations: 1 0 1n n  0 1 0 0 0  n n n K K n+1 K  ..  K  0 0 1 . 0   . .  . . . . .  .. ..  ......  0 1n λ 1 0    .  . . .   0 λ 1 .. 0   ...... 0 1      1n . . . . .  . . . .  ......  ...... 0 0   . . .  ......  λ 1  n n n n+1 . . . .  K K K K ...... λ 1n 0 for all λ ∈ K 1n

Sira Gratz Quiver representations Why quivers? Finite representation type Quiver representations Statement Gabriel’s theorem Examples But really, why quivers? Wild quivers

There are quivers, where we cannot even “describe” all the isomorphism classes of indecomposables, so-called wild quivers. For example, the 3-Kronecker quiver

• •

is wild.

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers?

Deﬁnition

Let Q be a quiver, and let i, j ∈ Q0.A path p from i to j of length l ∈ Z>0 is a sequence

p = (i | α1, α2, . . . , αl | j) such that

s(α1) = i

s(αk ) = t(αk−1)

t(αl ) = j.

For each i ∈ Q0 we deﬁne the lazy path at i to be a path (i || i) of length l = 0.

Sira Gratz Quiver representations The paths in Q are:

e0, e1, e2, e3, α, β, γ, αβ, αγ.

Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? Example

Consider the following orientation of D4:

β 2 α 1 0 γ

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? Example

Consider the following orientation of D4:

β 2 α 1 0 γ

The paths in Q are:

e0, e1, e2, e3, α, β, γ, αβ, αγ.

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? Path algebra

Deﬁnition Let Q be a quiver. The path algebra KQ is the K-algebra with basis given by the paths in Q, and with multiplication given by concatenation of paths.

Sira Gratz Quiver representations An element of CQ is a C-linear combination of the paths in Q, for example 2α + β ∈ C; 3e2 + γ ∈ C. We have

(2α + β)(3e2 + γ) = 6 αe2 +2αγ + 3βe2 + βγ = 2αγ + 3β. |{z} |{z} =0 =0

Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? Example

Consider the following orientation Q of D4:

β 2 α 1 0 γ

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers? Example

Consider the following orientation Q of D4:

β 2 α 1 0 γ

An element of CQ is a C-linear combination of the paths in Q, for example 2α + β ∈ C; 3e2 + γ ∈ C. We have

(2α + β)(3e2 + γ) = 6 αe2 +2αγ + 3βe2 + βγ = 2αγ + 3β. |{z} |{z} =0 =0

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers?

For a K-algebra A we denote by modA the category of ﬁnite dimensional A-modules. Theorem Let Q be a acyclic quiver. We have an equivalence of categories ∼ modKQ = repK(Q).

Sira Gratz Quiver representations Why quivers? Quiver representations Gabriel’s theorem But really, why quivers?

Theorem Let A be a ﬁnite dimensional K-algebra. Then it is Morita equivalent to a quotient KQ/I of the path algebra of a quiver Q by an admissible ideal I , i.e. we have an equivalence of categories ∼ modA = modKQ/I .

Sira Gratz Quiver representations