Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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In the standard coordinate basis of Z

, the Euler

form is represented by the matrix E = (a

) where

¼ 

 #f 2Q

j tðÞ¼i; hðÞ¼jg

Here 

is the Kronecker delta symbol. We define

the ‘‘Cartan form’’ of the quiver Q to be the

symmetric bilinear form given by

ð; Þ¼h; iþh; i

Note that the Cartan form is independent of the

orientation of the arrows in Q. In the standard

coordinate basis of Z

, the Cartan form is represented

by the Cartan matrix C = (c

)wherec

= a

þ a

Example 6 For the quiver in Example 1, the Euler

matrix is

E ¼

1 100

0110

0010

0011

and the Cartan matrix is

C ¼

2 100

1210

0 121

0012

The ‘‘Tits form’’ q of a quiver Q is defined by

qðÞ¼h; i¼

ð; Þ

It is known that the number of continuous para-

meters describing representations of dimension  for

 6¼0 is greater than or equal to 1  q ().

Let g be the Kac–Moody algebra associated to

the Cartan matrix of a quiver Q. By forgetting the

orientation of the arrows of Q, we obtain the

underlying (undirected) graph. This is the Dynkin

graph of g . Associated to g is a root system and a set

of simple roots {

ji 2Q

} indexed by the vertices of

the Dynkin graph.

Theorem 1 (Gabriel’s theorem).

(i) A quiver is of finite type if and only if the

underlying graph is a union of Dynkin graphs

of type A, D, or E.

(ii) A quiver is of tame type if and only if the

underlying graph is a union of Dynkin graphs

of type A, D, or E and extended Dynkin graphs

of type

D, or

E (with at least one extended

Dynkin graph).

(iii) The isomorphism classes of indecomposable

representations of a quiver Q of finite type are

in one-to-one correspondence with the positive

roots of the root system associated to the

underlying graph of Q. The correspondence is

given by

V 7!

i 2Q

ðiÞ

The Dynkin graphs of type A, D,andE are as follows.

Here the subscript indicates the number of vertices in the

graph.

The extended Dynkin graphs of type

D, and

are as follows.

Here we have used an open dot to denote the vertex

that was added to the corresponding Dynkin graph

of type A, D,orE.

Theorem 2 (Kac’s theorem). Let Q be an arbitrary

quiver. The dimension vectors of indecomposable

representations of Q correspond to positive roots

316 Finite-Dimensional Algebras and Quivers

of the root system associated to the underlying graph

of Q (an d are thus independent of the orientation of

the arrows of Q). The correspondence is given by

i 2Q

ðiÞ

Note that in Kac’s Theorem, it is not asserted that the

isomorphism classes are in one-to-one correspondence

with the roots as in the finite case considered in

Gabriel’s theorem. It turns out that in the general case,

dimension vectors for which there is exactly one

isomorphism class correspond to real roots while

imaginary roots correspond to dimension vectors for

which there are families of representations.

Example 7 Let Q be the quiver of type A

oriented as follows.

n – 2

n – 1

n n – 2 n – 1123

It is known that the set of positive roots of the

simple Lie algebra of type A

i¼j





1  j  l  n

()

tf0g

The zero root corresponds to the trivial representation.

The root

i = j



for some 1  j  l  n corresponds

to the unique (up to isomorphism) representation V

with

kifj i  l

0 otherwise



and



1 if j  i  l  1

0 otherwise



Example 8 Let Q be the quiver of type

, with all

arrows oriented in the same direction (for instance,

counter-clockwise). The positive root

i = 0



(where {0, 1, 2, ..., n} are the vertices of the quiver)

is imaginary. There is a one-parameter family of

isomorphism classes of indecomposable representa-

tions where the maps assigned to each arrow are

nonzero. The parameter is the composition of the

maps around the loop.

If a quiver Q has no oriented cycles, then the only

simple kQ-modules are the modules S

for i 2Q

where

k if i ¼ j

0ifi 6¼j



and S



= 0 for all  2Q

Ringel–Hall Algebras

Let k be the finite field F

with q elements and let

Q beaquiverwithnoorientedcycles.LetP be

the set of all isomorphism classes of kQ-modules

which are finite as sets (since k is finite dimen-

sional, these are just the quiver representations we

considered above). Let A be a commutative

integral domain containing Z and elements v, v

1

such that v

= q. The Ringel–Hall algebra

H = H

A, v

(kQ)isthefreeA-module with basis

{[V]} indexed by the isomorphism classes of

representations of the quiver Q,withanA-bilinear

multiplication defined by

½V

½V

¼v

hdim V

;dim V

½V

Here hdim V

, dim V

i is the Euler form and g

, V

is the number of submodules W of V such that

V=W ﬃV

and W ﬃV

. H is an associative Z

0

graded algebra, with identity element [0], the

isomorphism class of the trivial representation.

The grading H = 



is given by letting H



the A-span of the set of isomorphism classes [V]

such that dim V = .

Let C = C

A, v

(kQ) be the A-subalgebra of H

generated by the isomorphism classes [S

] of the

simple kQ-modules. C is called the ‘‘composition

algebra.’’ If the underlying graph of Q is of finite

type, then C = H.

Now let K be a set of finite fields k such that the

set {jkjjk 2K} is infinite. Let A be an integral

domain containing Q and, for each k 2K,an

element v

such that v

= jkj. For each k 2K,we

have the corresponding composition algebra C

generated by the elements [

] (here we make the

field k explici t). Now let C be the subring of

k 2K

generated by Q and the elements

t ¼ðt

k 2K

; t

¼ v

1

¼ðt

1

k 2K

; t

1

¼ðv

1

¼ðu

k 2K

; u

¼½

; i 2Q

Now, t lies in the center of C and if p(t) = 0 for some

polynomial p, then p must be the zero polynomial

since the set of v

is infinite. Thus, we may think

of C as the A-algebra generated by the u

, i 2Q

with A = Q[t, t

1

] and t an indeterminate. Let



= Q(t) 

C. We call C



the ‘‘generic composi-

tion algebra.’’

Let g be the Kac–Moody algebra associated to the

Cartan matrix of the quiver Q and let U be the

quantum group associated by Drinfeld and Jimbo to g.

It has a triangular decomposition U = U



 U

Finite-Dimensional Algebras and Quivers 317

Specifically, U

is the Q(t)-algebra with generators

, i 2Q

and relations

1c

p¼0

ð1Þ

1  c



1c

p

; i 6¼j

where c

are the entries of the Cartan matrix and



½m!

½p!½m  p!

½n¼

 t

n

t  t

1

; ½n! ¼½1½2...½n

Theorem 3 There is a Q(t)-algebra isomorphism



sending u

7!E

for all i 2Q

The proof of Theorem 3 is due to Ringel in the

case that the underlying graph of Q is of finite or

affine type. The more general case presented here is

due to Green.

All of the Kac–Moody algebras considered so

far have been simply-laced. That is, their Cartan

matrices are symmetric. There is a way to deal

with non-simply-laced Kac–Moody algebras using

species. We will not treat this subject in this

article.

Quiver Varieties

One can use varieties associated to quivers to yield a

geometric realization of the upper half of the

universal enveloping algebra of a Kac–Moody

algebra g and its irreducible highest-weight

representations.

Lusztig’s Quiver Varieties

We first introduce the quiver varieties, first

defined by L usztig, which yield a geometric

realization of the upper half U

of the u niversal

enveloping algebra of a simply laced Kac–Moody

algebra g.LetQ = (Q

, Q

) be the quiver whose

vertices Q

are the vertices of the Dynkin

diagram of g andwhosesetofarrowsQ

consists

of all the edges of the Dynkin diagram with both

orientations. By definition, U

is the Q-alge bra

defined by generators e

, i 2Q

, subject to the

Serre relations

1c

p¼0

ð1Þ

1  c



1c

p

¼ 0

for all i 6¼j in Q

,wherec

are the entries of the Cartan

matrix associated to Q. For any  =

i 2Q



i, 

2N,

let U



be the subspace of U

spanned by the

monomials e

...e

for various sequences

, i

, ..., i

in which i appears 

times for each

i 2Q

.Thus,U

= 



.LetU

be the subring of

generated by the elements e

=p! for i 2Q

, p 2N.

Then U

= 



Z, 

where U

Z, 

= U



We define the involution : Q

to be the

function which takes  2Q

to the element of Q

consisting of the same edge with opposite orienta-

tion. An orientation of our graph/quiver is a choice

of a subset   Q

such that  [



=Q

and

 \



=;.

Let V be the category of finite-dimensional

-graded vector spaces V = 

i 2Q

over C

with morphisms being linear maps respecting

the grading. Then V 2V shall denote that V is an

object of V. The dimension of V 2V is given by

v = dim V = (dim V

, ...,dimV

Given V 2V,letE

be the space of representa-

tions of Q with underlying vector space V.That

is,

 2Q

HomðV

tðÞ

; V

hðÞ

For any subset Q

of Q

, let E

V, Q

be the subspace

of E

consisting of all vectors x = (x



) such that



= 0 whenev er  62Q

. The algebraic group

Aut(V

) acts on E

and E

V, Q

ðg; xÞ¼ððg

Þ; ðx



ÞÞ7!gx

¼ðx



Þ¼ðg

hðÞ



1

tðÞ

Define the function " : Q

!{1, 1} by "() = 1 for

all  2 and "() = 1 for all  2



. Let h, ibe the

nondegenerate, G

-invariant, symplec tic form on

with values in C defined by

hx; yi¼

 2Q

"ðÞtrðx







Note that E

can be considered as the cotangent

space of E

V, 

under this form.

The moment map associated to the G

-action on

the symplectic vector space E

is the map

: E

!gl

EndV

, the Lie algebra of GL

with i-component

: E

!EndV

given by

ðxÞ¼

 2Q

;hðÞ¼i

"ðÞx







Definition 1 An element x 2E

is said to be

nilpotent if there exists an N  1 such that for any

sequence 

, 

, ..., 

in H satisfying t(

) = h(

t(

) = h(

), ..., t(

N1

) = h(

), the composition



...x



: V

t(

)

h(

)

is zero.

318 Finite-Dimensional Algebras and Quivers

Definition 2 Let 

be the set of all nilpotent

elements x 2E

such that

(x) = 0 for all i 2I.

A subset of an algebraic variety is said to be

‘‘constructible’’ if it is obtained from subvarieties

from a finite number of the usual set-theoretic

operations. A function f : A !Q on an algebraic

variety A is said to be a constructible function if f

1

(a)

is a constructible set for all a 2Q and is empty for all

but finitely many a. Let M(

) denote the Q-vector

space of all constructible functions on 

. Let

M(

)

denote the Q-subspace of M(

) consisting of those

functions that are constant on any G

-orbit in 

Let V, V

, V

2V such that dim V = dim V

dim V

. Now, suppose that S is an I-graded subspace

of V.Forx 2

we say that S is x-stable if x(S)  S.

Let 

V; V

, V

be the variety consisting of all pairs (x, S)

where x 2

and S is an I-graded x-stable subspace of

V such that dim S = dim V

. Now, if we fix some

isomorphisms V=S ﬃV

, S ﬃV

,thenx induces ele-

ments x

2

and x

2

. We then have the maps









V;V

!



where p

(x, S) = (x

, x

), p

(x, S) = x.

For a holomorphic map  between complex

varieties A and B, let 

denote the map between

the spaces of constructible functions on A and B

given by

ð

f ÞðyÞ¼

a 2Q

að

1

ðyÞ\f

1

ðaÞÞ

Let 



be the pullback map from functions on B to

functions on A acting as 



f (y) = f ((y)). We then

define a map

Mð

Þ

Mð

Þ!

Mð

Þ½1

by (f

, f

) 7!f

 f

where

 f

¼ðp



ðf

f

Here f

f

M(



) is defined by

f

)(x

, x

) = f

). The map [1] is bilinear

and defines an associative Q-algebra structure on





M(



) where V



is the object of V defined by



= C



There is a unique algebra homomorphism

 : U

!



M(



) such that (e

) is the function

on the point 

i with value 1. Then  restricts to a

map 



: U



M(



). It can be shown that



=p!) is the function 1 on the point 

for

i 2Q

, p 2Z

0

Let

(

) be the set of all functions in

M(

)that

take on only integer values. One can show that if

(

)andf

(

), then f

 f

(

)

in the setup of [1].Thus



Z, 

) 

(



Let Irr

denote the set of irreducible compo-

nents of 

. The following proposition was con-

jectured by Lusztig and proved by him in the affine

(and finite) case. The general case was proved by

Kashiwara and Saito.

Proposition 2 For any  2(Z

0

)

, we have

dim U



=#Irr



We then have the following important result due

to Lusztig.

Theorem 4 Let  2(Z

0

)

. Then,

(i) For any Z 2Irr



, there exists a unique

2



Z, 

) such that f

is equal to 1 on an

open dense subset of Z and equal to zero on an

open dense subset of Z

2Irr



for all Z

6¼Z.

(ii) {f

jZ 2Irr



} is a Q-basis of 



(iii) 



: U



!



) is an isomorphism.

(iv) Define [Z] 2U



by 



([Z]) = f

. Then B



{[Z] jZ 2Irr



} is a Q-basis of U



(v) 



Z, 

) = 



) \

(



(vi) B



is a Z-basis of U

Z, 

From this theorem, we see that B = t



is a

Q-basis of U

, which is called the ‘‘semicanonical

basis.’’ This basis has many remarkable properties.

One of these properties is as follows. Via the algebra

involution of the entire universal enveloping algebra

U of g given on the Chevalley generators by

7!f

, f

7!e

and h 7!h for h in the Cartan

subalgebra of g , one obtains from the results of

this section a semicanonical basis of U



, the lower

half of the universal enveloping algebra of g . For any

irreducible highest-weight inte grable representation

V of U (or, equivalently, g ), let v 2V be a nonzero

highest-weight vector. Then the set

fbvjb 2B; bv 6¼0g

is a Q-basis of V, called the semicanonical basis of

V. Thus, the semicanonical basis of U



is simulta-

neously compatible with all irreducible highest-

weight integrable modules. There is also a way to

define the semicanonical basis of a representation

directly in a geometric way. This is the subject of the

next subsection.

One can also obtain a geome tric realization of the

upper part U

of the quantum group in a similar

manner using perverse sheaves instead of construc-

tible functions. This construction yields the canoni-

cal basis of the associated quantum group (a

q-deformation of the universal enveloping algebra)

which also has many remarkable properties and is

closely related to the theory of cryst al bases.

Finite-Dimensional Algebras and Quivers 319

Nakajima’s Quiver Varieties

We introduce here a description of the quiver varieties

first presented by Nakajima. They yield a geometric

realization of the irreducible highest-weight represen-

tations of simply-laced Kac–Moody algebras. The

construction was motivated by the work of Kronhei-

mer and Nakajima on solutions to the anti-self-dual

Yang–Mills equations on ALE gravitational instantons

(see Instantons: Topological Aspects).

Definition 3 For v, w 2Z

0

, choose I-graded vec-

tor spaces V and W of graded dimensions v and w,

respectively. Then define

  ðv; wÞ¼



i 2I

HomðV

; W

Definition 4 Let 

=(v, w)

be the set of all

(x, t) 2(v, w) satisfying the following condition: if

S = (S

) with S

 V

is x-stable and t

) = 0 for

i 2I, then S

= 0 for i 2I.

The group G

acts on (v, w) via

ðg; ðx; tÞÞ7!ððg

hðÞ



1

tðÞ

Þ; ðt

1

ÞÞ

and the stabilizer of any point of (v, w)

in G

trivial. We then make the following definition.

Definition 5 Let LL(v, w) =(v, w)

We should note that while the above definition

and other constructions in this article are algebraic,

there are also more geometric ways of looking at

quiver varieties. In particular, the space

Mðv; wÞ¼

 2Q

HomðV

tðÞ

; V

hðÞ



i 2I

HomðW

; V

ÞHomðV

; W

has a natural hyper-Ka¨ hler metric and one can

consider a hyper-Ka¨ hler quotient by the group

U(V

). The variety L(v, w) is a Lagrangian

subvariety of (and is homotopic to) this hyper-

Ka¨ hler quotient. In the case g = sl

, the varieties

involved are closely related to flag varieties.

Let w, v, v

, v

0

be such that v = v

þ v

Consider the maps

ðv

; 0Þðv

; wÞ

Fðv; w; v

Þ!

ðv; wÞ½2

where the notation is as follows. A point of

F(v, w; v

) is a point (x, t) 2(v, w) together with

an I-graded, x-stable subspace S of V such that

dim S = v

= v  v

. A point of

F(v, w; v

) is a point

(x, t, S)ofF(v, w; v

) together with a collection of

isomorphisms R

: V

ﬃS

and R

: V

ﬃV

for

each i 2I.Thenwedefinep

(x, t, S, R

, R

) = (x, t, S),

(x, t, S) = (x, t)andp

(x, t, S, R

, R

) = (x

, t

)

where x

, x

, t

are determined by

hðÞ



¼ x



tðÞ

: V

tðÞ

hðÞ

¼ t

: V

hðÞ



¼ x



tðÞ

: V

tðÞ

hðÞ

It follows that x

and x

are nilpotent.

Lemma 1 One has

ðp

 p

1

ððv; wÞ

Þp

1

ððv

; 0Þðv

; wÞ

Thus, we can restrict [2] to 

, forget the (v

, 0)-

factor and consider the quotient by G

and G

This yields the diagram

Lðv

; wÞ



Fðv; w; v  v

Þ!



Lðv; wÞ½3

where

Fðv; w; v  v

def

fðx; t; SÞ2Fðv; w; v  v

Þjðx; tÞ2ðv; wÞ

g=G

Let M(L(v, w)) be the vector space of all

constructible functions on L(v, w). Then define

maps

: MðLðv; wÞÞ!MðLðv; wÞÞ

: MðLðv; wÞÞ!MðLðv  e

; wÞÞ

: MðLðv  e

; wÞÞ!MðLðv; wÞÞ

f ¼ u

f ¼ð

ð



f Þ

g ¼ð

ð



gÞ

Here

u ¼

ðu

; ...; u

Þ¼w  Cv

where C is the Cartan matrix of g and we are using

diagram 3 with v

= v  e

where e

is the vector

whose components are given by e

= 

Now let ’ be the constant function on L(0, w)

with value 1. Let L(w) be the vector space of

functions generated by acting on ’ with all possible

combinations of the operators f

. Then let

L(v, w) = M(L(v, w)) \ L(w).

Proposition 3 The operators e

, f

, h

on L(w) provide

it with the structure of the irreducible highest-weight

320 Finite-Dimensional Algebras and Quivers

integrable representation of g with highest weight

i 2Q

. Each summand of the decomposition

L(w) =

L(v, w) is a weight space with weight

i 2Q

 v



. Here the !

and 

are the

fundamental weights and simple roots of g , respectively.

Let Z 2IrrL(v, w) and define a linear map

: L(v, w) !C that associ ates to a constructible

function f 2L(v, w) the (constant) value of f on a

suitable open dense subset of Z. The fact that

L(v, w) is finite dimensional allows us to take such

an open set on which any f 2L(v, w) is constant. So

we have a linear map

:Lðv; wÞ!C

IrrLðv;wÞ

Then we have the following proposition.

Proposition 4 The map  is an isomorphism; for

any Z 2IrrL(v, w), there is a unique function

2L(v, w) such that for some open dense subset

O of Z we have g

= 1 and for some closed

-invariant subset K L(v, w) of dimension <

dim L(v, w) we have g

= 0 outside Z [ K. The

functions g

for Z 2Irr(v, w) form a basis of

L(v, w).

Additional Topics

To conclude, we have given here a brief overview

of some topics related to finite-dimensional

algebras and quivers. There is much more to be

found in the literature. For basics on associative

algebras and t heir representat ions, the reader

may consult introductory texts on abstract alge-

bra such as Lang (2002). For further results (and

their proofs) on Ringel–Hall algebras see the

papers of Ringel (1990a, b, 1993, 1995, 1996)

and of Green (1995) and the references cite d

therein. The r eader interested in species, which

extend m any of these re sults to non-simply-laced

Lie algebras, should consult Dlab and Ringel

(1976).

The book by Lusztig (1993) covers the quiver

varieties of Lusztig and canonical bases. Canonical

bases are closely related to crystal bases and crystal

graphs (see Hong and Kang (2002) for an overview

of these topics). In fact, the set of irreducible

components of the quiver varieties of Lusztig and

Nakajima can be endowed with the structure of a

crystal graph in a purely geometric way (see

Kashiwara and Saito (1997) and Saito (2002)).

Many results on Nakajima’s quiver varieties can be

found in the original papers (Nakajima 1994,

1998). The overview article (Nakajima 1996)is

also useful.

Quiver vari eties can also be used to give geometric

realizations of tensor products of representations

(see Malkin (2002, 2003), Nakajima (2001), and

Savage (2003)) and finite-dimensional representa-

tions of quantum affine Lie algebras (see Nakajima

(2001)). This is just a select few of the many

applications of quiver vari eties. Much more can be

found in the literature.

Acknowledgments

This work was supported in part by the Natural

Sciences and Engineering Research Council

(NSERC) of Canada.

See also: Instantons: Topological Aspects.

Further Reading

Dlab V and Ringel CM (1976) Indecomposable representations of

graphs and algebras. Memoirs of the American Mathematical

Society 6(173): vþ57.

Green JA (1995) Hall algebras, hereditary algebras and

quantum groups. Inventiones Mathemati cae 120(2):

361–377.

Hong J and Kang S-J (2002) Introduction to Quantum Groups

and Crystal Bases, Graduate Studies in Mathematics, vol. 42.

Providence, RI: American Mathematical Society.

Kashiwara M an d Sait o Y (1997) Geometric con struction

of crystal bases. Duke Mathematical Journal 89(1):

9–36.

Lang S (2002) Algebra, Graduate Texts in Mathematics, vol. 211.

New York: Springer.

Lusztig G (1993) Introduction to Quantum Groups, Progress in

Mathematics, vol. 110. Boston MA: Birkha¨user.

Malkin A (2002) Tensor product varieties and crystals: GL case.

Transactions of the American Mathematical Society 354(2):

675–704 (electronic).

Malkin A (2003) Tensor product varieties and crystals: the ADE

case. Duke Mathematical Journal 116(3): 477–524.

Nakajima H (1994) Instantons on ALE spaces, quiver varieties,

and Kac–Moody algebras. Duke Mathematical Journal 76(2):

365–416.

Nakajima H (1996) Varieties associated with quivers. In:

Bautista R, Martinez-Villa R, and de la Pen

aJA(eds.)

Representation Theory of Algebras and Related Topics

(Mexico City, 1994),CMSConf.Proc.,vol.19,

pp. 1 39–157. Provid ence, RI: American Mathemat ical

Society.

Nakajima H (1998) Quiver varieties and Kac–Moody algebras.

Duke Mathematical Journal 91(3): 515–560.

Nakajima H (2001a) Quiver varieties and finite-dimensional

representations of quantum affine algebras. Journal of the

American Mathematical Society 14(1): 145–238 (electronic).

Nakajima H (2001b) Quiver varieties and tensor products.

Inventiones Mathematicae 146(2): 399–449.

Ringel CM (1990a) Hall algebras. In: Balcerzyk S, Jo´zefiak T,

Krempa J, Simson D, and Vogel W (eds.) Topics in Algebra,

Part 1 (Warsaw, 1988), Banach Center Publ., vol. 26,

pp. 433–447. Warsaw: PWN.

Ringel CM (1990b) Hall algebras and quantum groups. Inven-

tiones Mathematicae 101(3): 583–591.

Finite-Dimensional Algebras and Quivers 321

Ringel CM (1993) Hall algebras revisited. In: Joseph A and

Shnider S (eds.) Quantum Deformations of Algebras and Their

Representations (Ramat-Gan, 1991/1992; Rehovot, 1991/

1992), Israel Math. Conf. Proc., vol. 7, pp. 171–176. Ramat

Gan: Bar-Ilan Univ.

Ringel CM (1995) The Hall algebra approach to quantum

groups. In: Gomez -Mont X , Montej ano L, de la Pen

aJA,

and Seade J (eds.) XI La tin Ameri can Schoo l of

Mathematics (Spanish) (Mexico City, 1993),Aportaciones

Mat. Comu n., vol. 15, pp. 85–114. Me´xico: Soc. Mat.

Mexicana.

Ringel CM (1996) Green’s theorem on Hall algebras . In:

Bautista R, Martinez-Villa R, and de la Pen

aJA(eds.)

Representation Theory of Algebras and Related Topics

(Mexico City, 1994),CMSConf.Proc.,vol.19,

pp. 1 85–245. Provid ence, RI: American Mathemat ical

Society.

Saito Y (2002) Crystal bases and quiver varieties. Mathematische

Annalen 324(4): 675–688.

Savage A (2003) The tensor product of representations of

ðsl

Þ via quivers. Advances in Mathematics 177(2):

297–340.

Finite Group Symmetry Breaking

G Gaeta, Universita

di Milano, Milan, Italy

Introduction

It is a commonplace situation that symmetric laws

of Nature give rise to physical states which are not

symmetric. States related by symmetry operations

are equivalent, but still nature selects one of them.

As an example, consider a ferromagnetic system

of interacting spins with no external magnetic field.

The ‘‘up’’ and ‘‘down’’ states are equivalent, but one

of the two is cho sen: the interaction makes states

with agreeing spin orientation (and therefore macro-

scopic magnetization) energetically preferred, and

fluctuations will decide which state is actually

chosen by a given sample.

Finite group symmetry is also commonplace in

physics, in particular through crystallographic

groups occurr ing in condensed matter physics – but

also through the inversions (C, P, T and their combi-

nations) occurring in high-energy physics and field

theory.

The breaking of f inite group symmetry has thus

been thoroughly studied, and general approaches

exist to investigate it in mathematically precise

terms with physical counterparts. In particular,

a widely applicable approach is provided by the

Landau theory of phase transitions – whose

mathematical counterpart resides in the realm

of equivariant singularity and bifurcation theory.

In Landau theory, the state of a system is

described by a finite-dimensional variable (the

‘‘order parameter’’), and physical states corre-

spond to minima of a potential, invariant under a

group.

In this article we describe the basics of

symmetry breaking analysis for systems d escribed

by a symmetric polynomia l; in particul ar, we

discuss generic symmetry breakings, that is, those

determined by the symmetry properties them-

selves and independent of the details of the

polynomial describing a concrete system. We

also discuss how the plethora of invariant poly-

nomials can be to some extent reduced by means

of changes of coordinates, that is, how one can

reduce to consider certain types of polynomials

with no loss of generality. Finally, we will give

some indic ations on e xtension of th is th eory, that

is, on how one deals with symmetry breakings for

more general groups and/or more general physical

systems.

Basic Notions

Finite Groups

A finite group (G,  ) is a finite set G of elements

, ..., g

} equ ipped with a composition law , and

such that the following conditions hold:

1. for all g, h 2 G the composition g  h belongs to

G, that is, g  h 2 G;

2. the composition is associative, that is, (g  h)

k = g  (h  k) for all g, h, k 2 G ;

3. there is an element in G – which we will denote

as e – which is the identity for the action of  on

G, that is, e  g = g = g  e for all g 2 G; and

4. for each g 2 G there is an element g

1

which is

the inverse of g, that is, g

1

 g = e = g  g

1

In the follow ing, we omit the symbol , that is, we

write gh to mean g  h. Similarly, we usually write

simply G for the group, rather than (G,  ).

Given a subset H  G, this is a subgroup of (G, )

if (H,  ) satisfies the group axioms (1)–(4) above.

Note that this implies that e 2 H whenever H is a

subgroup, and {e} is a subgroup. Subgroups not

coinciding with the whole G and with {e} are said to

be ‘‘proper.’’

322 Finite Group Symmetry Breaking

Given two elements g, h we say that ghg

1

is the

conjugate of h by g. The conjugate of a subgroup

H  G by g 2 G is the subgroup of elements

conjugated to elements of H, gHg

1

= {(ghg

1

h 2 H}.

Group Action

In physics, one is usually interested in a realization

of an abstract group as a group of transformations

in some set X; in physical applications, this is

usually a (possibly, function) space or a manifold,

and we refer to elements of X as ‘‘points.’’ That is,

there is a map  : G 7!End(X) from G to the group

of endomorphisms of X, such to preserve the

composition law:

ðgÞðhÞ¼ðg  hÞ8g ; h 2 G

In this case, we say that we have a ‘‘representation’’

of the abstract group G acting in the ‘‘carrier’’ space

or manifold X; we also say that X is a G-space or

G-manifold. We often denote by the same letter the

abstract element and its represent ation, that is, write

simply g for  (g) and G for (G). (In many

physically relevant cases, but not necessarily, X has

a linear structure and we consider linear endo-

morphisms. In this case, we sometimes write T

for

the linear operator representing g.)

If x 2 X is a point in X, the G-orbit G( x) is the set

of points to which x is mapped under G, that is,

GðxÞ¼fy 2 X: y ¼ gx; g 2 GgX

Belonging to the same orbit is obviously an

equivalence relation, and partitions X into equiva-

lence classes. The ‘‘orbit space’’ for the G action on

X, also denoted as =X=G, is the set of these

equivalence classes. It corresponds, in physical

terms, to considering X modulo identification of

elements related by the group action.

For any point x 2 X, the ‘‘isotropy (sub)group’’

is the set of elements leaving x fixed,

¼fg 2 G: gx ¼ xgG

Points on the same G-orbit have conjugated isotropy

subgroups: indeed, y = gx implies immediately that

= gG

1

When a topology is defined on X, the problem

arises if the G-action preserves it; if this is the case,

we say that the G-action is ‘‘regular.’’ In the case of

a compact Lie group (and a fortiori for a finite

group) we are guaranteed the action is regular.

(A physically relevant example of nonregular action

is provided by the irrational flow on a torus. In this

case G = R, realized as the time t irration al flow on

the torus X = T

Spontaneous Symmetry Breaking

Let us now consi der the case of physical systems

whose state is described by a point x in the G-space

or G-manifold X, with G a group acting by smooth

mappings g : X ! X. In physical problems, G quite

often acts by linear and orthogonal transformations.

(If this is not the case, the Palais–Mostow theorem

guarantees that, for suitable groups (including in

particular the finite ones) we can reduce to this case

upon embedding X into a suitably larger carrier

space Y.)

Usually, G represents physical equivalence of

states, and G-orbits are collections of physically

equivalent states. A point which is G-invariant, that

is, such that G

= G, is called ‘‘symmetric’’ for short.

Let  be a scalar function (potential) defined on

X,  : X ! R, possibly depending on some para-

meter , such that the physical state corresponds to

critical points – usually the (local) mi nima – of .

A concrete example is provided by t he case

where  is the Gibbs free energy; more generally,

this is the framework met in the Landau theory of

phase transitions (Landau 1937, Landau and

Lifshitz 1958).

We are interested in the case where  is invariant

under the group action, or briefly G-invariant, that

is, where

ðgxÞ¼ðxÞ8x 2 X; 8g 2 G ½1

A critical point x such that G

= G is a symme-

trical critical point. If G

is strictly smaller than G,

then x is a symmetry-breaking critical point.

If a physical system corresponds to a nonsym-

metric critical point, we have a spontaneous

symmetry breaking: albeit the phy sical laws (the

potential function ) are symmetric, the physical

state (the critical point for ) breaks the symmetry

and chooses one of the G-equivalent critical points.

It follows from [1] that the gradient of  is

covariant under G.Ify = g(x), then the differential

(Dg) of the map g : X ! X is a linear map between

the corresponding tangent spaces, (Dg):T

X ! T

The covariance amounts, with  the Riemannian

metric in X,to(

)(gx) = [(Dg)



](x); this

is also writt en compactly, with obvious notation, as

ðrÞðgxÞ¼ðDgÞ½ðrÞðxÞ ½2

(in the case of euclidean spaces ( = ) and linear

actions described by matrices T

, the covariance

condition reduces to (r)

x) = (T

)

[(r)

(x)]).

As (Dg) is a linear map, (r)(x) = 0 implies the

vanishing of r at all points on the G-orbit of x.

We conclude that critical points of a G-invariant

potential come in G-orbits: if x is a critical point for

Finite Group Symmetry Breaking 323

, then each y 2 G(x) is also a critical point for .

We speak therefore of critical orbits for .

It is thus possible (thanks to the regularity of the

G-action), and actually convenient, to study sponta-

neous symmetry breaking in the orbit space

=X=G rather than in the carrier manifold X

(Michel 1971).

If G describes physical equivalence, physical states

whose symmetries are G-conjugated should be seen

as physically equivalent. An equivalence class of

isotropy types under conjugation will be said to be a

symmetry type. We are thus interested, given a

G-invariant polynomial , to know the symmetry

types of its critical points. We denote symmetry

types as [H] = {gHg

1

}, and say that [H] < [K]ifa

group conjugated to H is strictly contained in

a group conjugated to K.

As we have seen, points on the same G-orbit have

the same symmetry type. On the other hand, points

on different G-orbits can have the same isotropy

type (e.g., for the standard action of O(n)inR

, all

collinear nonzero points will have the same isotropy

subgroup but will lie on distinct group orbits).

G-Invariant Polynomials

Consider a finite group G acting in X. (Many of the

notions and results mentioned in this section have a

much wider range of applicability.) We look at the

ring of G-invari ant scalar polynomials in x

, ..., x

By the Hilbert basis theorem, there is a set

{ J

(x), ..., J

(x)} of G-invariant homogeneous poly-

nomials of degrees {d

, ..., d

} such that any

G-invariant polynomial (x) can be written as a

polynomial in the { J

, ..., J

}, that is,

ðxÞ¼½J

ðxÞ; ...; J

ðxÞ ½3

with  a polynomial. (A similar theorem holds for

smooth functions.)

The algebra of G-invariant polynomials is finitely

generated, that is, we can choose k finite. When the

are chosen so that none of them can be written as

a polynomial of the others and r has the small est

possible value (this value depends on G), we say that

they are a minimal integrity basis (MIB). (Note that

some of the J

could be written as nonpolynomial

functions of the others, and the J



could satisfy

polynomial relations. For example, consider the

group Z

acting in R

via g :(x, y) ! (x, y); an

MIB is made of J

(x, y) = x

, J

(x, y) = y

,and

(x, y) = xy. None of these can be written as a

polynomial function of the others, but J

= J

.) In

this case, we say that the { J

} are a set of basic

invariants for G. There is obviously some arbitrar-

ness in the choice of the J

in an MIB, but the

degrees {d

, ..., d

}of{J

, ..., J

} are fixed by G. (In

mathematical terms, they are determined through

the Poincare´ series of the graded algebra P

G-invariant polynomials.)

We will henceforth assume that we have chosen

an MIB, with elements { J

, ..., J

} of degrees

, ..., d

}inx, say with d

 d

d

When the elements of an MIB for G are

algebraically independent, we say that the MIB is

regular; if G admits a regular MIB we say that G is

coregular.

An algebraic relation between elements J



of the

MIB is said to be a relation of the first kind. The

algebraic relations among the J are a set of

polynomials in { J

, ..., J

}, which are identically

zero when seen as polynomials in x. If there are

algebraic relations among these, they are called

relations of the second kind, and so on. A theorem

by Hilbert guarantees that the chain of relations has

finite maximal length. (This is the homological

dimension of the graded algebra P

mentioned

above.)

In the following, we will consider a matrix built

with the gradients of basic invariants, the P-matrix

(Sartori). This is defined as

ðxÞ:¼hrJ

ðxÞ; rJ

ðxÞi ½4

with h.,.i the scalar product in T



The gradient of an invariant is necessarily a

covariant quantity; the scalar product of two

covariant quantities is an invariant one, and thus

can be expressed again in terms of the basic

invariants. Thus, the P-matrix can always be written

in terms of the basic invariants themselves.

Geometry of Group Action

The use of an MIB allows to introduce a map J: x !

{ J

(x), ..., J

(x)} from X to a subset P of R

.If

the MIB is regular, P = R

, while if the J

satisfy

some relation then P  R

is the submanifold

satisfying the corresponding relatio ns. The manifold

P is isomorphic to the orbit space =X=G (the

isomorphism being realized by the J map) and

provides a more convenient framework to study .

As mentioned above, on physical terms we are

mainly interested in the orbit space up to equiva-

lence of symmetry type. The set of points in X (of

orbits in ) with the same symmetry type will be

called a G-stratum in X (a G-stratum in ); the

G-stratum of the point x will be denoted as (x)  X

(the G-stratum of the orbit ! as (!)  ). (The

notion of stratum was introduced by Whitney in

topology; a stratified manifold is a set which can

be decomposed as the disjoint union of smooth

324 Finite Group Symmetry Breaking

manifolds of different dimensions, the topological

(or Whitney) strata: M =

,withM

 @M

for

all k < j.)

It results that the G-stratification is compatible

with the topological stratification. Indeed, P is a

semialgebraic (i.e., it is defined by algebraic equal-

ities and inequalities) stratified manifold in R

; the

image of any G-stratum in  belongs to a single

topological stratum in P, and topological strata in P

are the union of images of G-strata in .

Moreover, the subgroup relations correspond to

bordering relations between G-strata: if [G

] <

], then (y) 2 @(x) and (with !

the orbit of x)

(!

) 2 @(!

There is a stratum, called the principal stratum 

which corresponds to minimal isotropy, open and

dense in X; similarly, the principal stratum 

open and dense in .

Landau Polynomial

In the Landau (1937) theory of phase transitions,

the state of the system under study is described by a

G-invariant polynomial  : X ! R having a critical

point in the origin, with at least some of its

coefficients – in particular those controlling the

stability of the zero critical point – depending on

external control parameters (usually, X = R

and

G  O(n); in particular, in solid-state physics G is a

crystallographic group). This should be cho sen as

the most general G-invariant polynomial of the

lowest degree ‘ sufficient to ensure termodynamic

stability; in mathematical terms, this amounts to the

requirement that there is some open set B containing

the origin and such that – for all values of the

control parameters – r points inwards at all points

of @B (i.e., B is invariant under the gradient flow

of ). If the polynomials in the MIB are of degree

 d

d

, then usually ‘ = 2d

The G-invariance of  and the results recalled

above mean that we can always write it in terms of

the polynomials in an MIB for G as in [3],

(x) =[ J(x)].

The discussion of previous sections shows that we

can study symmetry breakings for  : X ! R by

studying critical points of  : P ! R; in other words,

Landau theory can be worked out in the G-orbit

space  := M=G. The polynomial  – providing

a representation of the Landau polynomial in the

orbit sp ace – will also be called Landau–Michel

polynomial. (Louis Michel (1923–1999) pioneered

the use of orbit space techniques in physics and

nonlinear dynamics, originally motivated by the

study of hadronic interactions.)

In this way, the evaluation of the map  : X ! R

is, in principle, substituted by evaluation of two

maps, J : X ! P and  : P ! R. However, if, as in

Landau theory, we have to consider the most

general G-invariant polynomial on X, we can just

consider the most general polynomial on P.

Critical Points of the Landau Polynomial

and Geometry of Orbit Space

The G-invariance has consequences on the critical

points of . We have already seen one such

consequence: critical points come in G-orbits.

However, this is not all. Indeed, G-invariance

enforces the presence of a certain set (G) 2 X of

critical points, and conversely if we look for points

which are critical under any G-invariant potential,

these are precisely the points in (G); the critical

points on (G) correspond to critical orbits which

we call principal critical orbits.

The set (G) can be determined on the basis

of the geometry of the G-action. (A trivial

ex ample is provided by X = R and G = Z

acting

via g : x !x; any even function has a critical point

in zero, and albeit even functions can, and in general

will, have nonzero critical points, this is the only

critical point common to all the even functions.)

Indeed (Michel 1971): an orbit ! is a principal

critical orbit if and only if it is isolated in its

stratum.

For the linear orthogonal group actions in R

often occurring in physics, no nonzero point or

orbit can be isolated in its stratum. However,

we can quotient out the radial degener acy and

work on X = S

n1

 R

. In this case, a G-orbit !

in S

n1

which is isolated in its stratum corresponds

to a one-dimensional family {!

}ofG-orbits in

(call X

the corresponding submanifold in X);

the gradient of  at x 2 X

points along T

We can thus reduce to consider the restriction



of the potential  to X

. (See also the

reduction lemma of Golubitsky and Stewart in this

context.)

Correspondingly, if P

 P is the submanifold in

P image of X

, that is, P

= J(X

), we can reduce to

consider the restriction 

of  to P

As these become one-dimensional problems,

general results are available. In particular, one

can provide general conditions ensuring the

existence of one-dimensional branches of symme-

try-breaking solutions bifurcating from zero along

any such X

or P

; this is also known as the

equivariant branching lemma of Cicogna and

Vanderbauwhede.

Finite Group Symmetry Breaking 325