1. Introduction Quivers Hall Algebras
Representation Theory of Quivers:
Approach from Linear and Abstract
Algebra
Jon Scott and Daniel Lollar
University of Wisconsin-Whitewater
NCUR, April 07, 2016
A1
1
•
1
• E
2
•
α
A2
1
• EE
2
•
β1
α1
A
(1)
1
2. Introduction Quivers Hall Algebras
I. M. Gelfand
“All of mathematics is some kind of
representation theory”
3. Introduction Quivers Hall Algebras
Representation Theory
Definition
Representation is the way to present properties or the
structure of some object onto another object (linear
transformations 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.
4. Introduction Quivers Hall Algebras
Representation Theory
Research Methods
• Algebra.
• Analysis.
• Differential geometry.
• Algebraic geometry.
• Topology.
6. Introduction Quivers Hall Algebras
Why Quivers?
Definition
A quiver is just a directed graph, and a representation
associates a vector space to each vertex and a linear map to
each arrow.
11. Introduction Quivers Hall Algebras
What is an algebra?
Definition
An algebra is a vector space A over a field k equipped with a
bilinear 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.
12. Introduction Quivers Hall Algebras
Quiver and Lie Theory
Quiver Lie Theory
Dimensional Vector Positive Root
Category Rep(Q) Quantum Group
Triangulated Category Lie Algebra
***** *****
13. Introduction Quivers Hall Algebras
Cartan Matrix
Definition
A matrix C = (aij)i,j∈I is called a (generalized) Cartan matrix if
it satisfies the following conditions:
• aii = 2 for any i ∈ I
• aij are non-positive integers for i = j
• aij = 0 implies aji = 0
C is called symmetric if aij = aji.
14. Introduction Quivers Hall Algebras
Cartan Matrix
Example
If
C =
2 −1 0
−1 2 −1
0 −1 2
then the corresponding Dynkin diagram is
A3 •
1
•
2
•
3
16. Introduction Quivers Hall Algebras
Arnold’s Ubiquity of A − D − E Classification
#
" !
Dynkin
Diagrams
Semisimple Lie Algebras
and Lie Groups
Kac-Moody
Lie Algebras
Quantum
Groups
rrr
¨¨
¨¨
¨¨¨¨
Quiver Representations
Gabriel Theorem
rrrr
Representation Theory
of f.d. Algebras
Singularities of
Algebraic Hypersurfaces
Critical Points of
Smooth Functions
3-dimensional
Orthogonal Groups
Simple Groups of Lie Type
Chevalley Groups
¨
¨¨¨
r
rrr
17. Introduction Quivers Hall Algebras
Kac-Moody Lie Algebra
• C = (aij)i,j∈I: Symmetric Cartan matrix.
• g(C): Kac–Moody Lie algebra of type C.
18. Introduction Quivers Hall Algebras
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
19. Introduction Quivers Hall Algebras
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.
20. Introduction Quivers Hall Algebras
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.
21. Introduction Quivers Hall Algebras
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.
22. Introduction Quivers Hall Algebras
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.
23. Introduction Quivers Hall Algebras
Quiver and Lie Theory
Cartan Matrix C = (aij) E Kac–Moody Lie Algebra
g = g(C) = n− + h + n+
(Serre relations )
c
Universal Enveloping Algebra
U(g) = U(n−) U(h) U(n+)
(Serre relations )
c
Quantum Group
U = U− U0 U+
(Quantum Serre relations )
c
U+Hall Algebra H(Q)
c
Category Rep(Q)
c
Quiver Q of C
1
•→
2
•→
3
•
c
Diagram ∆(C) of C
1
•
2
•
3
•
c
24. Introduction Quivers Hall Algebras
General References
Notices AMS. 52 (2005)
QuiverRepresentations
Harm Derksen and Jerzy Weyman
200 NOTICES OF THE AMS VOLUME 52, NUMBER 2
Introduction
A quiver is just a directed graph.1
Formally, a
quiver is a pair Q = (Q0, Q1) where Q0 is a finite
set of vertices and Q1 is a finite set of arrows be-
tween them. If a ∈ Q1 is an arrow, then ta and ha
denote 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 each
vertex of Q and a linear map to each arrow (with
the appropiate domain and codomain). Then this
is 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 with
a collection
{Va : Vta → Vha | a ∈ Q1}
of K-linear maps. If V is a representation of Q, then
its dimension vector dV is the function Q0 → N
defined by dV (x) = dimK(Vx) for all x ∈ Q0. Here
N = {0, 1, 2, . . . } denotes the set of nonnegative
integers. The set of all possible dimension vectors
is NQ0. Here are a few typical examples of quiver
representations.
Example 1. A representation of the quiver
is a collection of two finite-dimensional vector
spaces V1 , V2 together with a linear map
Va : 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 can
view such a representation as a subspace
configuration.
Harm Derksen is associate professor of mathematics at the
University of Michigan. His email address is
hderksen@umich.edu.
Jerzy Weyman is professor of mathematics at Northeast-
ern University. His email address is j.weyman@neu.edu.
The authors are partially supported by NSF Grants DMS
0349019 and 0300064, respectively.
1
The 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 to
use the word “quivers” instead of “graphs”.
1 2
a
1
25
4 3
6
a1
a2
a3a4
a5
1
a
Derksen, Harm; Weyman, Jerzy
Notices AMS. 53 (2006)
aQuantumGroup?
Shahn Majid
30 NOTICES OF THE AMS VOLUME 53, NUMBER 1
A quantum group is in the first place a remarkably nice
object called a Hopf algebra, the axioms for which are
so elegant that they were written down in the 1940s
well before truly representative examples emerged
from physics in the 1980s. So let us start with these
elegant axioms, but with the caveat that it’s the mod-
ern examples and their further structure that really
make the subject what it is. A Hopf algebra H obeys
the following axioms:
1. H is a unital algebra (H, ·, 1) over a field k.
2. H is a counital coalgebra (H, ∆, ) over k. Here the
“coproduct” and “counit” maps ∆ : H → H ⊗ H
and : H → k are required to obey (∆ ⊗ id)∆ =
(id ⊗ ∆)∆ and ( ⊗ id)∆ = (id ⊗ )∆ =id.
3. ∆, are algebra homomorphisms.
4. There exists an “antipode” map S : H → H obey-
ing ·(id ⊗ S)∆ = ·(S ⊗ id)∆ = 1 .
There are three points of view leading indepen-
dently to these axioms. Each of them defines what a
quantum group is. For lack of space we will focus
mainly on the first of these.
The first point of view starts with the observation
that the functions k(G) on a finite group or the coor-
dinate algebra k[G] of an algebraic group form Hopf
algebras. For any finite set let k(G) be the pointwise
algebra 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 for
all a ∈ k(G),
(∆a)(x, y) = a(xy), (Sa)(x) = a(x−1
), (a) = a(e),
where e is the group unit element and x, y ∈ G are
arbitrary. We see that the group structure is encoded
in the coalgebra ∆, and antipode S. Similarly, for
every subset G ⊆ kn described by polynomial equa-
tions one has a “coordinate algebra” k[G] defined as
polynomial functions on kn, modulo the ideal of func-
tions that vanish on G. When k is algebraically closed
we obtain in this way a precise (functorial) corre-
spondence between such polynomial subsets and
nilpotent-free commutative algebras with a finite set
of generators. This is the basic setting of algebraic
geometry. When the subset G forms a group and the
group law is polynomial, the product map G × G → G
becomes under the correspondence an algebra ho-
momorphism ∆ going the other way. Likewise for the
rest of the Hopf algebra structure. Two examples are
as follows. The “affine line” is described by the coor-
dinate algebra k[x] (polynomials in one variable) with
additive coproduct ∆x = x ⊗ 1 + 1 ⊗ x corresponding
to 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 of
the Hopf algebra structure. Most familiar complex Lie
groups are likewise defined by polynomial equations
and have corresponding algebras C[G], as well as ver-
sions k[G] defined over general fields with the same
relations. Meanwhile, working over C, a “real form”
means the additional structure of a compatible
complex-linear involution making the coordinate al-
gebra into a ∗-algebra. In this case one can denote the
above two examples as C[R] and C[S1
] when taken
with x∗ = x and t∗ = t−1 respectively.
A general Hopf algebra H similarly has the struc-
tures ∆, , S but we do not assume that the algebra
of 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 not
physicist’s) sense of a noncommutative deformation
of a commutative coordinate or function algebra.
Much of group theory and Lie group theory proceeds
at 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 in
nice 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 an
operation d : H → Ω1 obeying the Leibniz rule
d(ab) = (da)b + a(db), ∀a, b ∈ H
and such that Ω1
= HdH. This is a bit weaker than
in 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 is
s.majid@qmul.ac.uk.
Majid, Shahn
26. Introduction Quivers Hall Algebras
Quivers
Definition
A 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
E
α β
•
2
E •
3
27. Introduction Quivers Hall Algebras
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 → Vi
for each i ∈ I satisfying
Vs(ρ)
E Vt(ρ)
Vρ
Vs(ρ)
E Vt(ρ)
Vρ
c c
θs(ρ) θt(ρ) θs(ρ)Vρ = Vρθt(ρ)
28. Introduction Quivers Hall Algebras
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 = i
0 otherwise
and Siρ = 0 for all ρ ∈ Q1.
αi = dim Si = (0, 0, . . . ,
i
1, . . . , 0, 0)
• V is indecomposable if V V ⊕ V , where V , V 0 .
Krull–Schmidt Theorem
Every representation V of Q is isomorphic to a direct sum of
indecomposable representations (unique up to isomorphism
and permutation of factors).
29. Introduction Quivers Hall Algebras
Rep(Q) and Problems in Linear Algebra
A whole range of problems of linear algebra can be formulated
in terms of representations of quivers
Equivalence of matrices
Q = •
1
•
2
Eα
Equivalence class of a linear transformation f : V1 → V2.
Classify the m × n matrices up to row and column operations.
Ir 0
0 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.
30. Introduction Quivers Hall Algebras
Rep(Q) and Problems in Linear Algebra
Similarity of matrices
Q = 1•
c α
Equivalence class of an endomorphism f : V1 → V1.
kn
kn E
Ekn
kn
c c
B
A
P P A = P−1BP
Similarity of matrices
Q = 1•
c 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 of
31. Introduction Quivers Hall Algebras
Rep(Q) and Problems in Linear Algebra
Equivalence of a pair of matrices
Kronecker Quiver: Q = •
1
•
2
EEβ
α
Equivalence class of a pair of linear transformations
fα : V1 → V2 and fβ : V1 → V2.
Weierstraß 1870, Kronecker 1890
km
km E
E
E
E
kn
kn
c c
B2
B1
A1
A2
P Q
A2 = Q−1B2P
A1 = Q−1B1P
32. Introduction Quivers Hall Algebras
Rep(Q) and Problems in Linear Algebra
Equivalence of a pair of matrices
Kronecker Quiver: Q = •
1
•
2
EEβ
αEquivalence class of a pair of linear transformations
fα : V1 → V2 and fβ : V1 → V2.
Weierstraß 1870, Kronecker 1890
kn• • knE
E
In
Jn(0)
kn• • knE
E
In
Jn(λ)
kn+1• • knE
E
( 0 In )
( In 0 )
kn• • kn+1E
E
0
In
In
0
Classification is solvable.
33. Introduction Quivers Hall Algebras
Rep(Q) and Problems in Linear Algebra
The five subspace problem
•6
•
1
c
α1
•5
s
α5 • 2
C
α2
•4
Uα4
•3
ƒ
ƒƒo α3
We cannot find a complete list of all indecomposable
representations
Classification is impossible.
34. Introduction Quivers Hall Algebras
Classification of Quivers
Only 3 Types !!!
• Finite type: The number of isomorphism classes of
indecomposable representations is finite.
• Tame type: There are infinitely many indecomposable
representations, but they come in nice one–dimensional
families.
• 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 the
underlining 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.
35. Introduction Quivers Hall Algebras
Auslander-Reiten Theory of Rep(Q)
Example
Q •
1
E •
2
E •
3
0 → 0 → k
d
d
d‚
d
d
d‚
d
d
d‚
0 → k → 0 k → 0 → 0
0 → k → k k → k → 0
k → k → k
· · · · · · · ·
· · · · · · · ·· · · · · · · ·
37. Introduction Quivers Hall Algebras
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).
