Quiver representations and ADE - University of Leedspmtdgh/lms2019/gratz/Quiver reps.pdf · 2019. 7. 24. · A quiver is a directed graph, where loops and multiple edges between two

Why quivers?Quiver representations

Gabriel’s theoremBut really, why quivers?

Quiver representations and ADE

Sira Gratz

Sira Gratz Quiver representations



Quiver

Definition

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

“Definition”

A representation of a quiver associates to every vertex a vectorspace, and to every arrow a compatible linear map.




Quiver

Definition


“Definition”

A representation of a quiver associates to every vertex a vectorspace, and to every arrow a compatible linear map.




A question from linear algebra

Question:

When is a square matrix diagonalisable?

Let A be a square matrix. When does there exist an invertiblematrix S and a diagonal matrix D such that

SAS−1 = D?




A question from linear algebra

Question:

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

SAS−1 = D?




The same question, in a diagram

SAS−1 = D ⇔ SA = DS

KnA

KnD

S

KnA

KnD

S






KnA

KnD

S

KnA

KnD

S






KnA

KnD

S

KnA

KnD

S





KnA

KnD

S

study representations ofthe loop:

•




Another question from linear algebra

Question:

Let (A,B) and (A′,B ′) be pairs of matrices, all of the samedimension. When do there exist invertible matrices S ,T such that

SAT−1 = A′; SBT−1 = B ′?





Km Kn

A

B

Km Kn

A′

B ′

Km Kn

A

B

Km Kn

A′

B ′

S T

Km Kn

A

B

Km Kn

A′

B ′

S T

Km Kn

A

B

Km Kn

A′

B ′

S T

Km Kn

A

B

Km Kn

A′

B ′

S T





Km Kn

A

B

Km Kn

A′

B ′

S T

Km Kn

A

B

Km Kn

A′

B ′

S T

Km Kn

A

B

Km Kn

A′

B ′

S T

Km Kn

A

B

Km Kn

A′

B ′

S T





Km Kn

A

B

Km Kn

A

B

S T

study representations ofthe 2-Kronecker quiver:

• •




Definitions and ExamplesA category of quiver representationsIndecomposables

Quivers

Definition


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.





Quivers

Definition

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


a set of arrows Q1;







Example

1 2

3

4

α

β

γδ

ε

ζ

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.





Example

1 2

3

4

α

β

γδ

ε

ζ

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.





Quivers

Definition

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


a set of arrows Q1;



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





Quivers

Definition

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


a set of arrows Q1;



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





Quiver representations

Throughout we work over an algebraically closed field K.

Definition

Let Q be a quiver. A representation (Vi ,Mα)i∈Q0,α∈Q1 of Q is acollection of vector spaces Vi of vector spaces over K, indexed byQ0, along with a collection Mα of linear maps, indexed by Q1, suchthat for all α ∈ Q1 we have

Mα : Vs(α) → Vt(α).







Definition

Let Q be a quiver. A representation (Vi ,Mα)i∈Q0,α∈Q1 of Q is acollection of vector spaces Vi of vector spaces over K, indexed byQ0, along with a collection Mα of linear maps, indexed by Q1, suchthat for all α ∈ Q1 we have








Definition

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






Example

C2 C

C

C100

[1 0

]

[21

]

[5][

3][

1 1]

[1 10 0

]





Finding representations

Question

Can we find all different representations of a given quiver?

Answer

What do you mean by “different”?





Finding representations

Question

Can we find all different representations of a given quiver?

Answer

What do you mean by “different”?





Morphisms

Definition

Let V = (Vi ,Mα)i∈Q0,α∈Q1 and W = (Wi ,Nα)i∈Q0,α∈Q1 berepresentations of a quiver Q. A morphism of quiverrepresentations ϕ : V → W is a collection of linear mapsϕ = (ϕi : Vi →Wi )i∈Q0 , such that for each α ∈ Q1 the diagram

Vs(α)

ϕs(α)

��

Mα // Vt(α)

ϕt(α)

��Ws(α)

Nα //Wt(α)

commutes.





Identity morphism

C2 C C

[1 0

]

[21

] [5]

[3][

1 10 0

]

C2 C C

[1 0

]

[21

] [5]

[3][

1 10 0

]





Identity morphism

C2 C C

[1 0

]

[21

] [5]

[3][

1 10 0

]

C2 C C

[1 0

]

[21

] [5]

[3][

1 10 0

]





Example

C2

[−1 00 5

]

C2

[−1 00 5

]

C2

[1 42 3

][−2 11 1

]

[−2 11 1

] [−1 00 5

]=

[2 5−1 5

]=

[1 42 3

] [−2 11 1

]





Example

C2

[−1 00 5

]

C2

[1 42 3

][−2 11 1

]

C2

[−1 00 5

]

C2

[1 42 3

][−2 11 1

]

[−2 11 1

] [−1 00 5

]=

[2 5−1 5

]=

[1 42 3

] [−2 11 1

]





Example

C2

[−1 00 5

]

C2

[1 42 3

][−2 11 1

]

[−2 11 1

] [−1 00 5

]=

[2 5−1 5

]=

[1 42 3

] [−2 11 1

]Sira Gratz Quiver representations




Example

C C2

[12

]

[23

]

C C2

[12

]

[23

]

C2 C

[1 1

][4 1

]

[13

] [2 1

]

[2 1

] [12

]= 4 =

[1 1

] [13

][2 1

] [23

]= 7 =

[4 1

] [13

]





Example

C C2

[12

]

[23

]

C2 C

[1 1

][4 1

]

[13

] [2 1

]

C C2

[12

]

[23

]

C2 C

[1 1

][4 1

]

[13

] [2 1

]

[2 1

] [12

]= 4 =

[1 1

] [13

][2 1

] [23

]= 7 =

[4 1

] [13

]





Example

C C2

[12

]

[23

]

C2 C

[1 1

][4 1

]

[13

] [2 1

]

[2 1

] [12

]= 4 =

[1 1

] [13

][2 1

] [23

]= 7 =

[4 1

] [13

]





repKQ

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

Objects: finite dimensional representations of Q over KMaps: morphisms of quiver representations





repKQ

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

Objects: finite dimensional representations of Q over KMaps: morphisms of quiver representations





Isomorphisms

Definition

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

ϕ−1 ◦ ϕ = idV ; ϕ ◦ ϕ−1 = idW .





Example

C2

[−1 00 5

]

C2

[1 42 3

][−2 11 1

]

C2

[−1 00 5

]

C2

[1 42 3

][−2 11 1

]−1

= 13

[−1 11 2

]





Example

C2

[−1 00 5

]

C2

[1 42 3

][−2 11 1

] C2

[−1 00 5

]

C2

[1 42 3

][−2 11 1

]−1

= 13

[−1 11 2

]





Indecomposable quiver representations

Definition

A quiver representation V in repK(Q) is called indecomposable if

V ∼= V1 ⊕ V2

implies V1 = 0 or V2 = 0.





Acyclic quivers

Definition

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

•

• •

•





Acyclic quivers

Definition


•

• •

•





Acyclic quivers

Definition


•

• •

•





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 ofindecomposable representations in a unique way (up toisomorphism and permutation of summands).





Example

C C2

[12

]

[24

]

C C2

[10

]

[20

]

[1] [

1 02 −1

] [1 02 −1

] [12

]=

[10

][

1 02 −1

] [24

]=

[20

]





Example

C C2

[12

]

[24

]

C C2

[10

]

[20

]

[1] [

1 02 −1

] [1 02 −1

] [12

]=

[10

][

1 02 −1

] [24

]=

[20

]





Example

C C2

[12

]

[24

] ∼= C C

[3]

[6] ⊕ 0 C

[0]

[0]

C C2

[30

]

[60

]

[1] [

1 12 −1

]





Example

C C2

[12

]

[24

] ∼= C C

[1]

[2] ⊕ 0 C

[0]

[0]

C C2

[30

]

[60

]

[1] [

1 12 −1

]




Finite representation typeStatementExamples

Rephrasing our question

Question

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

Answer

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





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.





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.





Gabriel’s theorem

Gabriel’s Theorem

An (acyclic, connected) quiver is of finite representation type ifand only if it is an orientation of an ADE diagram.

If Q is an orientation of the ADE diagram ∆, then the number ofisomorphism classes of non-trivial indecomposable objects inrepK(Q) is equal to the number of positive roots in the rootsystem of ∆.





Gabriel’s theorem

Gabriel’s Theorem

An (acyclic, connected) quiver is of finite representation type ifand only if it is an orientation of an ADE diagram.If Q is an orientation of the ADE diagram ∆, then the number ofisomorphism classes of non-trivial indecomposable objects inrepK(Q) is equal to the number of positive roots in the rootsystem of ∆.





Yesterday’s numerology

Type # roots # positive roots # simple roots

An n2 + n n2+n2 n

Dn 2n(n − 1) n(n − 1) n

E6 72 36 6

E7 126 63 7

E8 240 120 8





Today’s numerology

Type # roots # indecomposable reps # simple reps

An n2 + n n2+n2 n

Dn 2n(n − 1) n(n − 1) n

E6 72 36 6

E7 126 63 7

E8 240 120 8





Example

Consider the following orientation of D4:

• •

•

•

The number of positive roots is 12, so up to isomorphism we have12 indecomposable representations in repK(Q).





Example


• •

•

•

The number of positive roots is 12, so up to isomorphism we have12 indecomposable representations in repK(Q).





Indecomposable representations of D4

0 K

0

0

K 0

0

0

0 0

K

0

0 0

0

K






K K

0

0

0 K

K

0

0 K

0

K






0 K

K

K

K K

K

0

K K

0

K






K K

K

K

K K2

K

K

[11

] [1 0

][0 1

]





Non-example

The 2-Kronecker quiver

• •

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

We expect infinitely many isomorphism classes of indecomposablerepresentations.





Non-example

The 2-Kronecker quiver

• •

is a connected acyclic quiver which is not an orientation of anADE diagram.We expect infinitely many isomorphism classes of indecomposablerepresentations.





Representations of the 2-Kronecker quiver

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

Kn Kn

1n

λ 1. . .

. . . 0

0 λ 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . λ 1

. . .. . .

. . .. . . λ

for all λ ∈ K

Kn Kn

0 1 0 0 0

0 0 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . 0 1

. . .. . .

. . .. . . 0

1n

Kn+1 Kn

[1n 0

]

[0 1n

]

Kn Kn+1

[1n0

]

[01n

]







Kn Kn

1n

λ 1. . .

. . . 0

0 λ 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . λ 1

. . .. . .

. . .. . . λ

for all λ ∈ K

Kn Kn

0 1 0 0 0

0 0 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . 0 1

. . .. . .

. . .. . . 0

1n

Kn+1 Kn

[1n 0

]

[0 1n

]

Kn Kn+1

[1n0

]

[01n

]







Kn Kn

1n

λ 1. . .

. . . 0

0 λ 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . λ 1

. . .. . .

. . .. . . λ

for all λ ∈ K

Kn Kn

0 1 0 0 0

0 0 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . 0 1

. . .. . .

. . .. . . 0

1n

Kn+1 Kn

[1n 0

]

[0 1n

]

Kn Kn+1

[1n0

]

[01n

]







Kn Kn

1n

λ 1. . .

. . . 0

0 λ 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . λ 1

. . .. . .

. . .. . . λ

for all λ ∈ K

Kn Kn

0 1 0 0 0

0 0 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . 0 1

. . .. . .

. . .. . . 0

1n

Kn+1 Kn

[1n 0

]

[0 1n

]

Kn Kn+1

[1n0

]

[01n

]







Kn Kn

1n

λ 1. . .

. . . 0

0 λ 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . λ 1

. . .. . .

. . .. . . λ

for all λ ∈ K

Kn Kn

0 1 0 0 0

0 0 1. . . 0

. . .. . .

. . .. . .

. . .. . .

. . .. . . 0 1

. . .. . .

. . .. . . 0

1n

Kn+1 Kn

[1n 0

]

[0 1n

]

Kn Kn+1

[1n0

]

[01n

]





Wild quivers

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

• •

is wild.




Definition

Let Q be a quiver, and let i , j ∈ Q0. A path p from i to j of lengthl ∈ 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) oflength l = 0.




Example


1 0

2

3

α

β

γ

The paths in Q are:

e0, e1, e2, e3, α, β, γ, αβ, αγ.




Example


1 0

2

3

α

β

γ

The paths in Q are:

e0, e1, e2, e3, α, β, γ, αβ, αγ.




Path algebra

Definition

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




Example

Consider the following orientation Q of D4:

1 0

2

3

α

β

γ

An element of CQ is a C-linear combination of the paths in Q, forexample

2α + β ∈ C; 3e2 + γ ∈ C.

We have

(2α + β)(3e2 + γ) = 6 αe2︸︷︷︸=0

+2αγ + 3βe2 + βγ︸︷︷︸=0

= 2αγ + 3β.




Example

Consider the following orientation Q of D4:

1 0

2

3

α

β

γ

An element of CQ is a C-linear combination of the paths in Q, forexample

2α + β ∈ C; 3e2 + γ ∈ C.

We have

(2α + β)(3e2 + γ) = 6 αe2︸︷︷︸=0

+2αγ + 3βe2 + βγ︸︷︷︸=0

= 2αγ + 3β.




For a K-algebra A we denote by modA the category of finitedimensional A-modules.

Theorem

Let Q be a acyclic quiver. We have an equivalence of categories

modKQ ∼= repK(Q).




Theorem

Let A be a finite dimensional K-algebra. Then it is Moritaequivalent to a quotient KQ/I of the path algebra of a quiver Qby an admissible ideal I , i.e. we have an equivalence of categories

modA ∼= modKQ/I .


Documents

Quiver representations and ADE - University of Leedspmtdgh/lms2019/gratz/Quiver reps.pdf · 2019. 7. 24. · A quiver is a directed graph, where loops and multiple edges between two