2016--04-07-NCUR-JON (1)

Introduction Quivers Hall Algebras

Representation Theory of Quivers:Approach from Linear and Abstract

Algebra

Jon Scott and Daniel Lollar

University of Wisconsin-WhitewaterNCUR, April 07, 2016

A1

1•

1• -

2•α

A2

1• --

2•

β1

α1

A(1)1


I. M. Gelfand

“All of mathematics is some kind ofrepresentation theory”


Representation Theory

DefinitionRepresentation is the way to present properties or thestructure of some object onto another object (lineartransformations on vector space).• Representation theory of groups.• Representation theory of Lie algebras.• Representation theory of algebras.

• A: a group, a Lie algebra or an algebra.• A representation V (or module) of A is an algebra (or

group) homomorphism T : A −→ End(V)( or Aut(V)).• Element of A = operator on V.



Research Methods

• Algebra.• Analysis.• Differential geometry.• Algebraic geometry.• Topology.


Real Quivers in Life


Why Quivers?

DefinitionA quiver is just a directed graph, and a representationassociates a vector space to each vertex and a linear map toeach arrow.


Outline

Introduction

Quivers and Representations

Hall Algebras


What is an algebra?

DefinitionAn algebra is a vector space A over a field k equipped with abilinear multiplication satisfying certain compatibility axioms:

A× A −→ A(x, y) −→ x · y

• Associative: (x · y) · z = x · (y · z). Algebra of quivers.• Non–associative: Lie algebras.


Quiver and Lie Theory

Quiver Lie Theory

Dimensional Vector Positive Root

Category Rep(Q) Quantum Group

Triangulated Category Lie Algebra

***** *****


Cartan Matrix

DefinitionA matrix C = (aij)i,j∈I is called a (generalized) Cartan matrix ifit satisfies the following conditions:• aii = 2 for any i ∈ I• aij are non-positive integers for i 6= j• aij = 0 implies aji = 0

C is called symmetric if aij = aji.


Cartan Matrix

ExampleIf

C =

2 −1 0−1 2 −10 −1 2

then the corresponding Dynkin diagram is

A3 •1

•2

•3


Dynkin Diagrams

An q q q q. . . . . .

Dn q q q qq. . . . . .

E6 q q q q qqE7 q q q q q qqE8 q q q q q q qq

Euclidean (Extended Dynkin)Diagrams

An q qq q q q. . . . . .

. . . . . .@@

@@

Dn q q q qqq. . . . . .

E6 q q q q qqq

E7 q q q q q qq qE8 q q q q q q q qq


Arnold’s Ubiquity of A− D− E Classification

#" !

DynkinDiagrams

Semisimple Lie Algebras

and Lie Groups

Kac-Moody

Lie Algebras

Quantum

Groups

HHH

Quiver Representations

Gabriel Theorem

HHHH


of f.d. Algebras

Singularities of

Algebraic Hypersurfaces

Critical Points of

Smooth Functions

3-dimensional

Orthogonal GroupsSimple Groups of Lie Type

Chevalley Groups

HHHH


Kac-Moody Lie Algebra

• C = (aij)i,j∈I: Symmetric Cartan matrix .• g(C): Kac–Moody Lie algebra of type C.

• Root system of g(C):∏= αi|i ∈ I: The set of all simple roots.

Φ = Φ+⋃

Φ−

Φ+ = Φ+real⋃

Φ+img

• Classification of g(C):

Finite type: An, Dn, E6, E7 and E8. Affine type: An, Dn, E6, E7 and E8. Indefinite type: The rest.



• C = (aij)i,j∈I: Symmetric Cartan matrix .• g(C): Kac–Moody Lie algebra of type C.• Root system of g(C):∏

= αi|i ∈ I: The set of all simple roots.Φ = Φ+

⋃Φ−

Φ+ = Φ+real⋃

Φ+img







⋃Φ−

Φ+ = Φ+real⋃

Φ+img


Finite type: An, Dn, E6, E7 and E8.

Affine type: An, Dn, E6, E7 and E8. Indefinite type: The rest.





⋃Φ−

Φ+ = Φ+real⋃

Φ+img


Finite type: An, Dn, E6, E7 and E8. Affine type: An, Dn, E6, E7 and E8.

Indefinite type: The rest.





⋃Φ−

Φ+ = Φ+real⋃

Φ+img




Quantum Group

• g = g(C): Kac–Moody Lie algebra of type C.• Q(v): The field of rational function in the variable v.

Definition (Drinfeld-Jimbo, 1986)Uv(g): The quantized enveloping algebra over Q(v).

• Generators: Ei,Fi,Ki and K−i, i ∈ I.• Defining relations : include quantum Serre relations.


Quiver and Lie Theory

Cartan Matrix C = (aij) - Kac–Moody Lie Algebrag = g(C) = n− + h + n+

(Serre relations )

?Universal Enveloping AlgebraU(g) = U(n−)

⊗U(h)

⊗U(n+)

(Serre relations )?

Quantum GroupU = U−

⊗U0⊗U+

(Quantum Serre relations )?

U+Hall Algebra H(Q)?

Category Rep(Q)?

Quiver Q of C1•→2•→3•

?

Diagram ∆(C) of C1• 2• 3•

?

w


General References

Notices AMS. 52 (2005)

Quiver RepresentationsHarm Derksen and Jerzy Weyman

200 NOTICES OF THE AMS VOLUME 52, NUMBER 2

IntroductionA quiver is just a directed graph.1 Formally, aquiver is a pair Q = (Q0,Q1) where Q0 is a finiteset of vertices and Q1 is a finite set of arrows be-tween them. If a ∈ Q1 is an arrow, then ta and hadenote its tail and its head, respectively.

Let us fix a quiver Q and a base field K. At-tach a finite dimensional vector space to eachvertex of Q and a linear map to each arrow (withthe appropiate domain and codomain). Then thisis called a representation of Q . Formally, a repre-sentation V of Q is a collection

Vx | x ∈ Q0

of finite-dimensional K-vector spaces together witha collection

Va : Vta → Vha | a ∈ Q1

of K-linear maps. If V is a representation of Q , thenits dimension vector dV is the function Q0 → Ndefined by dV (x) = dimK(Vx) for all x ∈ Q0. HereN = 0,1,2, . . . denotes the set of nonnegative integers. The set of all possible dimension vectorsis NQ0. Here are a few typical examples of quiverrepresentations.

Example 1. A representation of the quiver

is a collection of two finite-dimensional vectorspaces V1 , V2 together with a linear mapVa : V1 → V2 .

Example 2. A representation of the loop quiver

is a vector space V1 together with an endomor-phism Va : V1 → V1 .

Example 3. A representation of the star quiver

is a collection of six vector spaces V1, V2, . . . , V6 to-gether with five linear maps Vai : Vi → V6 ,i = 1,2, . . . ,5. If all maps are injective, then we canview such a representation as a subspace configuration.

Harm Derksen is associate professor of mathematics at theUniversity of Michigan. His email address is [email protected].

Jerzy Weyman is professor of mathematics at Northeast-ern University. His email address is [email protected].

The authors are partially supported by NSF Grants DMS0349019 and 0300064, respectively.1The underlying motivations of quiver theory are quite different from those in the traditional graph theory. To emphasize this distinction, it is common in our context touse the word “quivers” instead of “graphs”.

1 2

a

1

25

4 3

6

a1a2

a3a4

a5

1

a

Derksen, Harm; Weyman, Jerzy

Notices AMS. 53 (2006)

a Quantum Group?Shahn Majid

30 NOTICES OF THE AMS VOLUME 53, NUMBER 1

A quantum group is in the first place a remarkably niceobject called a Hopf algebra, the axioms for which areso elegant that they were written down in the 1940swell before truly representative examples emergedfrom physics in the 1980s. So let us start with theseelegant axioms, but with the caveat that it’s the mod-ern examples and their further structure that reallymake the subject what it is. A Hopf algebra H obeysthe following axioms:1. H is a unital algebra (H, ·,1) over a field k .2. H is a counital coalgebra (H,∆, ε) over k . Here the

“coproduct” and “counit” maps ∆ : H → H⊗Hand ε : H → k are required to obey (∆⊗ id)∆ =(id⊗∆)∆ and (ε⊗ id)∆ = (id⊗ ε)∆ =id.

3. ∆, ε are algebra homomorphisms.4. There exists an “antipode” map S : H → H obey-

ing ·(id⊗S)∆ = ·(S⊗ id)∆ = 1ε .There are three points of view leading indepen-

dently to these axioms. Each of them defines what aquantum group is. For lack of space we will focusmainly on the first of these.

The first point of view starts with the observationthat the functions k(G) on a finite group or the coor-dinate algebra k[G] of an algebraic group form Hopfalgebras. For any finite set let k(G) be the pointwisealgebra of functions on G with values in k . We iden-tify k(G)⊗k(G) = k(G×G) , i.e., functions in two vari-ables. Then, when G is actually a group, we define forall a ∈ k(G) ,

(∆a)(x, y) = a(xy), (Sa)(x) = a(x−1), ε(a) = a(e),

where e is the group unit element and x, y ∈ G arearbitrary. We see that the group structure is encodedin the coalgebra ∆, ε and antipode S . Similarly, forevery subset G ⊆ kn described by polynomial equa-tions one has a “coordinate algebra” k[G] defined aspolynomial functions on kn, modulo the ideal of func-tions that vanish on G . When k is algebraically closedwe obtain in this way a precise (functorial) corre-spondence between such polynomial subsets andnilpotent-free commutative algebras with a finite setof generators. This is the basic setting of algebraic

geometry. When the subset G forms a group and thegroup law is polynomial, the product map G×G → Gbecomes under the correspondence an algebra ho-momorphism ∆ going the other way. Likewise for therest of the Hopf algebra structure. Two examples areas follows. The “affine line” is described by the coor-dinate algebra k[x] (polynomials in one variable) withadditive coproduct ∆x = x⊗1 + 1⊗x correspondingto addition in k . The reader can and should fill in the rest of the structure and verify that one has a Hopf algebra in fact for any field k . The “circle” is similarly described by the coordinate algebra k[t, t−1] (polynomials in t, t−1 with the implied rela-tions tt−1 = t−1t = 1 ) and multiplicative coproduct∆t = t ⊗ t corresponding to multiplication in k∗.Again, the reader should fill in and verify the rest ofthe Hopf algebra structure. Most familiar complex Liegroups are likewise defined by polynomial equationsand have corresponding algebras C[G], as well as ver-sions k[G] defined over general fields with the samerelations. Meanwhile, working over C, a “real form”means the additional structure of a compatiblecomplex-linear involution making the coordinate al-gebra into a ∗ -algebra. In this case one can denote theabove two examples as C[R] and C[S1] when takenwith x∗ = x and t∗ = t−1 respectively.

A general Hopf algebra H similarly has the struc-tures ∆, ε, S but we do not assume that the algebraof H is commutative as it is in the above examples.This is the point of view of noncommutative geome-try or “quantisation” in the mathematician’s (but notphysicist’s) sense of a noncommutative deformationof a commutative coordinate or function algebra.Much of group theory and Lie group theory proceedsat this level; for example a translation-invariant inte-gral

∫: H → k (in a certain sense involving ∆), if it

exists, is unique up to scale and does indeed exist innice cases. Likewise the notion of a complex of differential forms (⊕nΩn,d) makes sense over any algebra H. At degree 1 the space Ω1 of 1-forms is required to be an H −H bimodule equipped with anoperation d : H → Ω1 obeying the Leibniz rule

d(ab) = (da)b + a(db), ∀a, b ∈ Hand such that Ω1 = HdH. This is a bit weaker thanin usual differential geometry even when H is

?W H A T I S . . .

Shahn Majid is professor of mathematics at Queen Mary, University of London. His email address [email protected].

Majid, Shahn


Outline

Introduction


Hall Algebras


Quivers

DefinitionA quiver (directed graph)

Q = (I,Q1, s, t : Q1 −→ I)

is given by

• I = Finite set of vertices = 1, 2, . . . , n• Q1 = Finite set of arrows = ρ : s(ρ)

ρ−→ t(ρ)

Example

Q •1

-α β•2

- •3


Rep(Q): Category of Representations of Q

We now fix a base field k.• Object : a representation V of Q is given by a k-vector

space Vi for each i ∈ I and a linear map Vρ : Vs(ρ) → Vt(ρ)for each ρ ∈ Q1

• Morphism: θ : V → V ′ is given by linear maps θi : Vi → V ′ifor each i ∈ I satisfying

V ′s(ρ)- V ′t(ρ)

V ′ρ

Vs(ρ)- Vt(ρ)

Vρ

? ?θs(ρ) θt(ρ)y θs(ρ)V ′ρ = Vρθt(ρ)


Rep(Q): Category of representations of Q• The dimension vector :

dim V = (dim kVi)i∈I = (d1, · · · , dn)

• Simple representation: Si, i ∈ I

Sij =

k when j = i0 otherwise

and Siρ = 0 for all ρ ∈ Q1.

αi = dim Si = (0, 0, . . . ,i1, . . . , 0, 0)

• V is indecomposable if V 6' V ′ ⊕ V ′′, where V ′,V ′′ 6' 0 .

Krull–Schmidt TheoremEvery representation V of Q is isomorphic to a direct sum ofindecomposable representations (unique up to isomorphismand permutation of factors).


Rep(Q) and Problems in Linear Algebra

A whole range of problems of linear algebra can be formulatedin terms of representations of quivers

Equivalence of matricesQ = •1 •2-α

Equivalence class of a linear transformation f : V1 → V2.Classify the m× n matrices up to row and column operations.

(Ir 00 0

)m×n

ε1 : k→ 0

ε2 : 0→ k

ε1,2 : k→ k

r : the rank

The summand ε1,2 occurs r times in its decomposition.


Rep(Q) and Problems in Linear AlgebraSimilarity of matricesQ = 1•? α

Equivalence class of an endomorphism f : V1 → V1.

kn

kn -

-kn

kn? ?B

A

P P A = P−1BP

Similarity of matricesQ = 1•? α

Equivalence class of an endomorphism f : V1 → V1.Classify endomorphisms of a vector space up to conjugation.Jordan classification: dim V1 = n, ∃ a one parameter family ofindecomposable endomorphisms Jn(λ), λ ∈ k

Jn(λ) =

λ 1 0 . . . 0 00 λ 1 . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . λ 10 0 0 . . . 0 λ

n×n



Equivalence of a pair of matricesKronecker Quiver: Q = •1 •2--

βα

Equivalence class of a pair of linear transformationsfα : V1 → V2 and fβ : V1 → V2.Weierstraß 1870, Kronecker 1890

km

km --

--

kn

kn? ?

B2

B1

A1

A2P Q

A2 = Q−1B2P

A1 = Q−1B1P



Equivalence of a pair of matricesKronecker Quiver: Q = •1 •2--

β

αEquivalence class of a pair of linear transformationsfα : V1 → V2 and fβ : V1 → V2.Weierstraß 1870, Kronecker 1890

kn• • kn--

In

Jn(0)

kn• • kn--

In

Jn(λ)

kn+1• • kn--

( 0 In )

( In 0 )kn• • kn+1-

-(0In

)(

In0

)

Classification is solvable.



The five subspace problem

•6

•1

?

α1•5 QQQs

α5 • 2+

α2

•4 7α4

• 3SSSoα3

We cannot find a complete list of all indecomposablerepresentationsClassification is impossible.


Classification of QuiversOnly 3 Types !!!

• Finite type: The number of isomorphism classes ofindecomposable representations is finite.

• Tame type: There are infinitely many indecomposablerepresentations, but they come in nice one–dimensionalfamilies.

• Wild type: Classification is impossible (or “hopeless”).

Theorem (Gaberiel 1972, Dlab-Ringel 1973)Let Q be a quiver without loop and oriented cycle, Γ be theunderlining graph of Q.

• Q is of finite type ⇐⇒ Γ is Dynkin.• Q is of tame type ⇐⇒ Γ is Euclidean.• All other quivers are of wild type.


Auslander-Reiten Theory of Rep(Q)

Example

Q •1

- •2

- •3

0→ 0→ k

@

@@R

@@@R

@@@R

0→ k→ 0 k→ 0→ 0

0→ k→ k k→ k→ 0

k→ k→ k

· · · · · · · ·

· · · · · · · ·· · · · · · · ·


Positive Roots of Lie Algebra of Type A3

ExampleA3 •

1•2

•3


Dimension Vectors and Roots

• dim V = (dim kVi)i∈I =∑

i∈I(dim kVi)αi.Theorem (Gabriel 1972, Dlab-Ringel 1973, Kac 1980)If Q and g(C) share the same diagram, then• Dimension Vectors of Indec .

= Positive Roots• If α ∈ Φ+

real, then ∃ a unique indec of dimension vector α(up to iso).

• If α ∈ Φ+img, then there are (infinitely) many

indecomposables of dimension vector α (up to iso).


Auslander–Reiten Theory of E8


Root System of E8


Outline

Introduction


Hall Algebras


Looking at Woods Rather Than the Trees


Hall Algebras

Canonical BasesLusztig, Kashiwara 1990

Abelian p-GroupsSteinitz 1901, Hall 1959

Cluster AlgebrasFomin-Zelevinsky 2002

Hall AlgebrasRingel 1990

Derived CategoriesKapranov 1998, Xiao-Peng 2000

Homological Mirror SymmetryKontsevich 1994

Triangulated CategoriesToën 2006, Xiao-Xu 2008, Chen-Xu 2010

QuiverRepresentations

CategorificationGeometrization

OO


Kronecker Quiver K : 1 • • 2--

• Regular:kn• • kn--

In

Jn(0)

• Regular:kn• • kn--

In

Jn(λ)

• Preproj: kn• • kn+1--

( 0 In )

( In 0 )

• Preinj: kn+1• • kn--(

0In

)(

In0

)

• α1 = dim S1 and α2 = dim S2.

• δ = α1 + α2.

• Preprojective: nδ + α2.

• Preinjective: (n + 1)δ − α2.

• Regular: nδ.

• No real regular roots !!!

• Regular roots = Imaginaryroots nδ.


Auslander–Reiten Theory of the KroneckerAlgebra

(0, 1)

(1, 2)

(2, 3)@@R@@R

(3, 4)

(4, 5)@@R@@R@@R@@R

(5, 6)

· · · · · ·

(1, 1)6?

(2, 2)6?

(3, 3)········

(n, n)6?

······

(1, 1)6?

(2, 2)6?

(3, 3)········

(n, n)6?

······

(2, 2)6?

(3, 3)········

(n, n)6?

······

· · ·· · ·· · ·· · ·· · ·· · ·· · ·

· · ·· · ·· · ·· · ·· · ·

· · ·· · ·

(n, n)6?

······

(4, 3)

(3, 2)

(2, 1)

(5, 4)

@@R@@R

(1, 0)

@@R

@@R

· · · · · ·

(p, q) := pα1 + qα2


Thank You

2016--04-07-NCUR-JON (1)

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