An Introduction to Diagrammatical Methods in Representation Theory

VORLESUNGEN Fusdem

FACHBEREICH MATHEMATIK der UNIVERSITAT ESSEN ..

ii

'

Helt7

il-

V. B. Dlab

TO DIAGRAMMATICAL AN,INTRODUCTION METHODS IN REPRESENTATION THqORY' ,]

'

,Ausalbeitung: Riüard Dipper . ,,

l

I

e i f",

,l

1981 ,'

u/

+'' i/ 4 uu.iqi't i( -

/'t.< t '"

.1,: :i t'

r

1,,

VORLEST'NGEN aus dem FACHBEREICH UATHEMATIK

der UNIVERSITATESSEN

Iteft

7

V. B. Dlab

AN

INTRODUCTTON TO IN

DIAGRAMI.,IATICAL MEIHoDS

REPRESENTATION TIIEORY

Ausarbeltung:

Rlchard

19 8 1

Dlpper

Acknowledgements

These notes lectures

contaln

addressed

the materlar

to

the

graduate

presented students

rn a serres in

Algebra

of

at

the Essen durlng the surmer semester 1g7g. T should lLke to thank prof. c.o. Mlchler for hls klnd lnvltatlon to vlslt Essen, and the audlence for their partlclpatlon actlve ln the lectures. unlverslty

to

of

The alrn of

thl-s

some baslc

results

and conseguently tensor

algebras

brief of

the

course the

content

was to

expose

the

representatlon

theory

was restrlcted

to

and a few slmple

audience graphs,

of

a study

1llustratlons

of

of

the

theory.

As a result,

the reader finds a major overlap wlth the Me_ the Amer. ltath- society No. 't 73 and an omission of general theory of M. Auslander and I. Reiten.

molrs the

of

My special written tails

notes

are

due to

wLth great

r should

Department

thelr tatlon

excellent of

the

of

lr-ke to

Mathematics

typlng,

p.ichard

Dr. care,

and in many ways improvlng

Pinally, the

thanks

up the

of

contributed

supplylng

the

thank

Digper

original

alr

those the

de_

expositlon. secretalies

Essen University to

who has

missing

formal

who,

of by

presen_

notes.

Vlastimil

B. Dlab

CONTENTS Chapter

f

Valued

graphs

s1

Valued

52

The roots

Sg

Graphs wlth

Chapter

II

PAGE

graphs, of

1

Dynkln

and Euclldlan

graphs

graphs

valued

7

orlentatlon

Reallzatlons

of

1

11

valued

graphs

and its

20

repre9entatlons

s s s

1

X-reallzatlons

2

The Coxeter

3

Prejectlve

Chapter

ffl

Graphs

of

and representations functors

30

modules and extenslons

finlte

and of

The representatlons

of

s2

The non-homogeneous

representations

Dynkin

Euclld1an

Appendlx

graphs

56 of

60

of

85

graphs

'Blmodules Graphs

3f

graphs

The hornogeneous representatlons

s4 Ss

47

tame type

St

Eucll-dlan

20

of

Euclidian

and bimodules

type of

wild

100 type

ru)

A

Applicatlons

118

1

Algebras

118

2

Normal

3

Further

Appendix

B

Exanples

173

Appendlx

C

TabIes

180

form

problems

applicatlons

139 171

-1-

Chapter S 1

graphs,

Valued

graph

A valued rrith

Valued qraphs

r Dynkin

(I,d)

is

and

a finite

a set d ={(dii,dii)lar.

graphs

Euclidian

€N

u

(of

f

set

vertices)

together

€ I}(of

{c},i,j

vaiues)

L J

J L

satisfying i) li)

dlt=oforalli€I For every utj

= di1 fi

tj

Notice that join

€ f there

i

d, : i rJ

exists

for all

o L f a n d o n l y- i f

, If

d,. rl

j 1 Another valued graph ( l',d') (t,d),if Veltices,

( l,d)

is

a sequence 1 = ior that

the fi,

scalar

d:. )L

I o. fn this

i1,

multlples,

if n} c

For the rational

with

ft

g t

is

a subgraph of

(xi)l*iee,

s!'nnetric

bilinear

form

(1) :=

-f .i 'x. . y .

i,j

€ t

, there

ik = i of neighbours in

is

ccnnected.

-

i€r}withn= Bf

- I .

quadratic (r,r)

"l.

all

f

determj.ned up to

Of course we can nrite

=

definethe

lrl

by r (r,j)€f"f

d..f .x..;. rifixiYi

for

y I , I \ , .€ Q '

form tr*r'

,ä.

-

dijrj*i*j .1. ij

,

rvhere in edges

the second term the sum has to be taken over aLl (f,d) - clearly in Br and en der:end on i-;

the choice of the fj.'= , i € f , factors, j-f (i,d) is connected.

is

. Notice

space

{x=

wLth associated

for

neL_qhbours. A

N fo sone n €N.

vector

r ier

if

.i ,d.i i )

:

above are uniguely

A':=0"=

(x,1) =

*!

and wrlte

= 1 = tt... l1

....,

( frd)

case, we (di

(dii,dii)

connected,

I € f, 1n ii)

| = {1, ....r

Qf

-

d t.t l, = d , . ft ot r af lol ri , i a€ l l i , j € f lr c f . _. which are joindd by an edge, are called

valued graph

Br

€ f

i'j

i and j by an edge of value

or simply

€ N such that

fl

but are unique up to scalar

-2-

Eor k€I

let

= (k.)

k

€ 0f

definedby

k,

= o for

k I

I

€ f

and k* = 1.

1.1.1

Definition:

(trd)

a) Let

be one of

the

following

valued

sraPhs (lrl= n)' Arr: *o......a.1 1, 2 ) Brr:ffi......H

(2,1) C: n

o--O......H

\ )--......a---a

/ ? n=6

E.: o

E7'

? n=8

3r;

n=4

tt.

I

( 1, 2 ) '4'

(1,3) Gr: o---
4

Sn-t

n>4

örr-t

n> 5

örr-

t

( 1, 2 1

(2t1)

(2r1)

( 1. 2 ' , )

( 2, 1 1

(2t1)

.E---{-_o.....H

\

-7-n=7

E6

n=8

Ej

-/'

\

-4-

n=9

-8

n=5

x "41

n=5

-F 4 2

n=3

dr.,

n=3

"22

( f ,d)

Chen vre call

1.1.2

Lemma:

( f ,{)

Let

of the followlng

(rr,4r,,

i)

(i,j)

(r

ii)

Proof:

a

If

i,j

( | ,d) contains

one

dijdji

> 5 for a pair

x I

a triple

(i,j,k)

Euclldian

graph.

there are i,j

€ f.

Then

graphs as a subgraph:

€

ijk

d . . d . . -> 4 , rl lr.

Ä,,.,' or Ä,,r. thus F I

= 3 for

lä) ,

€ fxlxf

€ I with

(f1,d1),

Let i,j

,@w*h

= 2 for t r(resP.

= 3 and djrd*j

a subgraph of type for all

be Dynkin.

(resp. ( | i,gä)

urjuj,

iii)

not

.

:&+'i.h

€ f

2,9)

for

(graph)

Euclj.dian

. . i ! L W f L r r

I I u . . s . i i ' )

-J -

2 J .

( f ,d)

contains

a s s u m e d .r l . d .1 1 . -< 3 Because (f,d)

is

-t-

not

Dynkln,

i or j must have a nej_ghbour k € I

above dildkj

< 3. It

graph of type utjdjtg there

is

( tz,lzl

S t

a subgraph of type Et, all

{1,j}

with

€t

Ct or 60a,lrl

;

{k,f}

g I

F41 ot Fnr,

or it

must havg 6 branching

ls

of

type 0a

l,j

not Dynkln.

€ I

type Äa,

rrr

If

or !r,

because it

has only

branchlng point,

A bilinear vector

it

Leumas tet

varued graph

Proof:

(r,g).

Clearly

and take x'=

lt

Brr

(f',d')

is

the vector

it

a subgraph type agaJ-n because

lt

if

contalns

positive.

lrl

2 t €N.

if

O(I)

all

I

> o for

aII

€ en and positive definite.

the connected

then Brrispositivedefinite.

Assrxle B.

minimal hrith this

defined

E,

form o on the rational

a proper subgraph of

is

one branching

more then one

but not positlve

B, is positive,

ar1

a subgraph of type Er,

> o for

rf

a subgraph = 1 for

at reast

quadratic

g(x)

positive,

€ Of'

= 1 for

contaj,ns a subgraph of

is called. positive,

(f,d)

dttdtx

a subgraph of type ö,,

5 (xi) € QI' b" with er,(x,) z =(?t)

If

associated

(f',$')

contatns

contains

contalns

1t contains

contains

definite,

lf

it

not Dynkln.

space en, n €N,

senldeflnlte,

1.1.3

one,

form B wlth

€ 0n, positive

. rf

2. rf

( I,d)

Thus assurne drjdj,

a circle,

Otherwise

is

now

polnt,

( f,d)

case

t € N.

contalns

t €N.

I

Po1nt.

it

lf | :

( f,4)

If

thus.assume

, our graph must contaln

fn the latter

or döa,

a sub-

= 2, then t €N

By the

contains

withdlidir=

dXtdtt

I

it

I

Leti,j

, say j.

(f,{)

Er., or örr.

i,di'),

e r

{k,1}

I

( f

,

2foral1l,j 1s {i,j}

easy to see, that

is

not positive

property.

Let

= o. since for arbitrary by z., = iyrl

for

all

definite,

I

€ eI,

i € f,

-o-

always

satisfj-es

for

i

f

all is

Qr, (z) < er, (1)we can assume

€ f'.

In view of minimality

connected,

d., 'oJo I

there

o. oefine

d. ].'l o-o x+ *.. ro2'or

x

e ef Uy

l',

f'

all

xi

+ fi -o

-.

x?

-o

*i

for

dr.

f.

x.x.,

with

€ I,,

i

E

-Jo

Jo

a

Jo

'

di s f* x] lo lo

> o

I o. Because

and io € lr = *i

,a-oJo i€f

-

xi

= o othenrj.se.

and x.

Then Qf (x) = Qr, (I')

jo € f -

exists

= (*i)

of

that

= f., (x. d' .i f+ x., X+ roJo Jo r.o Jo Jo Jo

d. d. ro lo

i

39

x' ,2Lo

2

!

_r

' o 'i o * ?

=-

: = -rff#.

Proof

Thus skx = x + <xrk> b Furthermor" Let

be a connected

9)

I

qi,i>

e Qf, k,i

€ l,

by direct

calculation

(1)

- -2Yk *

(2) sr

"i =k si

I

= I

and either

= -2,

and

t x or

I "k = dik

€ N

for

i,k

L * k. We get the following from the definition ur* yjr

rä, +

= s. sk if

sk x < x.

i

if

+ dik

formulas

o, )k;

= dki

i

I k. easily

Bf,

hence for 1:

+ (.I,kr

and only

of

€ r,

+ 2yk > o hence

= O

(3) sr"ksi"kL-sksiI= (.1 ,i> (4) If "i

1) (.I,\>

d.* = d*. = 1 "k

I

"k

I

k +

+

i



i

then

= (

+ )i

"k"i"kI-si"kI=k - sk =i sk - i 1 = "j. "k "i "k I (5) From (3) we conclude easily: rf

sk I

f

yor

sil

I I,

only if dir = tri Define

"k "i

I

= 1.

Ru S Rf , -1

now

then

I

v € Z,

= "i

"k "i

if "k L

and

by

R_1,={-xeerlk€r} Ro,=ireErlk€r] Rv+1: = Rv U {skIlx Let v > O, I an I

s1ts>I},

€ Ru, x e Rv_1. Then, by definition

€ Ru_,|and some

O < x - I

€ Rv, k € f,

=

i,

i

€ f , such that

hence

> O.

* =

"i

!

v > O. of > I,

Rv, there i.e.

is

- 11 -

Let

i

f k € r

o t

sk x - x =

on \,, that trivj.al. of 1.

So 1et lf

by lnductlon induction

v:1.

= ( )k,

show now by induction v = o this

changes only

the

i-th

o.

Thus il.*

Next

we will

forsome

ls

and

by

trj.vial

LfI,

inplies

]="kI..y=l "f * dff i > k,

5 1

show again

I€f

is

component

:: 1 and drr

€ Rv_1. For v = O this

sk I

Nord

+.I,

kt

For

< I.

s* I

(2),

Then, by

I o.

yk I

o also

I

because

"if Because si

sk x < x.

I . L. We will "k 5 1 and d*, < 1, if x*

dif

Thusr

-

L

"k "i !> < O and

skl

"k "i this implies O > "i again by lnduction,

< O we conclude "i

"k

I

"i I

By induction -

f !-, thus

€ Ru_2. Furthermore

+ )

I = ( +
= ( o and

""'

sk1 x = kt,

.....

"*a_,

=

some 1 < t < lrl

is

"

x < o' By

""'

"*a "n., = ---.. hence x "o., "na

"*a_.,Ea.

Similar.

(with

The notation

pkt,

c € Wr) will

be kept throughout

1.2.6.

Lenma: Let

c = sk

.. ...

sk

^1

gkt

(f,9)

Proof:

< n,

have

is

N, cI

=

-u

by I.

the

is

1.2.4). Hence there

.

cr-*r,

,

roots

smallest

(I,d)

Then

r*r,

positive

all

n, a.ta

,:k

-9. -r 2. , . .

Because

saym€

Dynkin graph,lrl=

=K2"

of

transformatj.on

paper,

this

-a^.

c- IK_l

a list

be a

to a Coxeter

!*r,

-a" I ^r:^z^^n^ =*1 ,

'| < i

respect

€ Wf be a Coxeter transformation.

&., ,

is

there

of

integer

Dynkin, Thus for exists

(l,d)

where a.

such

t,hat.-1.-"i

c is

of

< m -

nn,

order

finite

O < x € Rr andy ' 1 < r

€ N,

1

mh E c" x h=l such that =

rre

-

r-'l

= pf O, 1 5 t,

"-t ES Dynkin type.

Prescribing,

the

calculation,

ä. (Lt)

(f,d)

px.)

) .'-Ä.;:-,

for

< n.

lernrna for

enough to prove

0 of

.-t

1 < t

see later in

(indicated

(sL

Oc, ("k.

< O for ä"(pt .-t )

Remark: l{e will

S3

then

= s, pL ^1 *i+1

-Dk-1 -

direct

pk.

n,

-1

By an easy tables

(

1 < t

) pr ^i " t- 1

prove

to

aIL

A vertex

respect

vertices 0 of

k1

(frd)

to i. €f

(t,d) .,

n, , which

is

kn of

said f

such

-

that

a sink

...

s. *i-1 is

is

k1

with

called

admissible admissible

and if

...,k'

k1,

l. 3 .1 .

(i) ii)

+

-

e

is

iii)

(iii)

(ii)

t

i

,

k,

It

sinks,

also

is

for

easy

sources, kr,

,..

to

show

connected

= {1,...,n}.

graph

Then the

\rith

following

cj-rcuj.ts

in

(t,d)

n of

inplj-es

n of n(i)

{1,...,n}

such that

< n(j).

: easy : Choose an admissible

n bY n (kr) :

It

is

= i'

i

easy to

sequence k1r...,k'

= 'l,...,n. see,

that

n(1),..,,n(n)

is

sequence.

alfows

an admissible

orientation

seguence for

sources.

j-s a permutation

(ii)

admissj.ble

(1.3.1

be defined

can also

no oriented

o +o

and define iii)

.....

admissible

There j

Proof:

are

o is

iii)

case k1 ,

equivalent.

There

ii)

respect

with

sinks).

be a valued

and let

are

i)

( f , d)

Let

fl,

statements

this

In

a sink

is

s. 0 = a. K1

I,erq!4.

orientation

(for

for

, and k.

1.

an admissible

sequence

.....

s, *r,

a

1 < n -

sequences

is

an admissible

that

-

to

sequence

Of course,

is

respect

1 < i

n, s. K1

|8 -

us,

to

sequence (f,d).

order of

f

sinks

i-n such a way that for

the

fj.xed

1,...,D

admissible

an

. k.l

to

_ 19 _

Note alsor if

k1,...k.

sequences for

the same admissible

C =

S,-

for

any pair

*r,

...

Ref erences:

sr_

*1

-

Srt

*r,

and kir...,k,

...

of vertices

't I I ] ,

t22l

Sr_r =

ki

i,j

are two adrnissible orientation

Cir

since

s. -i

e, then a---nd

s. -j

c-----ommuce

€ I which are not neighbours.

and [ 23 ]

Realizations

2

Chapter

S 1

K-realizations

2.1.1

Remark: Let K in

a

: =

Proof:

\J

'r (d.,dr) D is

i J

t' \ \.\

the

= t

(drd, )

= tr

dr)

tr(d)

= k+ o,

projection

. Then

€ K1.

the f

"r tp

HomK (

K-Linear.

by the

above property

O +

r

by tr

o and with

d€D

the

by

followed

cases, o + r

both

._HomK

of

DDD)

is

, it

t

(DMF, K)

by

easy

to

of

map is

. This

and'

an F-D-bimodule

is

K)

DMF,

of

HomD (

, f

must be injective, the

(DMF, K)

Thus let

DDD)

DMF, oDo)

see,

are

f

that

Thus Homo (

K-dimensions.

hence

as F-D-bimodule,

also

the

by

DMF, DDD)

Hom" (oM",

similar

and

surjective

bijective

therefore

HomK (DMF, K) as F-D-bimodule. Honk

dZ€D.

K1

tr:D*

D and

of

Hence, in

D onto

of

P 1691 tr+

HonD (DMF,

elements

of

projection

r

then

f inite,

Property.

all

equality

K hrith

homomorphism.

an bimodule

since

dt'

Otu,

19 €

for

K is

126]

Then define

: Horno (

course

is

for

desired

If

center

lsee

K1 on K'k k >t.

Again by

and (2.2.'l ) we get

E x t r ( t ; . . . s l * r l r . , s i . . . t l * r E . ) ! e * t 1( E r . s, i . . . s l * . , r . ) , Er., ti

...t1*,

Ft€ [ (M,s1+1...sn Q). rr

source. of course, 11 is not a direct

"k*1...s. s u m m a n do f s l

(2.2.i\ rherefore by (2.1 .6 b) ) din

r*+1 re

(x',

2.2.4

Proposition:

"*

...s|*f

It)

= o and rt

Let

X€

_ I

a)X=CC'xeP,whereP€

L (

is

injective.

M , 0 ) . Then

L

(M,Q)isprojective.

Ea

sl+1 !r)

s ; + 1 E r ) r=* r " n - f 1 . : = . . ; ; - . ; - " . ; ; - =

= ( s k ( d t usn; . . . Hence Extl

Q,k is a

"

=

-

Thus,

lf

1< t5i n,

39 -

X ls

lndecomposahle,

i.e.

c* I

1 < t o

(indecomposable),

if

x = c-r

Pr for

x = Cr I*

for

and 1 o,

13t

5n, and

then

nxtl

This

Proof:

= C-t 31, 12 or

X + O, then

extl ( c- x, x) ! b) rf

( M,Q )l I

L

(X, c+ x ) =

statements

Ft F. r .c c

as bimodule.

are straightforvtard

consequences

(2.2.3) and (2.2.5).

Let r:'O,

t (

M,

0) ) x = c-r

Pa (or

1 1t< n. Then the position

by : pos ({)

= n.r

+ t.

If

x is

cr ra ) with pos (X) is

a fi.nite

defined

direct

-41

of preprojective

sum

nodules

(or prelnjectl_ve)

[ ( lr,l, 0 ),

Xt €

preprojectlve

-

(resp.

then X will

preinjective)

indeconposable be called

also

and pos X is deflned

by pos (x) = max pos (Ii)

2.2.7LeIuta:

a) LetX,

Y€

Then Y ls preprojective €

I

of position

L ( M , A ), I

preinjecitve

of positlon

a) First

and Horn (!,

let

< p.

preinjective

of posltion

p,

< p.

X be indecomposable,

o< t o.

Changing

€ IN, q >r

x to

be inde-

Then by

(2'2'3)

vtecansuPPose

Therefore

(dj.n c+qx)a+o

Coxetertransformation Let

indecomposable

preprojective.

for

there

exists

q>k'

all

and

1 O and 1 o

C-I = q. Let

in

11 s 1n.

...s;_1ls)r+

tr.,t., ...

o, if

t"_., E")a+ o,

e€ {o,1}

in |

t> s, i.e.

if

, (c-ps)t=

t< s and

there

(C-egs)t+ O. By the remarks

such that

(2.1.3) and (2.2.1) Hom (par c-t!")+ o. Assume F - I' 3 " * o. with the remark in (2.2.'l) we get

r-1+ C '- "

t'!",+

Hom (x, c-(r-1* and therefore

o. NowX is i.njective (2.2.3\,

every epimorphic

every image of X in i.e.

e)n.=g-a-(r-1*t

indecomposabre. This implies a-(r+ by (2.2.3),

it*1)!"

thus in every case c

proceed with

hence

must be injective, "-(r-1* *,r"a be j.njective, because it is

'!"

C-(r-1+e

image of X (2.2.8),

t'E"

s instead

This shows finally

t considering

c*(r+n-1't"

'r"

= o. Now

neighbours

= O for all

of

s in

1< s1n.

b) Similar.

2.2.12 Corollary: and J=

( f,

b)

P=

c)

statements

1

some 1< s< n and vj.ce

d) o,

and e) 1< t-< n)

versa.

f

.

-45_

11.2.6)

shows, that

e) r a)

: Let

(f,

( f

a)

implies

e) .

, d) not be Dynkln. Then, by (1.1.2),

d) contalns a subgraph ( f,,

Euclldlan

or one of

the

following

(dt2,dzt (r1 ,91 ) = 1t -

r)

(12, 92) =1.

ii)

d.tZ dZ.t = 3

)

'3

(d12' d21)

whlch is

graphs:

wlrh d1

zdz.t25

(d23,d32)

. ,

.,

wirh

d32 = 2

and dra

(d12'd,1).

(13, g:

rir)

d'),

1 =',.

(dzy

dgz).,

r.

with

dtZ dZt = 3=d23 d32

Let Q'be

the restriction

be the restrictlon L (M there

"

1s a full 0')

(*, (e x) . =1 ' '

e

exact +

L ( M ,0

is

true

0').

clearly

L ( l/',

We will

1gi.-.j

) rr , , . o . =G l'1

0')

e and consider

L( M', 0').

inf,

otherwise

/o L

and similar

for morphisms.

L ( M' ,

as subcategory

show, that

modure, which for

(M,,

and

) by setting

otherlvise ,

).

d,)

( Ä{,,0,)

ernbedding

fori€r'

a preprojective this

( f',

ft A ) to

L(l{,0

X = (Xr,ror)€

Thus we omit of

( 1,, d'),

to

Q',) the rnodulecategory of

e: L(M',

for

(

of

of 0

is

O' )

L (M , e) contains

not preinjective.

Then, by d),

Assume

no preprojective

- 46 (M',

[

module ln

(l

be the Coxeter element of

, d) with

1 is a sink with projective

respect L'( lll',

in

F1 is not preinjective

resPect

to

and'

O

we can assume u = 1. Thus

ay changing orientation

u € f'.

Let c = sn...s.l

is preinjective.

0')

and

to

0

0')

and

in

L ( M ',

and F1 is

Q',

L( l\{ , 0 ) . By assumPtion, by

therefore,

0'),

(2.2.1O) Hom (Et, !) + O for infinite many nonisomorphic modules Y.€ L(l'l ', Q'), hence the same is true in L( M, 0 ) and F., is not preinjective by (2.2.10).

Therefore

Thus hre can assume ( f, the projective

that

= Ct Ia

assumeIt

by (2.2.3).

(1.2.7)

"-r1 given

all

in

the tables

(f

For x = (x.,, xr)€

formula

(s-(r'+1){,)), r foraU.

{o

for

i = 1,2,3').

di),

the Euclidian

graphs, that orj-entation

d1). Let l € f be a sink, c-1 =s1s2.

O2 we get

it

1)x., - d'rax., dt2x1-x2). is

(c-r(t))r>

easy to prove by induction, o for all

r > o,

that

hence c-r !

r:o.

Assume ( f,

d) = ( t 2 ' 42) wittr

(d23' d32) = (2,1\. source.

t

in Appendix C) .

'(1) = (( dztdt2-

Vtith this

, d) = ( f i,

(with the special

rio

d) = ( ft,

Assume(f,

Thus

we can choose a special orientation

we have seen for for

show

is not preinjective.

is enough to prove c-r

(1.2.8)

{ o

we will

d').

some r> O, 1< t< n. Then c-(r+1)1< O

for

(of course also, if

In

c

d) = ( f',

module Fl

Hence it As in

r:O.

Q

is not contained in J also for

P

Q).

l(M,

all

L (M ,0 ) again

in

Then

1€l

(d12'

d21) = (3 ' 1) aPd

beasinkand3€f

a-1 = - 1 s, s, and

bea

/o

-47(x) = (2xt * *2

c

xr, 3x1 * *2 - x3,2x, - x3),

I = (x1 t x2t x3) € e3. now it .

(c

-1

this

x)3 1t


O. Thus

Z ' = s-;r . . . s-]1 z + o . Consequently

(2.3.1)

yields

an exact

sesuence

-54-

O+4,-

sl- ... ^1

si,by -r

and,applying

(2.2.112

Z_ +1.

Comparison of Iast

q

exactness of s1 (i €r )

refr

-X

I

-

!'-Ek

the invoLved

dimension

the

shows that

thus we get the exact

homomorphism must be onto,

sequence

-L

9

If

preprojective,

Y is

-

o

-

we get

X +

I

L-

References:

Most of

$ 1 is

The Coxeter

functorshave

I.M,

valued for

and V.A.

celfand

all

graphs i.

algebraically 5 3 is

(f,

-.j

9.

been defined

Ponomarev in

[1]

vrithoriencation"l,

in

i), field

Ringel

done by C.M.

d)

closed

done by V.

simiJ.arly

where K.

the

I.N.

by

for

quivers

modulation

Rj-ngel in

t251.

Bernstein,

where d..

Furthermore

Dlab and C.M.

in

for [15].

(j-.e. = t

j.s given $ 2 see

= dli by an [11]

-

Chapter

fII

Graphs of

An abelian tion

category

series

A

A

and is

has only

objects, (il,o), if

for

then

of all

is

for

Artin

K',

K'

finite

to the

L(M'o)

denotes the category

type,

generated

exact

to

the

is

free x

assoy.

and if

A

modules over

an

so,

subcategorles

them .

l(M,n)

type,

embedding

one has to consider

also

of those which

L( M,0)

If

said to be of

if

in this

(f,d)

and only

(f,g)

again that

Our purpose in this q'peif

finite

and only

full

typs,

is

neither

tame (repre-

tvpe.

valued graph.

if

(representation)

modules over

finite

in-

K-realization

would be too special;

of

surn of

indecomposable

turo non commuting variables

Assume notr in the following

is of

direct

composi-

(representation)

exact

mod* Kcx,y>

equivalent

nor of wild

sentation)

unique

finite

a full

dirnensional

mod* Kr<x,y>

are representatj_on

is

definition

in addition to

has a finite

said to be of wild

example the category

isomorphic

of

said to be of

where

in

above, this

ring

object

isomorphisms)

there

finite

K'-algebra A

every

number of non isomorphi.c

is

-L(M,A),

over

tame tvpe

be again a valued graph with

sarne field

ciative (For

a finite

L(i,{,0)

mod* Kr<xry>

in which

is

(r,g)

Let

and of

an (up to

decomposable objects, if

finite

55 -

if

chapter (r,g)

is Euclj-dian.

cases more in detail.

is

is

a connected

to show that

L(M,n)

is Dynkin and of tame type

Furthermore we will

describe

-55-

s1

The representation

3.1.1

Theorem: Dynkin.

of Dvnkin graphs

L ( / l ,, ln )

Moreover

is

sable representations (f,g)

(f,d)

lndecomporoots

and the PosLtive

is not Dynkin, then, by

non-isomorphic

preprgective

non-isomorphic

preinjective

f (M,n)

therefore (f,g)

Let

L (M,0)

of

is

of

given by the dim function.

If

Proof:

between the

a bijection

is

there

(r,d)

if

and only

type lf

finite

of

qfldimX=x2o.

(M,0)

modules) in

c

the

is

set

of and

infinite' tyPe'

rePresentation

finite

x € L (M,0)

Let

the set of

(and equivalently

modules

cannot be of

be Dynkin,

Q-2.12)

be indecomposable'

Then

m-

haveorder

--,1

r . l l r f

hence I = o by c' under is invariant E crx € 0t r=O tt*1* ' o (1.2.1). Therefore there is an r > o with "t*' is projective' for some 1 < t < n = ltl crx ! 9t By (2.2.3)

Then

v :=

x=c-tBt

hence L{M,n)

is

is preprojective.

Again by Q.2.12)

L ( l ' ', lo )

preprojective,

nunber of non-isomorphj-c

a finite

module in

Thus every

preprojective.

has only

hence indecom-

posable modules. Now the second assertion

give

we will (f,d) of

will

now some applications

be Dynkin,

K

an infinite

by (1'2'5)'

follow

of

Q.3)

field

and

to Dynkin graphs. (M,o)

a

so let

K.realisation

(f,d) .

3.1.2 Proposition: in

L ( M ,o ) ,

Let

{t,-..,Ia

such that

and

Z

be indecomposable modules

-57-

J

dimZ= Then there

is

a sequence

o = Zo c 21 c tion

n

of

-..

3 x- T r ( t )

t !t-1

there

is

{k1,.

Proof:

Lt,

all

for

...

s kd_t S kd.

ld s 1d_t

, 1 < t < d.,

/2. / lkt

are indecomposable

d ='1,

assertion

the proposltion for

to € {t r...,dl

aLl natural ,

rrithout

dim Z - dim X1 = 1 indecomposable with Then, by

of root

is

system of Dynkin graphs says, as sum of roots, also

trivial.

numbers

O nX

e

Wl

for

weget k+i€

=,t.

-

(xr) € ef .

for xi"., ! f ' ' r. € ( d i m X) . The !,Ieyl group x

,*

factor

if

nE if

{-L g eFlcx

representation

.!. € Qr* i)

For

representation

every

belongs

61=-x1,6i=Xi+dikxk,

vector

{x e Qflc*

to

XX ,=

transformation,

defect

by

of

R (it{,a)

of

0f"

by the

eft

type

R (M,a) )

i.".

Or,

rr X€Q'-.

eQI'

space

we write an

and

(in

respect

of

A regurar

dimension

be denoted

with

basis

the

subcategory

vector

(Xi)

defi.nitions.

series

will

the

natural

x€0'

if

The full

Now consider

the

basic

homogeneous,

representations

wrlte

the

then

there

is

an epimor_

=

-64-

equation

be non homogeneous possessing

E € R(M,o)

Lenrma: Let

3.2.4

L>2

and

i)

c+rE :

o < r < r

c+rE,

and all

E

the

of

orbit Then

be 1'

c

transformation

under the Coxeter

dim E

in

the elements

the number of

Let

n.

an

non-

are mutualry

isomorphic,sirnpleregularnon-honogeneousrePresentatj.ons. nX = O

il)

for

and x7c+e.

x7E,

Extl (C+rE, x) = o

iii)

with

there

Because

only

I

- {,

Let

I

this

q

has to be simple

E g Nf,

din

= n (E) > o,

there

is

that

there

is

E + X,

which must be an j-somorphism, because

By (2.2.2) x I

E, I

nX

(O

implies

we can assume, that

7 C*E

and consider

obvious. by defi-

a monomorphism

nX ) O

simil-ar

is

thus,

nition,

regular;

implies,

the orbit

by (2.2.'t\,

Now i)

representation;

regular

thus

a monomorphisrn

E) = dim E

must be an lsomorphism.

be a simple

!'

elements) '

number of

a finite

of

of an equation for

hence

not homogeneous,

factors'

(as element

q

of

Now dim c+le = cr(S

therefore

iii)

factor

on composition

(CIearIy, because E € R(itl,a)'

n (c+In)

Because s - c+lg.

ii)

additive

so by definition

is

E'

by (1.2.1).

can contain i)

is

n

must be a monomorphj-sm E

regular. I > 2

and

) o,

nX

representations

simpre regular

all

for

must be an composition

R(M,a))

representations

si-mple regular

all

x7c*t*ln-

and

nE ) O

Because

there

= o

Extl (4,c+rs) x7c+rn

Proof:

for

x7c+r+1n-

xlc+rE, iv)

with

X,

representations

regular

sinple

all

I

is

XsC+E.

r = O.

Thus assume

an extension

simple

_ 55 _

-O.

O-I-3-E and

Z_ contains

Now

X7y,

E 3 Y lv)

hence

a subrepresentation

because

Z *E

nX = O,

By ii)

is

the

identity

isomorphic

I

XlE,hence

=nX+nE=nE

nZ

Xfly=O on

E,

to

)O E.

andso

i,e.

the

sequence

splits.

Similar.

Now given is

which

E,

an equation

orbit

of

of

possesses

C+rE.

dim E,

an equation 2 < l- (

If

-

is

!r

observe

the

cardinality

crn

that of

the

put

n Kercrl=KnOg

can be seen directly).

So let for

X € R(M,e).

each Euclidian

(M',4')

of

(M,n1

Now, our graph

proof

and for

defined

will

each

by the

consist ,,

(t)

non zero

,

of

considering,

a contraction components of

n

(t)

,

- 74 -

and in decornposing

R(X)

into

indecornposable representations

of

( M ', a ' ) . /+I n'-'X

that

Note,

i.e.

R(x) ,

(t)x

n of

restriction

are

the

positive that

composition similar, direct the

notation,

contraction F * F),

-

2*-l*

or

llorv if

n(1)

nt5)

contaj.ns rf

n

(1)*

N o \ , ra s i n

for of

a Dynkingraoh,

must

% is

thus

(3.1), that

occur

in

to

the

and these

show,

isomorphic

isomorphic

t

n(t)x the

nonhomogeneous

E(t).

to

c+E(t)

for

some

for

graph

1 -

R(X)

into

the

types

set

p < n).

if

A2 t

representations.

a generating

is

Ep+1)

g(1),

direct of

consider

(wj.th

2

the

the

realizatj-on

sum of direct

type the

there

(o,j),

indecomposable summands are

('l,o).

of

of

I,

T' (yo) )

must be a direct (2.2.1)

n(1)'

we see,

!

is

summand

additve

( 1 , . 1)

and

T'(y-ol

= g(1).

type

Evidently

(namely

must be a direct

because

surnnands v

there proof

o,

isomorphic

to

summand Vl that

and

n(1)u

E of

a o

Thus

( 1)

o

de-

tables.

the

an equation

= n(1)x,

a submodule

the

to

defined

Dynkin

type

a o,

natural

(1,'l).

dimension

those

is

The dimension

{O,1),

the

of

R(X).

is

and decompose

components

denotes

we will

s x

on1y,

to the

= o

T' (yo)

E(2)

(1,o)

n(1)'v

summand

no regular

is

the

dimension

di.agram.

T" (!t)

of

of

their

=

representations,

of

by

the

and

< o,

are

n

11) t)\ = {E'" LZ, E'-'

:

n(1)

of

we refer

There

by

(t)'

a realization

particular

V1

if

,

determined

n(t)x

course,

First,

for

is

R(x),

summand

determined

(M' , e' ) .

to

roots

of

Ar, (n > 2)

%

(t)

one

if

Ä . , . ,, Ä . , r :

(of

= n

(t)'R(x)

summands are

implies,

For

uniquely

(M'rQ')

Furthermore direct

n

is

x

.

type

(1,O).

-75-

+ im19n+1 = x1

imtQ2 module

Ft

would

(x =

be a direct

be regular

(3.2.2

sunmand of

type

(1,O),

preimage

of

fore

the

From this

it

i).

{E = IZ}

ConsiCer

the

dlmension (o' l),

types

(1'1)

must occur

would by

d.

(2,1).

type

hence

(2,'l) .

type

be surjective,

xt

of

set the

with

82.

summands of n x > o,

rf

n x

or

(2,1) .

cannot

be of

would

negative

I

R(I)

2.

of

%

The

(o,1)

type

has a direct must

1q2

(1,O).

X

*.

(.1,O),

are

isomorphic

and

E(2).

,(1,Z'l

As above,

type

Xp+l.

n = 2*-.1

R(X)

< o,

be not

defect

for

a summand

If

be

g+S(1).

!

equatj_on

diagramm

there_

cannot

an equation

(1,o)

T' (v1 )

be a module

is

n(2)

not

has a direct

tqp+1

T"(yj)

3 n.

Vl

If

under

that

direct

T,(yo)

of

sur]ective,

the

or

sununand v1

cannot

shorrs that

of

and

ipZ f

to

would

R(x)

easily,

contraction

X

hand,

a generating

is

and

I,

projective

the

other

thus i*tqz

othen"rj,se

summand of

on the

follows

The same argument Err

(xi,iaj).

So

Vl

to

C+E,

be

is

wour.d be not

of it

regular

(3.2.2).

and

Ecnt

similar

-

/ tl 8D,.,: {E*" = l,

to case

8..

EQl = Frr";

..Fn-2(l

l r" a s'enerarins "n

set

$tith

equations

The proof For

for

E(2)

= 2*-1*

n(l)

E(1)

is

conslder

the

similar

and to

contraction

=

I(2) gn.

case to

2n*-1*.

11,2)

1
O,

The type tive type

a summand (O,1)

defect, (1,1)

would

thus, by

% lead

because

(3.2.2').

of

type

(O,1)

to

a submodule

x

i.s regular,

or of y-

( . 1, t ) X

occurs.

of

posl_

must

be of

- 76 -

Therefore

=t.,

T'(Yo)

o

*o,..+Fp.,.

f.,_Z(

p+2,

ft

^n O +

the modul,e would

o *

...

be a submodule

woul-dn't

of

be regular

_

Fp...".-r(;

by

T,(Vo)

of n hence

,

(3.2.2).

oosirive

of

defecr

and

I,

X

p = 2 and

Therefore

t2l

r'(v^) = E' If

(

n I

O,

( 1 , o ).

of type

T"(Vl)

cDn:

is

to

= -2

n'-' Di--!

F2 ...

r,

p + n-1,

If and

X

is

a direct

then

not

Vl

summand

'o * o ... odo for

"p

the

defect

regular.

of p = n-'l

Hence

BDn

C^ <e

set

*

+n

^i .-t

with

/1\

= -l

nt"'

,

^^-^'der

'i-. r

P4

äö-F: F"FO*.F Inn

a generatj.ng *

have

-

3g+u(2).

f,in t , I' , !n ( 1 ) = 1 r .3 , !F ( 2 ) =

t)\

must

so r" (vl)

negative

r'(v1)

simil_ar

is

R(X)

1 < p < n-1.

some

and

then

tLhr reE

6-0 7 rn_2_ld

...

equations *

F:2* "^^:; id = E ,L "(3)= o,, 3

n(1)

-

. . . . tnr_ 2

O-O /

-1*-2*+3*

*

+n

cu vor nr Ltrraa9cL tr ivol l n

tLoU

tLhr reE udrioaygt dr ranm

the

dimension

orf u

tL-y'lhJ ca

^a3 -

'\

1

/ 2lz In

the

decomposition

direct nF

t.,ri+i-^\.

e! wrrLrrrgl;

rf and

n(l)x

> o,

T,(V^) !

summand vt

1

(using O

^

R(X),

of

summands are

.

O O, ö t,

O

the ^

natural

1

of type

rt I

n ( 1 ) : o,

nx

of

type

GzZt

1.

summand ru

to the Dynkin graph of

the contraction

in

M

tion

Direct

every Euclidian o

and orientation

there

Thus we can formulate

!([,1,n).

3.2.8 Theorem:

for

(r,il)

Let

the product of

H(M,0)

R(t) , d.""tibed

in

calculation

orientation and

ß.2.7) .

shows also:

h

exists

theorem.

graph with n.

uniserial

with

a generating

the following

be an Euclidian

rtt and admissible

(l,g)

graph

Then

K-modula-

R(M,o)

subcategories

is

- 83 -

3.2.9

Coro1lar: h

Let

n

the nunber of

be the number of vertices elements

representations.Then all .

slmple

glven

regular

by the

L "

a generating

in

OShS3

(f,d)

of

set

and

of regular

andthenumber

non homogeneous representations

I

of is

formula

L = n+h_2 The number

h

is

the graph

of If

0a

is

Ärrtn > l)

an orbit

representatlon

vrhere

Let

1-

independent

of

of

a simple

x € l(M,n).

sion type, if

the

length

or

regular of

of

in

the case

2. nonhomogeneous

C.,

then

0+.

T h e n I € l(M,a)

dim X € N.,

except

h = 1

when

under the action

denotes

O

otherwise

is

of continuous of discrete

dimen-

di.mension

type. 3.2.10 Corollar:

The mapping

between all

indecomposable

mension type and all Proof: Iar Ler

dirn : L(M,n1 - qf

By (3.2.1) representations {s(t) llsr

lr

(c,3)

for

map

u , 3

is

T(A')

is

Z J'

easy to see that

ß and

(h11,).

s o c T ( A ) = e -y J

.L (hrl,) L'

of course

X ----+

o

T(c,ß)

I O

t(4)/_^^ ' s o c -r,(^Ar , , = ( E -X S

and

a homomorphism then

T

r

is

exact,

T ( O , G G , O )= I ,

For the proof

of

(orcc,o)

ii)

property,then L(ix,Y])). in

i)

a is and

note that

projecti-ve.

the property in ii). If

so

to the whole of

construction

steps

L !j-:Ly,e,

O o

A = (V",W",ro), where

tations

(in

(f'...-.',

s

induces a c-linear

by resrricrion to soc T(A), rhis induces 3,", and we can go backwards, i.e. T is ful-L.

(the extensj-on of

and

gX__-_-_____)o

J'.I

I O. Obviousl-y

withS=I If

is

(f..,) Xr -r4

O

ü

JtT(a,B)

Wä

J ',lsisr *

"l

I

where

g : Wc -

over G):

----------> gZ------->

O-->T(A)

o ------)

(where

diagram

is

an exact

Lr,(rM"),

i.e.

furl

also

not monomorphic, is

ernbedding;

to represen_

trivial).

By

im T c l(tx,y]). (FF,O,O)

Therefore

u e [({X,y})

T

every

is

injective

erement of

in irn T

is indecomposable satisfying

In(FttO), must have this

a n d V 'ts^o^ c- ,ur i O X forsome J,L€N JiThis means that g c a n b e e m b e d d e di n o z. Now arrsocUIOy

can be reversed

and we get a preimage of

U."

-93-

Renark:

Of course

submodules

modulation f{(l{,a) glven

image of

T

can be deterroined

cNr

be an Euclidian

i{ and admissible

graph not of type

orientation

we can assume of course that in

radical

the tables. of

also

for

proper

or

i'

wlth

of Extl (x,y) .

(f,d)

Let now

the

Let

nf

Q. rn order is

ö

to investigate

one of the orientatlons

be the generatj_ng element of

rrith r resDect respect

Of

i.,.,

to

.Qa. Let r-a+

5

N-,

the

i F L^ the !L^ source -^a fF b. io

E. o

8

(t,d)

if

6

is

of

-7 E^

ö

type

4

-41 F.

'42 'l

G" n

3

G2z

and

,

n.t 1 Notice

that

restricted Ii

= tt,i

otherwise

to

f'.

is

otherwise

short

to

root

the maximal

First,

non negative

for

in

the

of

Now the Lermnafollows

is

by

rirhere di = o

,r^

investigatiän

a root

cases

of

=d

and

shows,

(f',d),

in

fact

Fery

Bn, CDn, Fn1 and GI r

root. I,I

€ Of'

components. This

For the notation

case by case.

f,

(1,,d,)

defined

rhen a direct

restricted

the maximal

a Dynkin graph

= (yi) €or

Let I

(4f = (.t,i)).

3.3.4 Lenrna: I

Proof:

= f

f\{io}

long

say gives

and short

easily

x 2 1, a partial roots

inspecting

if

x _ I

ordering

see for

has on

ef'

example [ 4 | .

the Euclidian

graphs

-

(M',o')

Define the K-realizatlon let

yr = (Yir

y

to

restriction

of

defined by

Yi = Yi

{i,i}cr'g

f

(l',d')

of

!r r where

tyPe

of dimension

Y = (Yrrrar)

(3.1 -1) . Let

l',

and

by restriction

be the up to isomorphism uniquely

e L(M',Q')

i9'r)

rePresentation

deternined

94-

is

L'

the

€ t(M,a)

= O, j(9i = j9r. tf "ro is an inand .t0t= o otherwise. Then of course ! i € f'

if

of

(M,O).

we will

appty

(3.3.3)

= X

injective,

decomposable representation In the

following

Notice

that

F.

-i-

and

I

is

with

because

L(M,n).

F,

, Ye

i

€lisasource

_I

o (2.1.3) .

3.3.5 Lemma: i) ii)

Hom(X,Y)= O = Hom(Y,I). =F

EndX=F. 10P

iii)

Ext'(x,Y)

= ^N-

and

aredi-visionrings-

EndY=G

except if

A..,

is a bimodule of type

( r , g ) l = or" .l n. dö rr.H .t" cNr ," "r .r n" Proof:

i)

factor

of

morphic ii)

Because

Of

!.

End Y 3 End v',

= Ei -o

In particular

image of course

= O, I Yi ^o

it

be a submodule or

an ePi-

Y.

End X = F,^ th"

t""oit

= F

a division

is

for

follows

g Fi for G ancl (2 .2.4) . In fact 'o r | jo a 0' must have the sarne length

iii)

as composition

c a n n o t occur

cannot

{ .,-

!

as

=

; iö*.j

d.,

J'o

y' € of

y;. r

Because

from (3.1.1) '

some j o e f ' ,

By i) and (2.1.4)

d i n ^, , E x r 1- ( x- , y ) = - n Il t x- , v- l

ring.

'.

(2.2.12)

where the root

-9s-

Inspectlng (f ,d)

every

is

= tt^

of

case it

type

iCrr, h"r"

I'

is

of

e Q'

of

gives

this

of

) = 1, therefore EndY=EndYr

Let

This

is

fir.,

T:

type

= ,ro, c'

and But

some O < r € N, therefore

is

the maxirnal

is

the unlque

w €

wn, implies

i.e.

dim,ON) = 1, and

d"=ir"a.

""

tells

us that

there

of course

T

is a furl

is described

(3.3.3

in

we can speak about

exact

is of the same type

rMc

ii).

Because all

modules of

if

A e g*(füC).

and only

if

Then

T(A)

is

A

is

of

continuous

of continuous

continuous

L = dln

ker

(f ,d) of

be not of

continuous

type

4

is

V, - dim ker or) 6örr. fh.r,

dimension

nf = x + I

that

Therefore

pF + (din

type

because

of continuous

if

of

is

of

and only

if

dimension

T

type.

in (3.3.3)

= dirn V".

rüC

(f ,d)

dimension

dimension

Let A = (V,W, q) . Then by the construction dtun ( T A ) = d t u n w c . d i r X rrith ! + L.din

Notice

1

j € fr

l,=w(i),

Proof:

is

I,

(f,d)

type.

type

A

length

except j_f

o

are Eucridian

3.3.6 ternma: Let

Let

if

=

dim(cN)

a rong root.

for

L*(FMG) -!(M,o).

graphs

di.mension

except

the knowredge ot tt"

tro

=F1 . Now f,t=4.fr,

GNF. AIso the funageof

involved

of

hence

11r,4,),

(3.3.3) cNFo. Then

enbedding as

is

Y' = C-rrF,

type

=

#c

as above).

11',g')

Dynkln graph.

simple long root

CNf

NF = 4. Now

now ln every case

6örr. H"."

Is of

= 2

dllr(NF)

,Jo

type

root

that

dln

(jo e f '

uth(Nr),/f*

of

follows

is type

type

dim V, = dim !tO.

not of if

and

4,

type

ftrr.

and onLy if

-

96-

ditn(TA) = dln

W - ( d l n _f + d f u o X ) = d l n W ^ . _nl-- , u

i.e.

lf

contlnuous

(fral)

f4

u.

Is of

iCrr.

fh.r,

type,

if

ls

flC

and only

of

dlmenslon

type

i.,,,

Let

type.

and

of

ls

A

d1n VF = 2 dim WG. But thls

if

and only type

be of

continuous

= dlm WC(+E Y + 2 dtn x) = dim wc.lr,

din(rA)

lf

ls

dimenslon to

equivalent

because here

9r = 2 x + YNotice

that

A€ I-*(E$G)

r(A) ls

€ t(M,o)

lndecomposable,

composable, 1f and only

lf

lndecomposable,

ls

because

resp.

A

End(A) 3 Sna(f(a))

lf

and only

are lnde-

T(A) is

lf

a local

ring.

3 . . 3 . 7L e m m a : T ( f l * ( E J , ! G ) 1c R ( M , n ) .

Proof: Recall in

the

R(M,a)

functor o(t)

there

- up to

+ 1

of

the

in orbits

(3.2.7).

A, direct

(see 2.1.7) extensions.

(1 < t s h)

0(t)

inspection

(o s r(t)

isomorphisms dlrnension

(in

sion ring

non homogeneous representations

regular

is a unigue representatlon

type

R(M,o)),

as composition factor '

slmple

dlvided

n.lt) ) e p(t) r, l

continuous lt

notion

Ct

= (' A l Jt), the

(3.3.6).

Obvious by

H is

and

uniquely with

B(t)

- regular

Then

nxt(e(t) rg(t) ) = i.e. !rHH,

a fu1l

exact

o(t)

subcategory of

closed

be

of

length

occurs just

L11e(t) i) [(M,0)

B(t)

rp.dule

ena(g(t) )

4

orbit

-

= A(t)

and cornposition

4(t)

ß.2.7).

in every

A{ + o. Let ao

determined

socle

E(t)

hrlth

every element of

i.e. of

shows that a+r(t)

< r*) c

under the coxeter

once

is a dlvi= 3(t) under

+

-97-

3.3.8

Theoren: T induces an equivalence. between (1) (h) A *. . .* e x f l ( M , 0 ) = : C ( r r , t , e. )

Proof :

We have to shobr T(H,t(F!Ic))= C(M,a). By (3.3.7), (3.3.6)

and (3'3'3

ii)

it

is

enough to show thatanindecornposabre

ä € R(M,a)

of continuous

if

and only

if

ls

J.somorphic to a dj.rect

dlrnZ ="=

contains

sum of

sum of .

let

copies

"! E fr

sabLe direct

direct

=o

q.

of

itri) ) , hence

hence Z, contains

some simple

I t s t < h),

so every

direct

with

as above,

E

Furthermore every

because jo

restricted

= a"(z) j e r

Z_ is of to

image of regular io

u.

defect W is

and

is

all

Let

zero.

i€r.

submodules of thj-s j-s true

Then

q

is

has negati.ve defect.

-

,o.

öc(l{) *j^

*

w,

c(M,n).

rn particular,

can be seen.

ac(Z/u) - - z. of

regurar,

non homogeneous module

z cannot be contained in q

for

be a indecompo-

E

non homogeneous. But then

regular

Z, but this

such that

19",9'),

ur=i,

forall

In particular

W1 * O, as easily

neishbour

an epimorphic

i.e.

summand of

ö"(u)

by

isomorphictoa

regular,

summands of

nto = O,1w = (W1,

e1a(t)

is

!-

(3.2.2).

defect

s'nmand of

,

. ttoiice that di.m U = J.I o n, = (nr)eqr, because dim z eN.nf.

Z e C(M,n). Because

indecomposabre

bour

z = (zt,iq:1

* = Ei

of

where

Z_ have non positive

ZrU

such that

".a € L(M,n)

,/u"L"

and

C(M,0),

"]

=o,iti

usz

g

of

U = (Ui,iüj)

andU,

isasource,

2, wltnJ=_:,

for

?

-otr^

direct

g

copies

modure

contaj_ned in

g =

a submodule

fQrio+jer

Because io

First

g-

dimension type is

(zil , anddefine

1ti=iei

all

flnhUC)

,nur, is

öc(L). ;r.;, -o

;",

< O. So there

is

O. Note that

!'

of

-o

*j

hence

= o w

is

by (3.2.2),

impossible

always a root

Ler

at

least

neigh_

= dim W

the Dynkin diagrarn

-

98 -

Nou we have to dlstlnguish

11',9').

cases:

several

/

BDrrr Drrr E5' E.7, Eg, F42 and G22.

a) Type

Here y'

dim u = z, - o - - oI

hence

are at most

zi -o

sumnands in

=,,

tro

rmPlies

- Tnis

that

of

g.

into

By maximality

of

I'

all

gher:e

a direct these equals

restrlcted

to

I'

Furthermore

the

first

dim U

Y, because

= 1r

ö"(io)

a decornposltion

modules.

to

must be isomorphic

Also

ac(U) = - z.

and

indecomposabele

sum of

Rr,.

the maximal root of

is

z. . vr. to ör, ana 6örr.

fr,

b) Type Again

is

I'

the maxlmal root

component of every

root

(w' = (wi) € Rf ,),

because

there

just

are

be isomorphic

Here

at most

is

to

and again

U' which

must

Y' € Rf,.

of

by maximality

!

y,

of

summands i.n a decomposition

,j

*.i = t

1, hence

din U - 21

+ O. llow

ri

-

vn-1 3 1

1, and

= 1, because only

ti-r

for

n-1

dim U = 2.,

= 1, hence

JO

JO

roots of

(f',d'),

1. All

first

component

d) Type Again

' I

U. But

*l'=

2. So lt'

all is

of

a decomposition

gSI

root

the maximal short

is

I'

jo = t

in

Rf,

g'n.

c) Type

t.i

in

Rr,.

in

< I'

(f',d').

of

v = (vr)

roots joined

t

are just

= 1, hence

,l_

Therefore

Rr,'

by an edge with

and there tl

Furthermore

")- O = ,1,

1o' Now summands therefore

Jo

which are greater than g'

and therefore

= L',

I'r

have

i.€.

asdesired. io.r. I'

is

the maximal short

.1 = 1, therefore

dim U = zl'

root

of

(f',d').

L. As above

*i

Furthermore = 1, hence there

are

I

-99-

just is

z, only

dlrect one root

has defect e) Type

surulands in greater

zero.

a decomposition

than

U. In

namely the naximal

I, W g y.

As above

of

R, r there root,

which

F4t. that

-Notice

every

root in Rf, greäter than has defect O, where

O + a e A, then !' of

=O}

U

[ UH =, H€ S H+S

a representation

+boer)r{"-le)=(a

to the kernel If

is

Let

homogeneous modules with

by (:.3.1).

H(FMF)

we will

dfun S = n.

with a^

and

3

1 _____-___>2 6, thls (frdr)

so let

we may also (M,O)

proves

be a proper

assume that

restricted

to

-

the

lemma in

(r,A)

subgraph of is

11"4')

of Euclidian

connected. i e f,,

l,et

11',4').

the second case. type.

(M"Q,)

Let

j € I\f,

be

be neighbours.

Conqider

pi € L( l,{', e, ) _c L( /t{,a) . of course (din pr) o, i * "-r so *i = (Qim a-t !i). is arbitrarily large, because = dirn p. -. p r ) dlm C-r *E, ö.(dim nf, where m denotes the order of

c

modulo the radical

By (2.2.4)

End (c-r

= Fi,

so "-r

polnts.

(note that and

Bi

or

di\

depending

= -eltc-t

!i,gj)

Exr(Fj,

c-r

= -"ltgl,

Ei)

on the orientation

respectj-vely.

pi,Fj)

o, and this

We consider

only

so we have to prove (din

,M)

the

dim

acts

"lA centrally K

is

the graph

assume in

(f,4):

to be infinite.

with

are arbitrarily F = Frr

!,Iith

we get a full

exact

representatj.on

a common subfield dj-m and

of

*G

type

of

realization

F and

= g < -.

G, whiihIn parti-

G.

For finite

must be rnodified

than the category

(a,b)

xr

we need some notation.

paragraph

this

now nothing

K

fi

case.

of wird

pM6, and dirn *F = f ( -, contained in the center of F

ments and proofs is

K

is

"i

= dii

first

on

$ 4 we wilr

[ (FMG)

t(rMc)

(dirn Mc) > S. First = b, dim MG = u,

Let

As in

noer' that

fj

dimensions

=extl(c-r pi,Fj) = ll,, and GNF rgc "j, enbedding [,r(FMc) 59. Therefore

Of course

induced

1. Thena>5,

K-codimensi.on

that

is

the F::-orbits

and only

Firstletb=

endornorphism of

= vrhr

v1F, and vrF induces

a subbimodule

u € U, then u'

of

are F-bimodules,

two automorphisms

Consider F\. : = hu € U', So

i.e.

o and

t

U O (v1 + vr)F

of F. S

f\..

- 113-

+vr)

h.(u+v,

=u'+

(v.,+vr)f,=u,

forcjlg o= r.

Similar

and consider

U O (v.,+vliro)

forcing

d = ho, hence

copies

of

f o r s c n r af , e r

l s i < a. Now Iet

forall

F < FMF. Then,

for

a certain

ho€

F,

fr e Fr

+ (v., + vrho)fi = u, + v.,ho + vrhoho

nono - hoho.

F = K'

was arbltrary, of

= u'

+ v., + vrho)

h.(u

=v.ho

h.v.

+v.rho +vrht

Therefore,

must be comnutative,

sorne one dlmensionar

because

and

F-F-bimodule.

ho.

is

FltF

a direct

proves

This

F sum

the

propositlon.

3.5-5

corollarv:

rf

ab > 5, then

t(Fl'rc)

is

of wird

representation

tyPe.

Proof:

F = G = K'

Tf

sum of

copies

proved in

is

of some one dimensional

(3.5.3).

In particular

F = G = K, because

if

fy the assumptions of induction

K

enbedding from some wird

4,8 i) ii)

and with

category if

e L(FMG) vrith the following

true,

it

N^ : = Extl(B,A). ro. l . 1

For, by induction, by (:.3.3) L(Fnc).

this

if

is

into

is

a direct

has been pro

f = S = 1, i.e. let

fMG

prove the corollary enough to find

a full

satis_ by exact

L(FMG). so we are finished

we find

two incongruous

components

properties:

L e t E n d o = F . l. E n d B = G . , , t h e n d i r n Let

Flb

on M. So

we will

Of course

(3.3.3),

is

centrally

ii).

and if

F-bimodule,

this

acts

(:.5.4

over max {f,g}.

by induction

a commutative field,

*F.,,

dim

*G.,

< max{f,g}.

T h e n d i n. . _ FrN .' > - 5 - ..

L ( c N ^ ) i s of wild representation type, and '1 "t t h e r e i s a f u l l e x a c t enbedding from t,r.,*n., ) into

114 -

-

L (X) , $rhere X

purpose consider

For this

componentsof dlrnension type infinite.

By changing

a > b, hence < -1,

sion

of

so

E

End E

then

is

X3

by induction serial

composition )(o

An factors

is

L(fl'tc) i.e.

t (X)

and

all

A'

rhar B.,

und

is

a subfield

bounded.

So we can choose

End -n A

(rxt11e-,A. ^'o "o ) )

din

3.5.6

This

proves

Theorem: orientation

o. Let

a K-realization i) ii)

of finite

non isomorphic

with

order < f

*(End L)

an uni-

downwards),

(3.5.4) . Let in

extension

components in

B(Brr,Ar,) = B(2 Io,n.

-2n,


5. Now set

satisfy

4

=

\, "o

B = B-, "o

the conditions

i)

and

and ii)

the corollary.

(r,d)

Let

n + 1

n

End -o' X , the K-dim'ension of

of

o End An = F.,, then A, B and Fl above.

number

are incongruous

B'

X3€X\ {X1,X2},

L (E!{G) . Proceeding

in

be a non trivial

n > 1. Furthermore

End -n A

(in L(X)),

E

(in this din

of type if

of tame type if

be a connected valued graph utith admissible K

be an infinite

(f,d).

Then

and only and only

if

if

field

and

(M,o)

be

L(M,tl) is (r,q) (f,9)

.

exten-

a non trivial

natural

length

of

ehe K-dimension of nxtl (En,An)

Because is

for

every

is

X

e X. Then

It,Iz

components

for

incongruous

By (3.5.4)

the socle of

In,In_j,...,X1,ä

I"

of

End X2. Furthermore if

of

choosen such

* t {L,...,In} -a bct'(Yn,Xo). Again

is

incongruous

in

Let

there exists

X2. Now X2

are

set

we can assume that

necessary,

max{fr9J = f.

we can construct

object

where

by

a subfield

and

E

if

hence, by (2.1.4)

X1

the

(see (2.1.7) ).

*o

orientation

f > g, L.e.

B(I1,I2)

is

is

a Dynkin graph,

is an Euclidian

graph,

4

- 115 -

and iii)

of wild

type,

Dynkln Finally 1.)

-of

graph

course

it

nor

fleld

and

is

a

graph.

remarks: the

L(M,o)

process in

whlch satisfy

K,NK,

i-s an extension

is

resurts

to the

case,

a K-reari-zation

comes in

(3.5.5)

of

where

same varued

of

----)

(3.5.3).

of

K, and we get

[(M,n).

of course, K,

a full

exact

Here an extension

embedding

field

of

K

play.

some words seem to be in $reen tame and wild

do this

gories

this

cation

of

in

order

categories.

some chance to crassify

all

some speciar

concerning

all

finitely

category

indecomposabre

embeclding.

11,...,rd

it

l_-_->

K-algebras. over

K, let

modL K

M,- e...O

L__j,

"

t{-

(lh

d+2-times 21

acts

as the

(d+2) x

rn fact

For wird

For let C

we

cate-

(d+2)-matrix

by € nod'i R)

R

be

be an

be a full

Define

T : M-

is

wourd suppose the classifi..

S : m o d * K1 . 1 , " 2 > _ _ - _ > C

T : modt .Rc-->

bet-

there

objects.

cases of bimodures.

dimensional

and

the distlnction

For tame categories

seems to be hopeless,

K-algebra generated by abelian

rnay lead to a bimodule

the condltions

fierd

mod* K' 1"1,"2>

where

neither

(f,d).

The induction

will

(f,d)

if

an Euclidian

i-s easy to extend

finlte

graph

3.)

and only

we urant to make some general

K is

2.)

if

exact

-

/"

11 "r=['1

9J":...

['

is

easy to

have in

o

";,

\o

It

fact

see that a full

T

exact

defines

In particular, is

R, so every ring

4.)

of

if

R

an object finite

For simplicity supposed that

isomorphism

dimensionaL

So we

K, then

over

endomorphism ring K-algebra

here only

i-nvolved

birnodules.

ing binodul.

embedding.

occurs

isomorphic

to

as endomorphisrn

in C.

over a common central involved

with

dimensional

we treated all

exact

c.

1s finite on C

sorne object

full

embeddi-nq

ST : mod't R deg p(t)

such that

cn (R.

even Posters of all

for

a polynomial rf

n is

degree

€ I has degree k < n,

Let

I

of degr..

= rr 1t2;+irr(t2).t,p(t)i=r1

than n in

smaller

all

prinitive

ideal

= t2-a,

ft)2-ft2(t2)t,

is

i

hence zero.

a polynomial of

is

hence zero-

I,

over

right C so

the annihilator

ideals

as polynomial

of

ideal R,z,

is

only

Thus either

even or

o + a€lR or p(t)

*iah

Artinian

= t or p(t)

a maximal is

€Rlt2l

Therefore

= (t2-a)(t2-ä)

ring,

r.rords,

other

Let p = p(t)R

of R [t2].

it

R. Then "/,

a simple */-.

of

are maximal.

of R. By the above p(t)

irreducible p(t)

- p(t)i

I,

eBtt2i=

t have non zero coefficients.

< R be a maximal

where P denotes

n in

than

"^"tf.t = ip(t)

even 2irr(t2)

dimensional

finite

consequentlY r., (t2),r2(t2)

n is odd, then 2rn (t2)=ip(t)+p(t)

odd poqters of

only

Let O < k < n-1 be maximal

- p(t)'t

center of R. consider ip(t) (i2=-1 ) . rf

Of course

t).

€I.

o < k < n-l.

arl

'

o + g(t)

Then tp(t)

hence c* € lR for

principal-

R is

I of

ideal

hre see easily,

ej-ther 9(t)

for

= t,

somea€c\lR'

or

-

t2 - a€R.

Let a € G and conslder tz-a

= (t-b) (t-c)

for

U+E = o and bc = -a. of

this

equations

same b,c

we see that

p(t)

= t.

and So = "/p

2.1

is

End* (So) = O.

p(t)

= t--a,

right

ideal

)

O < a €R.

tS")

p(t)

= t2-a,

o > a € R. Then (t2-a)

ideal

and

*rn

= (t2-a) (r2-ä),

a maximal'right simple

=

the -

R/o module.

overJR, and

by the

"/p

the

complex

R is

also rnaximal-

R/"-rnodule

The sinple ". Il.

a € 0\lR.

ideal

"P /Dn - m o d u l e .

So we can paranetrlze R-modules

- simple

determined

is

= n.

ena*

p(t)

over O

\,iä).Ris amaxi.mal

R/f and S. =

ring

ideal

dimensional

Then I = (t -

(2,2)-matrix

Sa has endomorphisrn ring 4.)

also maximal as right

R-module,.one

uniquely

'/o = Y!(2,F.), the

as right

is

over p = (t2-a)RcR,

D

part

cases:

a simple

with

and imaginary

O < a € n.

four

up to lsomorphisn

3.)

the real

Then P = p(t).R D

Assume

€ 0. Thi.s forces

Separating

Sd we have to distinguish 1.)

-

Ilt

Then r=(r2-a)R is

over p = p(t)R

R7, and S. = i"

. M(2,C) and End* (S") = C.

isomorphlsm

classes

of

the

simple

numbers a € iD, whose imaginary

j.s non-negative. Again with

(3.4.3)

we can describe

t-lul

(M =

ncn o na6)

Of course

this

can be interpreted

normal form problern of

(n,m)-matrj.x

now fl(M) and therefore

as solution palrs

of

the

(A,B) over {D

part

th"

- 152-

!,/ith

the

definition

followinq

and (A',

Br)

if

similar, (of

comnlex matrices

of

there

correct

But the

(A.B)

Call (P,Q) of

a pair

is

invertible

such that

size)

PAQ-I = Ai and pEO-1 = E', conjugated

sirnilarity:

d e n o t e s t h e complex

where E,E'

to B,Br resoectively.

matrix

interpretation

matrix

following

seems to b e m o r e

interesting. Note that There

CAC

is

all

= COE Ch

a natural

Hon6 (vn 6 for

O

vector

Now L(l,l) is

0-C-bimodule.

"t

Homn (r*Gnrw6) ) " Honn (vAoAh%.CA,

spaces VC, WC over

the categary

of

where g,

tp' : Vn O

if

are automorphisms

there

Ro'

isomorphism.

C*,

A

O

Ah.

O

(VC, .ryC,o) ,

- WC äre called

nAC

,

O .

triples

all

I{A)

equivalent

q of Vn and p of WC

such that

wc

vcoc%encc

I

I s8 1 0I1

lp I

J

vcoch

isomorphism

;"äT" notation Two

of

similarity

lR-Iinear

are calLed

q:VC'VCandP:Wn

(pt

if

wc

onoc

above this in

is

Horn* (Vn I

transformations

similar

.t

there -

exist

ü,ü'.

equivalent Co*, vc o

regular

WC suchthat

to

the

following

Homn(a0n' I{n)) C%

{

C-linear

:

HomC (ncc, wc) transformations

- 153 -

vcoah

"otn,(11cc,

I

I

I

I

I n"'' I

J

(ncc'P) Homc

J

ut

vc e ch.

wc)

Honn (*0n, wn)

conunutes. Identifylns lrith

the

vC o

resbdctLons

the

following

CalI

a real

of

normal

2)-blocks

has the

if

exlst

there

((2n,

2nl

and

real

nurnbers we get

forrnally

1 < I

cornplex if

< n) of

its

every

partition

into

form

tkt'

^*,

\-on. (2n,

(naturally)

on'\

( "*'

Ttro real

the

!{C)

problem:

form

(1 < k < h,

h0a,

Vn, WC to

(2m, 2n)-matrix

(k,l)-block (2,

and Homn

Cf*

2rn)-matrices formatly

bkl

€ R

)

A,Ar

complex

( 2 m , 2 m )) r n a t r i c e s

are

said

regular p,e

to be O-similar square

matrices

such that

PAQ-1 = g' With

other

words we consider

transforrnations

normal

forms

betereen complex vector

of

real

spaces \,ri.th respect

to complex similarity. By the

above it

i.nto matrix

remains

terms.

to

translate

the

results

for

t (M)

154 -

-

by (:.3.1)

Flrst, only

an (2n, 2m) matrix

n = m+l (preprojective

if

matrix),-

- or if

n = m+1, and dually

So, if

matrix)

one indeconposable

(2n,

if

or m = n+1 (preinjective just

is up to sinilarity

in these cases there

indecomposable matrix

can be indecomposable

one

m - n (honogeneous matrix).

m = n*11 we have to find

just

2m)-matrix.

Let

Eo=

(Ä:)

c,ß €R, and consider the

for

?)

"-=(S

(2(m+1), m)-matrix

IE

'

p-

=

\

'-.'.

i

P*

with

space

In that

order there

to

transformation

a

(n+1)-dimensional { v . ' | , v . ,i , v , v r i ,

. . rw6*1 , w,o.,1i} (i2

{w1,.

prove is

I

an lR-linear

V into

{ w . ,r w 1 i , w r t w r i , . .rv*},

I

I

.--

to lR-bases

respect

{v1,.

II

I\ o ".". '.''o

|

describes

C-vector

\

..

\\ ' " o l

Then

\

^u

'.'.

|

(over lR)

\

I rE ' , . ' . /

andEsB=(ä-l)

.rw6*1}

are

no non-trj-vial

c-vector '..,v*,v.i}

= -

of

an m-dimensional sPace W, and

1) , where

C-bases of

indecomposability

g of

P*,

decomposltion

v and w,

we have

respectively'

to

show

VC = Vö e Vö

,

- 155-

WC = Wö O Wf such that

tp decomposes into

gt.

W".

-

Vt

This

is

gt'

Wt,

Vtt +

:

trivial

for

m = 1r for

without

R-linear

loss

maps

of generality

V'

= V,

hence tp" = O, Wn =,OT'!VI O W"C = rp(-V)-O Wf,, where denotes

aWI

subspace of

'l For m > let

generated

,

the

least

C-vecror

indecomposable.

V be the O-subspace (ü = (O), if

m = 2),

of V generated

by

and fr be the O subspace of W

fhen 6 = el?

by w2,...,r^/m.

(by induction,

i.e.

tp(V) . But 6-[VJ- = WC, hence

W contalning

Wö = tOl and g is

u2,...,vm-.'

of p(v),

the O-closure

: V - fr is

indecomposable

m > 2 , a n d b y + _ h eO - d i r i r e n s i o n s , 1 f r n = 2 ) .

if

Now e(V) = e(V')+e(V"),e(V')5W',9(V',)etv" i.mpli-es O(V)= (p(v) n W') O [a(V) n W,']. Nore that lalgest

p(V).

C-subspace of

Let o = o1 + u,

ut € 9(V) nW', 12 €to(V)nW". Then ui and ui

= *1 * x'

with

€p(v) nwr,

t

fr is

rhe € fi c

A(V)

,

= u,,i + uri €ff , x2 €a(v) flwr'

Now'

W = Wr O lrl" and W', W" are O-subspaces of w, hence = x1 , uri

rli

= x, and u.,i,

Thls irnplies fr c the other that

(ffnw')

inclusion

is rl is

injectlve.

(finW"),

e

being

trivial.

in

fact

,p-1 tfi)

is

g-1

(ff) c v,.

p,

Ofcourse

(Vnv"). i.e.

the whole of v,

O (fifl!{,'),

has full e

By induction

rank,

(o-1 (fi)n v.), we nnay

fr c W'. Again by i-njectivity

Now e(v1),

e ff,

a(v*i)

the JR-subspace of V generated

{v,,vrr92ir...,vm-1rv*-1i,v*} ,p-1rfi1 is

o

(say) fr = finW',

of e we see that

so fi = (ffnw')

S o t p - l t f i l = ( , p - 1( f i ) n V ' )

and, as above, ? = tVnV') assume that

u r i € r p ( V ), h e n c e u 1 , u 2 € f r .

by

, so the G-closure q=lfil so y = e-1fi1

w = e T V fS W ' , i . e . v = v ' , w = w ' .

.

v,

,

of

- 156 -

t

Let

I.

= Pfr

I^

is

indecomposable.

(the

transposed

Next vre want to describe H (M), M = that

O

nOC

the elements

generate using simple 1.)

C%.

we get

=

It

C O C - OC C 6 = 1 @ 1 - i

U = (O,O,p) , tp : lI I M -

Corresponding we get

the

9.

a)

t S" O

+ S.

q O,

O-base

Sa is

a)

of

O < a €lR

to

real

choosen

as

of

Mu r correspondinq as

C-sDaces

to

(p.,

and

the

above):

( a = o):

"\ t -

\o < O or

by

JI.J1

vC = wC = S"

( ""=+ s

be calculated

t , o ( v ,o e . , ) - r o ( v , 6 e r ) ) . i

l*"

b)

}ln can

tv., ,v.,i.....o*,.r*i)

t o l , r . ,e e r ) + o ( v . , & e " ) )

matrices

(where

the lR-bases

"/

l

ß > O :

/.

1ln

I'

I'r--l

az1

\o

t

o

q

1

ß-o

1

I

O,

?

o

1/

ß\

I

/Er I

E\

oul

E1

I

of

158 -

g

To

c)

corresponds

o\ -,)

lt \" that

Notlce

from the all

of a)

left

and the

) := "Jt For

€ r ß : o,

€

by

)

(Note that

't

rf

Q : vc + wc

VC

and

respect

is

a

tp

to

mean b7 irreducible series

etc.

Using

(3.3.1)

uniserial and it.s

(Vö,wü)

pair

are

(9(vö) I !'fö

a subtransformation

of

.

I

matrices

two

map betlteen

Vö , wö

if

and if

respectively,

u € H(M)).

to

lR-linear

\.re say the

WC,

or

/

corresponds

"il

B > o

where either

u"u)

("''

_

\"-l

'.c

given

o:

d,
2)

ä.,

,/2-.....-p\ T. \ / '- p + 1 _ . . . . . . . . - n Q. (x) =

-

t ( x , - x r ) " , where

the

edges

-

i

< p < n+l

I

sumnation

runs

all

through

j

'" '''-t\r, 6---=' 1 ) , lfY

,/'

n -l

\1-1

""'-r'/''

A

"tI"..._,>" =

where riqht

given

N is

of

of

K=center

fixes f

lrl

f

E'-' F O -- . . . O

oo oo-....o

E'-'

to. -.. -o oo o o .- - . . o

o. -. -..o

o-. -. .oF oo o"':

"o

oo o... - -.o

10-....o 1

o o -- - . . o

o 11" "'1

o--.. -.o

o" ".ol oo

o...o-11

u...

- r.9

: oFo" "o

(1)

v f

o. " "'o 1t

FF'.."F

r

{1)

dim C-

o 1 0 .- . ' o oo o......o Z=

ni.,

the

F on

by a field

automorphism

c'

and

UF

"N action

o" ' :"o

F of

(vrhich F).

-184-

if

p



l>

l>

^ts'

ö

N

@NH

r9

e

oooo\ +++ lr'!

lr'l

lrt trt

@NH

e

El

e o

e o

rlt

El

o

e

nl

4

e

e

hc

rd

El

@

hl

o e o

e o

o

e

fd

(E qt

e rd

El . (E)P E--E

lo

El

9^

lkl

lI Ee tl ^le It vY IE l'

o |Il

rd

(E

o

B

IE

(,

El

o

e. o l-. hi9 Ox

"jF El:

E

e |{

rd 6

N

o

El r. r9 :-1 rl

UP -14

t1

eP EIY eEl T

l(E

e

o n lr'l

r{

o

I |

El

"l

e El

le IE

U

I I lo

EJ

lh t' I

o

I

PO

lo lH'

lP.

HP NP

IF (rts

H

N

FI

NN

IE

N

NP

UJN

Nts

N

NH

NN tsP

oPo

PO

POO

oH

n

r,l

PO

lr

. P

H

oPo

N

POO ll (!P

@N

tll NP

HP

otsH

P

POP

NH

NP

-L97-

-41

(L'2')

a -

"

1 -

c

$l)

= 3(2xr- *r)2 + (3xn- 2xr)2 + 2(2xr- 3xr)2 + 6(2*r- *1)2

llr

=1-

2_ 3-2-

trn=o

-a.c =1->1+1-+-4+-2 = F-

M

F -

F-

Fl -

F, wirh [Fl:F] = 2;

dim

c'Et"

o I

ooF1FlFl

FOF FeF FeP FloFl

-1

-1

.r

o 1

I

'1

FA

100-1

1

o 2;\t o T j. 2\1-

o10-1

0

r

C-

lll

E'-'

"+,(1) = A(2) c+ E(2)

.r

u

n

r

v

11210

101-2

t'

L

-z

r

(2)

o1111

=t^

a(1) =

(1)

2222r

I

^4 -oA

n

u

"(2)

'1

r

L

oo211

dim

'1

OFF

(1'1)Fl

( 1)

cr

w

- 1a a

a

0

-

-

lvö

F nr. (2,r)

3 _

4 _

1

olx)

= o(2xT

.r

=1-2-i-4-z;s=(il;

*r)2

+ l(2xs-

*12

5;

+ 2(3x2-

2xr)2 + (3xn- 4xr)2,.

\-/ ä"

=1-+1->-3