Representatio Quiver Infinite Type

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REPRESENTATIONS OF QUIVERS OF INFINITE TYPE

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1973 Math. USSR Izv. 7 749

(http://iopscience.iop.org/0025-5726/7/4/A03)

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Izv. Akad. Nauk SSSR Math. USSR IzvestijaSer. Mat. Tom 37(1973), No. 4 Vol. 7(1973), No. 4

REPRESENTATIONS OF QUIVERS OF INFINITE TYPEUDC 519.4

L. A. NAZAROVA

Abstract. In this paper we describe the quiver representations (see ManuscriptaMath. 6(1972), 71103) which do not contain die problem of reducing a pair of matri-ces by similarity transformations.

The concept of a quiver representation was introduced by P. Gabriel in [ l ] . Inthe terminology of [ l ] , a quiver is a set of points which are connected by (directed)arrows. We say, as in [ l ] , that a quiver representation over a field k is given if toeach point i of the quiver there is assigned a vector space V(., and to each arrow go-ing from the point i to the point / there is assigned a linear mapping a., of Vi intoV.. If each V. is finite-dimensional, the representation is called finite. In this pa-per we will consider only finite representations.

Two quiver representations are called equivalent if there exist vector space iso-morphisms Vi - K(, i = 1, . . . , n, such that ^ = ^. for all i, j .

In [ l ] , the concept of the direct sum of representations and the concept of an in-decomposable representation are introduced in the natural way.

Quiver representations arise naturally from the representations of algebras overfields. We remark that similar questions were also considered in [2] and [3]. It iseasy to see that the problem of describing the representations of a fixed quiver can beinterpreted as a matrix problem in the sense of [4],

Indeed, if we fix bases in the spaces V 1 ? , V n, then to each operator cu.:V.-tV. there corresponds a matrix with entries from the field k. Note that the num-ber of columns of the matrix of . . is equal to the number of rows of the matrix of . (and is equal to the dimension of V.);the matrices of . . and a., have anequal number of rows, and the matrices of .. and a, . an equal number of columns.

It is not difficult to see that if two representations (V., a. ..) and (V., ..) areequivalent, then there exist nonsingular matrices . such that .. = ~ .

750 L. A. NAZAROVA

1.

Representations of this quiver correspond to the problem of reducing one matrixby elementary row and column transformations. It is well known that by means ofsuch transformations any matrix can be reduced to diagonal form with ones and zerosalong the main diagonal.

2.

0Representations of this quiver correspond to the problem of reducing one matrix

by similarity transformations.3.

Here we obtain the well-known problem of a pencil of matrices, which was solvedby Kronecker and Weierstrass (see [?]).

4.2

4

The problem of describing the representations of this quiver can be interpretedas the problem of classifying quadruples of subspaces of a given space. This prob-lem was solved by I. M. Gel'fand and V. A. Ponomarev [6] under the assumption thatthe ground field is algebraically closed. Essentially the same problem (in a some-what different formulation) for an arbitrary field was solved by the author in [7] (seealso [8]).

REPRESENTATIONS OF QUIVERS 751

5.

Here we obtain the classical unsolved problem of reducing a matrix pair by simul-taneous similarity transformations. This example shows that the problem of complete-ly classifying the representations of an arbitrary quiver is, at the present time, hope-less.

In various questions in the theory of representations and matrices (see, for ex-ample, [9], [10], [ll]) one considers the problem of completely classifying representa-tions in those cases where this problem does not contain the above-mentioned problemof reducing a matrix pair. Such a statement is also natural for quiver representations.

In this present paper we give necessary and sufficient conditions for the problemof describing quiver representations not to contain the matrix pair problem, and forsuch quivers we indicate a method for constructing all indecomposable representations.

Note that the above class of quivers contains the class of quivers of finite type,i.e. having a finite number of indecomposable representations .0) Representations ofquivers of finite type are described in [l]. Thus it will suffice for us to describe rep-resentations of quivers of infinite type not containing a matrix pair.

1. In the definition of a quiver appear directed arrows. Actually, the structureof the representations of a quiver is essentially unchanged if in the quiver we changethe directions of any number of arrows to their opposites. We have no natural proofof this fact,(2) but it is easy to see that in all situations considered in this paper thereversal of arrows makes no essential difference, i.e. if in any quiver under consider-ation we change the directions of arrows, the type of the quiver is unchanged. There-fore (like Gabriel in [1]) we will use dashes instead of arrows in the notation for aquiver in most cases.

We will henceforth consider only connected quivers, i.e. those for which any twopoints are connected by some set of arrows. It is evident that the problem of des-cribing the representations of a disconnected quiver splits into several independentproblems.

(') The converse is , of course, not true, as shown by Examples 2, 3 and 4 above.( ') After writing this paper, the author learned that it was proved in [l8j that if two quiv-

ers not containing cycles (the definition will be given later) differ only in the direction of ar-rows, there is a natural one-to-one correspondence between their indecomposable representa-tions.

752 L. A. NAZAROVA

The following theorem is proved in [ l ] :A quiver has a finite number of indecomposable representations if and only if it

is a quiver of one of the following forms:

An 1 2- n 1 n, n> 1;D

n 1' 0 1 n 4 n 3, i > 4 ;

EB 2' 0 1 2;

1

;, 2' 1' 0 1 -2 3;

g 2' 1' 0 ,1 2 3 4.

We will prove the following theorem.

Theorem. There exist the following 6 types of quivers having infinitely many in-decomposable representations such that the problem of describing these representa-tions does not contain the problem of reducing a matrix pair by simultaneous similari-ty transformations:

I.

1 txt 2 cc2 3 n-1 . ,

II.

2,

III.

rv.

l'_!i_OJ!i_

V.

REPRESENTATIONS OF QUIVERS

(n-1)

753

3'-2-2'-' -2-b-2-l-2i-2-2i-3

VI.

k- - k

2' " 2 ' a i Q t j . 2 3 g< 4 "

In this section we will prove that the problem of describing representations ofquivers of infinite type not belonging to IIV is as difficult as the problem of reducinga matrix pair.

Following [ l] , [7] and [12], a quiver of type I will be called a cycle, the quiver

a triad, and the quiver

1 0 3

a tetrad.

Lemma 1. The problem of describing the quiver representations of

754 L. A. NAZAROVA

0contains a matrix pair.

Indeed, to this quiver corresponds the following matrix problem: two matricesa. and o. are given, where a t is square and a 2 has as many rows as a.^; a^ ad-mits similarity transformations, and when we perform some transformation on the rowsof OLj, we must perform the same transformation on the rows of a 2 ; the columns ofa j admit any nonsingular transformations. Let

Then it is not difficult to see that and admit only simultaneous similarity trans-formations. This proves the lemma.

Lemma 2. // a (connected) quiver contains a cycle of any length and at least oneadditional dash, the problem of describing the representations of this quiver containsa matrix pair.

Suppose we have

Put j = a2 = = a._ j = a. j = = a^ = E. It is easy to see that the matricesa . and form the problem of Lemma 1. This proves the lemma.

In view of Lemma 2, we can now consider quivers which do not contain cycles.For this purpose we need the concept [4] of a representation of a partially orderedset and its connection with quiver representations.

Let 5R = |flj, . . ., an j be a finite partially ordered set.

By a representation of 51 over a field k we mean a mapping which assigns to


each ai a (finite-dimensional) matrix Ai over k, where all of the A{ have the samenumber of rows.(3)

Two representations \A{ \ and \B i \ are called similar if the corresponding sets ofmatrices can be obtained from one another by a set of elementary transformations ofthe following form:

1) elementary transformations of the columns of any A r,2) elementary row transformations on all matrices of a given set simultaneously;3) addition of columns of A{ to columns of A . if a{ < a. in the sense of the or-

dering in 3.The concepts of indecomposable representation and direct sum are introduced in

the nat.ural way (see [4]).By the width of a partially ordered set 3t we mean the maximal number of its ele-

ments which are mutually noncomparable.For sets of width 3 there was suggested in [4] an algorithm for solving the re-

sulting matrix problems, the algorithm permitting either an increase or a decrease inthe width of the set under consideration in a finite number of steps. The algorithmconsists of the following.

Let A = \A \ be a set of m-rowed matrices defining a representation of a partial-ly ordered set 3 = |a

1 ? , a^) of width 3. From 31 we construct a set 31^, where ais a maximal element of 3t. The set 31^ consists of:

1) all elements of 9t except a;2) the two-element subsets ( , a() such that a, a and a and > , then > (, );4) (, ) < (, ) if each of , is less than one of , .It is proved in [4] that there is a natural one-to-one correspondence between all

indecomposable representations of " and all indecomposable representations of 5except for a finite number of representations of 31 whose form is indicated. Hence ifwe know all representations of 3l

a, we can find all representations of 31. The opera-

tion of passing from 31 to 31^ will be called differentiation with respect to the maxi-mal element a.

It is essential for us to note that although the indicated algorithm applies onlyto sets of width 3, it carries over without any changes to sets of any width if the setof elements not comparable to a (with respect to which we differentiate) has width 2.

It turns out, in analogy with the above-mentioned differentiation with respect to

(3) Some of the A . may be "empty", i.e. matrices may actually be assigned to only someof the elements of it. If each A . is nonempty, the representation is called exact.

756 L. A. NAZAROVA

a maximal element, that it is also possible to differentiate with respect to a minimalelement. This way of differentiating was suggested by A. G. Zavadskil. We will ex-plain the basic idea behind his method.

Let


a, aa .

0

0

Then it is not difficult to see that the matrices and form a representation of thequiver

Qand the assertion of the lemma follows from Lemma 1.

Lemma 4. // even one dash is added to a tetrad, the problem of describing therepresentations of the resulting quiver contains a matrix pair.

Indeed, the possible cases are:

2

1) 12 3

0 3; 2) 1 0 4; 3) 1

2

-0- 5.

In case 1) our assertion follows from Lemma 1; in case 2), from [10]. In case 3)we reduce the matrix a 5 to the form a 5 = (E|0). We then obtain the matrix problemconsidered in Lemma 3. This proves Lemma 4.

Lemma 5. The problem of describing the representations of the quiver

1" 2'

! ' gi

contains a matrix pair.

We have the matrix problem

1

758 L. A. NAZAROVA

Let

We obtain the representations of the quiver 3) from Lemma 4. Lemma 5 follows fromLemma 4.

Lemma 6. // a quiver contains two triads, it has the form III indicated in the theo-rem.

Indeed, it is not difficult to see that if a quiver contains two triads and does nothave the form III, then, putting certain matrices equal to , we reduce the problem tothat of the representations of the quiver considered in Lemma 5.

Lemma 6 immediately implies

Lemma 7. A quiver whose representations do not contain a matrix pair can con-tain at most two triads.

We now consider quivers containing neither cycles nor tetrads and containing atmost one triad.

Lemma 8. The problem of describing the representations of a quiver obtainedfrom

by adding any dash contains a matrix pair.

If a dash is added to 1, l ' or 1", this would contradict Lemma 6; if to 0, Lemma 4.Suppose a dash a , is added to the point 2 (the points 2 and 2 are analogous

to 2). Put

Then the representation problem of the quiver under consideration reduces to that ofthe partially ordered set


In accordance with [4], differentiate 21 with respect to the point a 2 . The set 2I^2contains the subset \a j , 6 2 , c 2 , c } , (>1? Cj)|, the order relation on which is the sameas on the set considered in Lemma 3 (namely, c 2 < c , and the remaining elementsare mutually noncomparable). Hence by Lemma 3 the set 21^

2 contains a matrix pair,and by the results of [4] the set 21 also contains a matrix pair. This proves the lem-ma.

Lemma 9. The problem of describing the representations of a quiver obtainedfrom

3' -- 2' 1' 0 1 2 3 , , .. .

by adding even one dash contains a matrix pair.

If a dash is added to 1 , our assertion follows from Lemma 8; if to 1 , 1, 2 or 2,from Lemma 6.

Clearly, the situations are analogous whether we add a dash to 3 or 3'. Supposefor definiteness, that we add a dash a 4 to the point 3. Reduce the matrices

/|0 \ / |0 | 0 | 0I j, = ( | 0 ) , , ^ 0 ] . , , = 0 | [ | 0 ),

| / \ | | | 0 | | 0

where the vertical and horizontal divisions in matrices having the same number ofrows and columns and admitting only simultaneous transformations are the same. Af-ter the indicated reductions, the columns of a.j are separated into 4 vertical bands;the columns of d p into 3. As a result we obtain the representation problem of thepartially ordered set

So = {a, bi^bz^i b3, cx

760 L. A. NAZAROVA

1by adding even one dash contains a matrix pair.

If a dash is added to l ' , 1, 2, 3 or 4, our assertion follows from Lemma 6; if to1", from Lemma 8; if to 2', from Lemma 9.

Suppose we add a dash a 6 to the point 5. Reducing the matrices a'2, a 5 , a 4 ,a^ and a 2 as in the preceding lemma, we obtain

/ f o | O | o \a4 = [ | | ,

\o I o | | /

[E I 0 1 0 I 0

3 =

0o

00

000

0

00

000

f | 0 1 0 | 0 [ 0 1 0

a, =

000

00

0

0

00

000

000

o | o | o | | | ^ = (10).

Then the columns of a j are separated in connection with the partition of the columnsof 2 into 6 vertical bands; and the columns of a' j, into 2. The problem is reducedto the representations of the partially ordered set

= {a, bx < b2, ^ < c2 < c3 e } .

Differentiate this set with respect to the point b^. Consider in &j,2 the subset

X = {bv c3, c4, c5, ce, a, {a, cx), (a, ca)}.

It is easy to see that the order relation on X is:

C6, a < (fl, C (fl,

Thus, in the terminology of [4], X has type (1, 3, 4) and is isomorphic to the set considered in the preceding lemma. Hence ^ , and EJ,2 and S contain a matrixpair. This proves the lemma.

Using the lemmas we have proved, we can conclude that a quiver whose repre-sentation problem does not contain the matrix pair problem either has one of the types


An, D^, Eg, E7 or E g listed in Gabriel's theorem, or one of the types IVI of our

theorem.Indeed, if a quiver contains a cycle, then by Lemma 2 it coincides with this cy-

cle (i.e. has the form I). If a quiver contains a tetrad, then by Lemma 4 it coincideswith it (i.e. has the form II). If a quiver contains at least two triads, then by Lemmas6 and 7 it has the form III. And if a quiver contains no cycles or tetrads, at most onetriad and no subquiver of a type listed in Lemmas 810, it is easy to see that it canhave one of the forms A

n, D

n, E f i, E ? , E g or I-VI of our theorem.

Hence the theorem is proved.

2. VCe now turn to the classification of the indecomposable representationsof quivers of types IVI.

A quiver representation is called faithful if:1) V. 4 W for any i;2) a{.: Vi -> V. is a noninvertible operator (i.e. singular matrix) for all / and ;'.The meaning of this definition is as follows: if a nonfaithful indecomposable

representation does not satisfy 1), then it is a representation of some subquiver. Nowsuppose that a representation does not satisfy 2). In this case it can be viewed asa representation of the quotient quiver with respect to the set of dashes which cor-respond to nonsingular matrices.

Let be a quiver, and a subset of the set of all dashes of .By the quotient quiver of with respect to we mean the quiver obtained

from in the following way:a) dashes belonging to are discarded;b) any two points which were connected in by a dash belonging to are iden-

tified, i.e. merged into one.It is easy to see that if a quiver has one of the forms IVI, then any of its sub-

quivers and quotient quivers either also has one of the forms IVI or is a quiver ofone of the forms A , D , E^, E 7 or Eg, whose representations are described in [lJ.Therefore it will suffice for us to describe only the faithful indecomposable represen-tations of the quivers IVI.

I. Consider an exact indecomposable representation of the cycle( 4 )

(4) It is easy Co see that the following discussion is unaffected by changes in the direc-cion of arrows.

After writing this paper, the author learned that the classification of cycle representationwas given in [l5j.

762 L. A. NAZAROVA

Reduce the matrix cij to the form

(E\0a, = \

0 | 0We will now reduce the matrix a 2 in such a way that the form of j is preserved.Since the columns of a j are " l inked" with the rows of 2, we must partition therows of 2 into two kinds corresponding to the partition of the columns of a.j . Hencewe can reduce by means of arbitrary column transformations, arbitrary transformationsof rows belonging to one horizontal band, and additions of rows of the upper horizon-tal band to those of the lower. In other words, we obtain (to within transposition) therepresentation problem of the linearly ordered set of two elements. We will solve thisproblem, i.e. we will reduce a 2 and then reduce a , in such a way that


corresponding to the above form of the operators .. Consider the set consistingof all e\. We say that two elements e\ and e* are connected if one of them is theimage of the other under the action of one of the ..

From what has been said above it is clear that each element of the set is con-nected to at most two elements. On the other hand, it follows from the indecompos-ability of the original cycle representation that must be connected. Hence cancontain at most two elements each of which is connected to only one element. In oth-er words, among all the rows and columns of the matrices a{ there can be at most twozero vectors. Consequently, if were greater than 2, then at least one of the .would be nonsingular, contradicting the faithfulness of the representation.

Hence only cycles of length 1 and 2, considered in examples 2 and 3, possessfaithful indecomposable representations. Their indecomposable representations, asobserved in the Introduction (Examples 2 and 3) are well known (see [5], [3]).

II. Consider the representations of a tetrad, i.e. the quiver

As we observed in the Introduction, the representations of this quiver were describedby I. M. Gel'fand and V. A. Ponomarev in [6] under the assumption that the groundfield is algebraically closed. We will show that the classification of the representa-tions of this quiver (with no restrictions on the ground field) also follows directlyfrom the results of [7], and we will give, using [7], the explicit form of these repre-sentations, which we will need later.

Reduce in the following way the matrices

000

0

0

0

0

0

0

0

00

0

1470000

0000

00

000

000

2580000

00000

0

369

0

0

Excluding from consideration the representations ? = (1), 4 = (1), we obtain thatthe unreduced cells 19 form the matrix problem solved in [7], p. 1362 = 1306. Henceby substituting for the cells 1-9 the solutions of this matrix problem we obtain from[7] all indecomposable representations of a tetrad.

764 L. A. NAZAROVA

Let us introduce the following notation. Denote by the identity matrix ofdimension ;

V0 En-J

is a nonsingular matrix in Frobenius normal form whose characteristicpolynomial is irreducible or a power of an irreducible polynomial, except for fit) -= (1 - t)n.

]n is a matrix in Jordan normal form with characteristic polynomial /(t) = (1 tY

In this notation the indecomposable representations of a tetrad have the forms(E

o and E'

o are "empty" matrices):

(En\ /O

1 ) is obtained from 1) by the exchange a ? -+ a 4 ; ( 5 )

2 ) is obtained from 2) by the exchange aj - a2;

3 ) is obtained from 3) by the exchange a, > a 4 ;3") is obtained from 3) by the exchange cij - a 2 ;3 ) is obtained from 3) by the exchanges


4)/.\ / 0 \ /* \ /\,\

4') is obtained from 4) by the exchange al - a 2 ;5)

. n>

5 ) is obtained from 5) by the exchange a ? - a 4;6)

6 ) is obtained from 6) by the exchange a, *-* a 4 ;6") is obtained from 6) by the exchange j -+ a 2 ;6 ) is obtained from 6) by the exchanges aj - a 2 and a^7)

0\ /'n\ (On

7')0

U J a ^ U J - a < = U J 8)

o i \ / o i \ / o \ / i o . . . o \0 , 0 , = 0 0 , a 3 = ^ + 1 > a 4 =

0 0/ \ n 0/ \J+I/ V En

8 ) is obtained from 8) by the exchange a , - a 4 ;

/ 0 IX /0 IX /Oa 1 = O'l, et,= 0 O l . a , - ^ 1 , 0 4 = *! , n > 0 ;

VO 0/ \ n + 1 0/ \ n + 1 / W

9') is obtained from 9) by the exchange a.j a 2 ;

766 L. A. NAZAROVA

10)

En \,

En

rVo"E

n

E~n

0

0

00

'\ 4 =

10 ) is obtained from 10) by the exchange a , > a 4 ;11)

/,.. G/H

- \\

(3 = \ En

A.

En

0

1\0

b01

11') is obtained from 11) by the exchange12)

a 2 ;

= I , a4 = I Jn\

13)

= I 0 I , 3 = \E,0 . 0 . =^0

0

b10

,n>0;

( 6 ) When = 0 we consider 10) to be a representation of the form a? =(1), 10 ) the forma 4 = ( l ) , 11) the form a 2 = ( l ) , and l l ' ) the form a 1 = ( l ) .

REPRESENTATIONS OF QUIVERS 76714)

0

E~n

En

0

0E~n0

""

1\

~o~

1

"

"

"

[ 1 0 . .

Jn

0

. 0 00

101

15)

0 0.1 0

00

0En0

10

I o00

00

00

0000

10000

3 =

0F3+1+

00

10 . . . 000

, =

10 . . . 0EnEn00

000

p3

En+i

16)

, =

0

000

000

+1

0

10000

00

+

00

0000

1 )0000

0F3

+

00

0 . . . 0 100

+\

+

(X 4 =

1 0 . . .p*

En+i00

0 000

En+iEn+i

768 L. A. NAZAROVA

17)

000

00En0

Cl =

En

En000

00En

El0

00001

18)

a, =

a, =

En

0000

00En00

0En000

000

0

a, =

En

En

0

0

0

0

0En

El

0

10

00010

61

0E

n

000

000E

n

o

En

En

00

I o

0

0

p\F*C-n0

00

61001

E1

p 2

0

0

0

00En

En

0

0000

61 )

n>0.(7)

The representations of a tetrad can, of course, also be viewed as the represen-tations of a partially ordered set {flj, a2, a^,


set S over a field k is defined if there is given a finite-dimensional vector space Vover k and to each point i S there is assigned a subspace Vi of V in such a waythat V . C V. whenever i < /'.

To obtain a matrix representation as defined earlier from a representation in thislatter sense we choose a basis of V, and then for each i we consider Vi -

u =j>14 =^>M = 1;P = 2 ' : f t , = p

u =

ft4= 1; P = 9 : ^ > = ^ = ; = 3 : p14 = fe = 1; = 9 ' : Pa ^ = p 2 3 = l ; - 3' : ^ =p 2 4 = 1; = 10 :p1 2 = p14 = p2 4 - 1; - 3" : p 2 4 = pa = 1;

P =

1 0'

: Pa = Pi3= A = 1;

_ 3- D __ , _ ^ = 11 : 8 = As = A* = !

p . - - - - 11 : /J12 = p 2 3 = p 2 4 = 1;P = 4 : p 2 3 = p 2 4 = p 3 4 = l ;

^ Pit P24 P34 l - ,

P = 5' :p 1 3 = p23 = p 3 4 = l ; ^ = 14: _plf =_pu = J ; _ _ = 6: p 2 3 = p23 = 1; = 15 ^ = = p = Pi* = Pw = Pst = ^

= '

: P u = ~ = l; = 16: A2 = P13 = P23 = Pu = P24 = P34= U _ c-. __ _ 1'. = 17: p 1 2 = p 1 3 = p 2 3 = p 1 4 = p 2 4 = p M = 1; L " ~ , = 1 8 :

= 3 = = 4 = =

= 1 - 6": pM = p M = 1; = 7 : ,/ = p t/ = 0;We now consider a partially ordered set ?I to whose representations, as we will

see later, the representations of the quivers IIIVI will reduce.We will henceforth depict partially ordered sets graphically, with one point less

( 8 ) Of course, any matrix representation of a partially ordered set can be obtained in sucha fashion, if merely the columns of the matrix . are linearly independent for each point i.This restriction is not essential, since any matrix representation differs from the above by afew trivial direct summands.

770 L. A. NAZAROVA

than another if it is joined to the latter by an ascending sequence of dashes:

a-

Lemma 11. // a point of a partially ordered set S is not comparable only witha linearly ordered subset = 1 < 2 < < < i o/ S, then any indecomposable rep-resentation of S which is faithful at has the form = (1); = (1), a = (1) (1 < i x | . Let be a representation of S.By row transformations we produce the maximal number of simultaneous zero rows inall matrices corresponding to the elements b.. Divide the rows of all the other ma-trices of into two horizontal parts, the lower corresponding to the zero rows of B,the upper to the nondegenerate rows of B. By appropriate column additions we pro-duce in the matrix X and the matrices of C all zeros in the upper band. We then re-duce the rest of X to the form

o | oAfter this the matrices A{ are split into three horizontal bands. These bands obvi-ously form a linearly ordered set, since by adding the middle band to the upper we canrestore the previous form of X, adding a suitable linear combination of the columns ofthe matrices of B.

Since X is a linearly ordered set, it is not difficult to see that by the indicatedtransformations we can, while preserving the reduced form of the matrices of andX, reduce the matrix A = (A ^\A

2 | . - .\A ) in such a way that each row and each col-umn will contain at most one one, the remaining elements being zero. From this weobtain the assertion of the lemma.

A point of a partially ordered set will be called trivial if it satisfies the hypoth-esis of Lemma 11.

It follows from Lemma 11 that in studying representations of partially orderedsets we need only consider sets which do not contain trivial points. By a set S de-


rived from a set S we will understand a partially ordered set obtained from S by dif-ferentiating with respect to some point and deleting the trivial points from the set ob-tained.

Consider the successive differentiations of 21 (each time the little circle indi-cates the point with respect to which the differentiation is carried out):

Hence 2I ( 7 ) ~ 2.Let be a representation of the set 21. If is faithful at the encircled point of

21, on differentiating we obtain a representation T' which is either a representationof 21 or one of the trivial representations by which 21 and 21' differ (see [4]). Inthe latter case, is lost after a finite number of differentiations. In the first case,if is faithful at the encircled point of 21, on differentiating with respect to this

772 L. A. NAZAROVA

point we obtain a representation T" which in the nontrivial case is a representationof SI". Since the dimension of is finite, we can conclude that after a finite numberof steps turns into a representation of one of the sets 2Il , i < 6, which is nonfairhfulat the encircled point. It is not difficult to see that if we delete the encircled pointfrom each set 5 , t: = 1, , 6, and perform successive differentiations with the newset 55 (beginning with another point), we obtain for each such set the same picture asfor SI (i.e. we obtain 8 7 ~ 8 " ' ) . The same thing happens if we perform a series ofdifferentiations with the set obtained by deleting any point of 1 , and so on. In-deed, on deleting the encircled point of SI, SI', SI", SI^4' or SI we again obtain oneof the sets of the same type. After deleting the encircled point of SI ( 3 ) or 2I ( 6 ) weobtain a set with "upper part" :

{av ,, ,,, 4, bv bs, b3, bx >a1,b1^ a2, 62 > a2, b2 > a,, bs > a,, b3 > a4}.

We differentiate this set with respect to i>2 and so on.It follows from what has been said above that after a finite number of differenti-

ations any representation of ?I vanishes or turns into a representation of a tetrad(at a tetrad the process ends, since it is impossible to differentiate a tetradthere isno point such that the set of points not comparable to it has width 2, which saysessentially that a tetrad is the only "nondifferentiable" subset of the set SI and allits derivatives).

For our purposes the following lemma is essential.

Lemma 12. An indecomposable representation of the set SI which is faithful atthe points al, a2, a ? , aA remains faithful at these points after all differentiations,carried out in the order indicated above.

Indeed, let X be a faithful indecomposable representation of a set S, where Sis a subset of 21 or of one of its derivatives containing the points e l f a2, a^, 4 Suppose that X' ceases to be faithful at one of the a.. From the matrix interpreta-tion of differentiation it is not difficult to see that the set 5 obtained from S by de-leting a. must have a faithful indecomposable representation (which is constructedfrom X by deleting the matrix of a. and those rows containing two ones, one ofwhich belongs to aj. It is also easy to see that if we delete from SI or from one ofits derivatives the point a. of the tetrad, we obtain a set of finite type. Consequent-ly S is also a set of finite type; hence it is one of the faithful sets listed in [14],

Thus it suffices to show that if S is a subset obtained from a faithful S con-taining three points of a tetrad by adding the fourth point a., then in S one need nev-er differentiate with respect to a point not comparable to a.. If S has the form(1, 1, 2), (1, 2, 2), (1, 2, 3) or (1, 2, 4), this is evident, since the added point

.

Editor's note. A faithful set is one that has a faithful indecomposable representation.


must be comparable to all points not belonging to the tetrad (otherwise 5 would con-tain (1, 1, 1, 2)). If

S = { au bx > a^, b2 > a2, b3 > a4},then the added point a, could be comparable as follows:

i. e. we would obtain.the set

ia1,b

1>a2, b2> a2, fr2>a3, f? 3> %, b3>a4},

which, according to our hypothesis, should be differentiated with respect to b2 (notby, or else the lemma would be violated). The other points of the set are handled anal-ogously.

The lemma implies that any indecomposable representation of 2 which isfaithful at the points of a tetrad reduces in a finite number of steps to an indecompos-able representation of the tetrad, and the latter is uniquely determined if we stipu-late the points with respect to which we differentiate. Representations which are non-faithful on a tetrad are easily described, since any such representation is a faithfulrepresentation of some faithful (in the sense of [14]) subset of ?I.

Let X be a representation of an arbitrary partially ordered set S, and X' the cor-responding representation of S , where S is the derivative with respect to a maximalpoint a. In this situation it is essential to distinguish two cases:

1) X is not faithful at all "new" points (i.e. at all points contained in S butnot in S).

2) X' is faithful at some "new" point.Now assume that U is an indecomposable representation of S which is not

faithful at all "new" points (i.e. which is, in essence, a representation of the setS\a), defined by a set of subspaces Vi of some space V. In this situation it is easyto indicate an explicit form of an indecomposable representation X such that X' = U.

Indeed, from the matrix construction corresponding to differentiation [4] it fol-lows that X is obtained from U by adding to the existing set of subspaces the sub-space V

a = V. The representation X will be faithful at the point a if ^i

774 L. A. NAZAROVA

which are faithful on a tetrad. Let MQ denote the set of those representations in each of which can be converted into a representation of a tetrad by a finite number ofdifferentiations corresponding to case 1). It follows from what has been said abovethat if 5 is an arbitrary set containing a tetrad jflj, a 2, &y ^ 4 ! and is a represen-tation of S which reduces to a representation of this tetrad after a finite number ofdifferentiations corresponding to case 1), then can be recovered from = (Vj, V2,Vy V4! if we put Vi = V for i ^ 1, 2, 3, 4. Thus, for example, from the representa-tion of the tetrad we can construct a representation, belonging to MQ, of the set

! = {1, 2, 3, 4, (1, 2); (1, 2 ) > 1, (1, 2)>2>,

if pl2 1 0, or the set

8*3.34 = {1, 2, 3, 4, (2, 3), (3, 4); (2, 3 ) > 2, (2, 3) > 3, (3, 4 ) > 3, (3,4) > 4},

if p2i 4 0, p^^4 0, and so on.Taking the above into account, it is easy to show that any indecomposable repre-

sentation of MQ is a representation of a set isomorphic to one of the setsS31)S92= {1,2,3,4, (1,2), (2,3); (2,3) > 2 , (2,3)^3, (1 ,2)^1, (1,2)3*2),S3={1,2,3,4, (1,2), (3,4); (1,2)^1; (1,2)^2, (3,4)^3. (3,4)^4),

$4={1,2,3,4, (1,2), (2,3), (3,4);(1,2)^1, (1,2)^2, (2,3)5*2, (2,3)^3, (3,4)^3, (3,4)^4),

and the tetradal representations corresponding to the sets 8 j , 332, 8^, 4 havethe following forms:

For pP={\0, 10', 11, II ' , 13, 14, 17, 18},

t h e c o r r e s p o n d i n g r e p r e s e n t a t i o n s o f B j b e i n g d e n o t e d b y 1 0 1 2 , 1 0 ' 1 2 , 1 1 1 2 11 ' t 2 >1 3 1 2 , 1 4 1 2 , 1 7 1 2 a n d 1 8 1 2 .

F o r O 2 ,

P = {10', 11', 17, 18},

1 812,23

the corresponding r e p r e s e n t a t i o n s of * 2 denoted by 1 ^ 2 2 j , 11 j2 23 ^ 1 2 23

, 2 3 For 83,

P={\7,\8},the corresponding representations of S, being denoted by 17 1 2 j 4 and 1 8 1 2 ^ 4.

For S 4 ,^={17, 18},

the corresponding r e p r e s e n t a t i o n s of 33 4 be ing denoted by 1 7 1 2 2 } J 4 and 1 8 1 2 2 3 ?


Let j denote the subset of consisting of those representations which do notbelong to MQ, but which are converted to MQ after one differentiation.

If X e M j , then X e /ô> a n d ^ iS âltâ^ a t some "new" point. Assume firstthat X' is a faithful representation of 8 j . Then it is clear that the point (1, 2) is"new" and that X was a representation of the set

M = {1,2, 3, 4, (3,4), (3,4)2*3, (3,4)2*4},and for the tetradal representation corresponding to X we must have p 3 4 ^ 0. Hencefrom the representations of 8 j only 17 1 2 and 18 j 2 can occur as X , and each ofthem can be obtained by differentiating a faithful representation of 8 } 4 . We denotethe corresponding representations X eMj by ^34(12) a n d ^34(12)

Now suppose X' is a representation of 8 2 . Then the "new" point can be (12)or (23). Assuming that it is (12) (without loss of generality), we obtain as X the re-presentations 1 7 3 4 ) 2 3 ( l 2 ) a n d 1 834,23(12)'

Representations of 8 3 cannot occur in the role of X . Indeed, if (12) were a"new" point in 8 ? , then X could be a representation of the set

{1, 2, 3, 4, (3, 4), (3/1)(3, 4 3 , (3, 4)>4, (3?4)>(3, 4)),

which is impossible, since then the corresponding tetradal representation would have/ > 3 4 > 1 . ^

If X' is a representation of S 4 , the " n e w " point can only be (23). We obtainthe representations 17

1 4 ( 1 2 3 4 ( 2 3 ) a n d 1814,12,34(23)*The set 8 , 4 is isomorphic to 8 j , the set

823,34= {1,2, 3,4, (23), (34), (23)2*2, (23)2*3, (34)2*3, (34)^4}is isomorphic to 8^, the set

4.2.34= 0 , 2 , 3, 4, (12), (14), (34),(12)2*1,(12)2*2,(14)2*1,(14)2*4. (34)2*3, (34)^4}

is isomorphic to 5 = { 1 . 2, 3, 4, (12), (23), (14),

(2)2*1. (12) ^ 2 , (23)2*2, (23)2*3, (14)2*1, (14)2*4},an'd 8 5 is isomorphic to 8^.

Thus we may assume that each representation of Mj is a representation of oneof the sets 8 j , 8 2 or 8,., and we denote the corresponding representations of S jand 8 2 by 1 7 1 2 ( J 4 ) , 1 8 1 2 ( } 4 ) , 1 7 1 2 > 2 3 ( 3 4 ) , 1 8 1 2 > 2 3 ( 3 4 ) ,

1 7 i 2 i 23.14(34) a n d1 8 1 2 2 3 14(34). These representations can be reduced to the following forms:

776 L. A. NAZAROVA

17(34)

01

J_0

6U

= (Alt A A3, ) =bj_

bl i

1712.23(34) 1 = (, 3, ) =

0 1

b_

b

I812,23(34) 1

17i2,23,14(34) =

010

b01

; J

Alt =

0^

b1

b11

o

lo

b1

l o


A23

0 )

10_0

1,14(34) (, AZf A3, /14) I , j ; Al2

10

01

_0_0

0

I 1

In each representation, 10 and 10' (11 and 11 ) must be of the same dimension.Now consider the set ^ consisting of the representations in which are not

contained in MQ or Mj, but which convert into representations in 1 j after a few dif-ferentiations corresponding to the first case. All such representations can be obtainedby adding several points and assigning to them the whole space. Thus to the repre-

sentation 17i2(34) ^ t n e s e t 1 w e c a n a c ^ o n e o r t w o ^ t n e P i n t s ^2, 23, 14,but not all at once, and to each of these points we assign V. After doing this with allof the representations in Af j , we obtain the following representations belonging to *l ( :

1712(34),12' 1812(34)12' 1712(34)12.23' 1812, (34), 12,23'

It is easy to see that these account for all representations in

0',

Now consider the set M2 of those representations which do not belong toor j , but which convert into representations in ^ U

alter one differentia-tion (it is clear that in this differentiation each such representation is faithful atsome " n e w " point).

Arguing as above, we see that M2 consists of four representations:

778 L. A. NAZAROVA

17l2l34)23(H), 1712 1 812(34)23

17l2(34)23(14)'-

| \ /


5* = {^

C 2>

at, ci, %> cs, A. h, h\ h

ca < a2, c2 < o3,

2, fta3, c3

are obtained from those of 8 by duality. Indeed, if S is a partially ordered set and Xa representation of it, i.e. to each point i S there is assigned a subspace V. ofsome space V, then we can construct the dual representation X of a set S whichis inversely isomorphic to S by passing to the space of functionals V and letting Vjstand for the functionals vanishing on V..

Thus we have indicated all those representations of ?i which contain only pointsoccuring either "not below a tetrad" or "not above a tetrad".

We now consider representations of 2 which include both upper and lower pointsof 21. Note that on differentiating the "bottom" of 21 its " top" does not change.Hence any representation containing both upper and lower points of ?I is converted,after a finite number of differentiations with respect to lower points, into a represen-tation containing only upper points. Let X be a representation containing both upperand lower points and such that X' contains no lower points. Then in the tetradal rep-resentation corresponding to X we must have p.. 0, which can be seen by. con-sidering the explicit form of the construction of differentiation with respect to a mini-mal element. However, of all the representations belonging to and actually con-taining upper points only two possess this property (up to a renaming of the elements):13 j 2 1412 Relative to these we can construct the representations 1 3 1 2 |-j ofthe set

. = . 3

4 0 . 2), (1, 2); (1,2)> 1, (1,2)> 2, (1, 2 ) < 1, (1, 2) < 2 >

and 141 2

of the set

= = 1' 2

.

3

4> 0. 2). O, 4); (1, 2)> 1, (1, 2)>2,(3, 4)

780 L. A. NAZAROVA

100*

0

01 0 ..

Jn

En0

. 0. 0

00

0101

0

0

11000

10 . . . 0

00

0]100

00

0

01000

_0 _0

It is not difficult to see that ?I has no other representations containing both up-per and lower points.

Hence any indecomposable representation of 21 which is faithful at the points ofa tetrad agrees up to a renaming of the elements with one of the following representa-tions: 1 18,

10i2, 1015, 11 1 2, Ili2. 131 2, 14j2) 1812, 1712, 10 1 2 ) 3, 1' ' 1 2 , 2 3 > I " 1 2 . 2 3 . I ' l 2 , 3 4 l ' " 1 2 , 3 4 i I ' M , 23,34 I o 1 2 , j 3 1 3

)- 2 3 , 1 8 l 2 ; ( 3 4 ) , 2 3 ,

18 I 2 ( 3 4)23,(14), 1 7 1 2(34)23(14) 1812(34)23(14).

12,12' 12,34' '12,23,34(14). 1 Ol2,23,34(14)

or their duals.In the following section we will need, in addition to 2 , the representations of

the set


31 ( s , t) = { 1 , 2 , 3 , 4 , ( 1 , 2 ) 1 , ( 1 , 2 ) \ ( 1 , 2 ^ , ( 1 , 2 ) s ,U2)1, (~2)2, - , (2)'. ( ) ' < (T^T)'"1 < . < (T72)2

(1, 2 ) ' < 1, (1, 2 / < 2 , (1, 2f> 1, (1, 2 ) 1 > 2 , (1, 2 / < ( l , 2)*< . . . < ( 1 , 2)%

As above, it is easy to see that any representation of 9t(s, /) is a representation ofa set isomorphic to one of the sets

9 1 ( 0 , 0 ) = { 1 , 2 , 3 , 4 } , 31(1,0), 31(0,1), 31(1,1), 31(0,2), 3i(2, 0)

and is isomorphic to one of the representations 118 or

1 0 l S , 1 0 ; j f 11,,, 1 1 K , 13 1 2 , 14 1 2 , 17 1 2, 18 ] 2 , 1-, 1-,

3. Consider a quiver of type III:

("i-l)"

l'Jl_0-^_i_!u_2 . . . ^ (,-)-^-

We will reduce the matrices

, a l t a {, a2, . , a m _ l (using, for example, the re-sult of [1] concerning the representations of >

n). We will then explain which trans-

formations , . and preserve the reduced matrices.

L

It is not difficult to see that, as a result, there arise the representations of thepartially ordered set %l(s, s) considered above. Any representation of 3l(s, s) is infact a faithful representation of a subset of ift(s, s). It is also easily seen that anysubset of width 4 of this set is isomorphic to 3t(&, /) with 0 < k < s and 0 < / < s.Concerning sets of smaller width, it is evident, first of all t that among these subsetsonly those of the form (1), (1, l), (1, 1, l) or (1, 1, 2) possess faithful representa-tions (see [14]); and, second, it is easy to verify that the quiver representations ob-tained cannot be faithful. Consequently it follows from 2 that any faithful repre-sentation of the quiver II is obtained from one of the subsets isomorphic to 9t(l, 0),31(0, 1), 51(1, 1), ?l(2, 0), 91(0, 2) or (0, 0) ((0, 0) is a tetrad).

Furthermore, it is easy to verify that from the indicated subsets faithful repre-sentations of the quiver III can be obtained only when m = 1 or 2.

Thus we must consider the representations of the quivers

782 L. A. NAZAROVA

III,:

1'- ' .

2 i _ 2

For the first quiver we obtain the matrix problem:

2'

I a"i

1

O i

The reduction to the representations of 51(1, 1) indicated above is obtained by the fol-lowing reduction of the matrices j and jt

a'i = ^s | a j = ^ 4

0

0

0

0

0

0

A-12

0

0

0

0

0

A,

0

0

0

0

0

At

0

0

0

0

Alt

,

It is not difficult to see that if we substitute for the matrices j , 2, A j , A 4, 1 2 ,A-J-J one of the faithful indecomposable representations corresponding to the faithfulsets yi(k, I) listed above and exclude the nonfaithful representations of the quiver,we obtain the following series of indecomposable representations:

1)

-

1}J}-

where A j , A 2, ? , AA are the tetradal representations 8, 8 , 9, 9 , 10, 10 , 11, 11 ,

REPRESENTATIONS OF QUIVERS

12, 13, 14, 15, 16, 17, 18, except for 8, 8 ', 10, 10', 11, l l ' , 13, 15 when = 0;2)

783

A, A,

0

0

AM

0

0

0

Ai

0

0

0

A,

where the matrices ^ A 2, A ^t A4 and [ 2 stand for the indecomposable represen-

tation 1 7 1 2 ( 3 4 ) or 1 8 1 2 ( 3 4 ) ; -3)

As\ A,

0

0

0

0

At \ A-

}1}

a t

where the matrices A l t A 2 , A

4 andor

stand for the representation

Let us consider the quiver III 6 . As in the previous case, if we write down thematrix problem corresponding to III6 and exclude its nonfaithful representations, weobtain the following series of faithful indecomposable representations of III 6 :

1)

a i .-

/Is

^ 4

0

0

0

,

0

0

0

-4,

0

0

0

0

where the matrices

A 2 , A 3 , A^ and A 1 2 stand for the representation 17

o r 1 8 1 2(34)'1 2 ( 3 4 )

784 L. A. NAZAROVA

2)

A, A.

0

0

A-12

a.

0

A1

0

a,. *

0

a

0

where the matrices A ] t 2, ^, 4 and A J-J stand for the representation

We now consider the quivers IVVI. We first remark that while for the quiversIIII we wrote out only the faithful indecomposable representations, for the quiversIVVI we will not distinguish faithful and nonfaithful representations. The fact isthat each of the quivers IIII contained, speaking generally, many quotient quiversof infinite type. Hence the faithful representations comprised a small part of all rep-resentations. Quivers of types IVVI do not have quotient quivers of infinite type(see [l]), and therefore all but a finite number of representations of each of thesequivers are faithful.

IV. Consider the quiver

On reducing the matrices a 2 , a 2 a n 95. 12 14 1%. 65. l i t o 16--,;

2,23' ^ 2 , 2 3 ' 1 512,23(34)' 12,23(34)' 2(4),23' .23()>|C 1

12(34)23(14) l 12(34)23(4) *

Let us show by example using the first set how to write down the representationsof the quiver IV from the listed representations of the partially ordered set. The cor-responding series have the form

' 1,23 yi2.23'

t o ,

where A l t A 2 , ,, A 4 and A 1 2 stand for one of the representations corresponding

to the first set; (4) is a matrix such that 4 |( 4 )) is nonsingular (for differentsuch (4) the representations are isomorphic).

Beside the series listed above (and the trivial representations mentioned earlier),the quiver IV also has a finite number of representations corresponding to the faith-

786 L. A. NAZAROVA

ful subsets of width less than 4 of the set (2, 2, 2) . Each such subset is isomorphicto one of the sets ( l) , (1, 1), (1, 1, 1), (1, 1, 2) or (1, 2, 2), and their representationsare easily written down in accordance with [14J.

V. Consider the quiver V:

j2' V

Reduce the matrices a , , a 2 , a , and a 2 . In the remaining matrices a j , a j and we obtain the representation problem of the partially ordered set

( 1 , 3 , 3 ) = {Differentiating (1, 3, 3) with respect to w, we obtain

As in the previous case, there is a one-to-one correspondence between the indecom-posable representations of the set (1, 3, 3) and those of quiver V, except for a fi-nite number of trivial representations. Using Lemma 11, it is not difficult to list allindecomposable representations of (1, 3, 3)' containing the points (I/JVJ), ( j f 2 ) ,(u 2vi)' u\ a nd f and their corresponding quiver representations. As agreed earlier,we now seek the representations of the set

(1, 3, 3)' = {alt a2, a3, a4, bv bt, b3, clt Cj, c3; q < ait cx < a s , ^ < Os, Cj < a 3 ,

where ax = u3, 0^= u2vlt a3 = u ^ , a 4 = v3, bi = u3vlt b2 = UJUJ, b3 =

cx = u2, C2 = !!, c3 = t>2.

To find all indecomposable representations of V we must find all indecomposablerepresentations of the set ( l , 3, 3) or (1, 3, 3)'. Since (1, 2, 3)' is a subset of 21,all representations of V are obtained from the corresponding representations of SI.


We list all faithful subsets of width 4 of the set (1, 3, 3)': all these contain thepoints a p a2, 3 and a 4 , and, in addition,

1) V. ) C l;

2) b2; 2') c,;3) b3, 3') c3;4) ftlf &2; 4') c1 ( c 2;5) bt, b3; 5') c2, c3;6) b

v b3; 6') c l t c3;

7) 61, b2, b3; 7 ' ) c i . c. cai8) blt ci,9) ft2> c2;10) 63, c3;11) *i, c3;12) 63, c,;13) ={a

u a2, a3, a 4 } .

Each of the sets 1)3) has 10 series of indecomposable representations, each ofwhich is obtained after suitably renaming the points of

1012,

> 1 3 1 2 , 1 4 1 2 , 1 7 1 2 , 1 8 1 2 ,

each series of representations of the sets 4) and 5) is obtained by suitably renamingthe points of

10l2,23i Ill2.23> 1'1,23 112,23> 1'12,23(34) '

1 18l2,23,(14)i 1 ' 12(34).23,14> 18l2(34),

the series of the set 6) is2 34> ^ 1 2 3 4 ! 2 3 3 4 series of the set 7) is

1 8 12,23,34' 1712,23( 14),34' 1 8 12,23( 14), 34* The representations of the sets l ')-7')

are obtained dually. The representations of 8)10) are obtained from 13 1 2 -p; those

of 11) and 12), from 1 4 J 4 j ^ .As an example, we show how the representations of quiver V correspond to those

of the first of the indicated faithful sets:

a, =0

-r l 3= I 0 0

a, =

I U, V, ,

000

An100

A,0

0

A300

000

000

0

0

788 L. A. NAZAROVA

where the matrices A p A 2, ? ) 4 and A 1 2 stand for one of the listed indecompos-

able representations corresponding to the first faithful set.Beside the above series (and trivial representations), quiver V also has a finite

number of representations corresponding to the faithful subsets of (1, 3, 3)' of widthless than 4. As in the previous case, all these are isomorphic to one of the sets (1),(1, 1), (1, 1, 1), (1, 1, 2) or (1, 2, 2), and their representations are described in [4],

VI. Consider the quiver

1"

- 1 .

ii-2 3 4 5

Reduce the matrices a'2, a2, a^, and a 5 to the form

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

>

In the remaining matrices cij, . j and j we obtain the representation problem of thepartially ordered set

(1, 2, 5) = { f 1 < t ; ! i < u 3 < i 4 < t i s , ux < u 2 , w}.Differentiating (1, 2, 5) with respect to u2 and then (1, 2, 5) with respect to i/j,we obtain

(1, 2, 5)' = {Uj < v2 < u3 < v4 < uB, w < (,) < (wv2) < (wv3)


(1,2,5)

V)

As in the previous cases, we have reduced the representation problem of quiver VI tofinding the representations of the set (1, 2, 5)". Using Lemma 11, we find all repre-senta t ions containing the points f j , w, Vj, (wv^), v^buv^), v^iwv,), wv

4 , v^wv-^iand 4(u/f?). We then pass to a set 3 which is a subset of (1, 2, 5)" and isomorphicto H:

3t = { (awj, ((awj), f3), (ayy2), aw,, ((owj, o j , ((aw,), v3),(Ot(woJi, ((wv2), u j , aw3, vv (wvj, ((owi), 2), 3, (aw,),

Corresponding to ?l we introduce the notation

y5 = ait wvi = 3, 3 (^) = av wv2 = a l fa7 = &3, (aWi) t4 - i?2, (aw,), = ^ ,BWJ o 4 = g2, ow3 = g l t w3 = fv w>2 = /= g 3

= c 3 ,

The indecomposable representations of (1, 2, 5) ^3 correspond to the exactsubsets of (1, 2, 5)". We list the faithful subsets of 21

of width 4: all these contain

790 -. A. NAZAROVA

c3;7) gi, & 2 ; 7') fu c2;8) 6,, fc3; 8') cu c3;9) gi, & 3 ; 9') /,, c3;10) b

u b2, h; 10') c , c2, c3;

) g i , 6 2 , t>3; 11') f,, c 2 , c 3 ;12) b

u c

x;

13) fe2> c2;14) 63, c 3;15) gu /,;16) g,, c,;17) /,, 6,;18) 6,, c3;19) g,, c3;20) b 3 , c x ;21) 63, A;22) gu b

x- 22') c l f fu

23) g,, 61, 62; 23') fu cu c2,24) = { a l f a2, a3, a 4 }.The representations of the sets 1)21), l ')ll') and 24) are described as in

the case of quiver V. The representations of 22) are 1"^ 12(34) 2 a n c * ^12(34) f2>those of 23) are

23' 1(34)223' ' 1(34),23(1442'

The representations of 22') and 23') are obtained dually.We give, as in the previous cases, the representations of quiver VI correspond-

ing to the first set. The form of all matrices except for al = w, a^ and ai hasbeen indicated earlier. In this case the matrices a j , a j and a } are


000000

000000

00

0000

\00

000

00

000

C000

00

000010

00000

dj = -^ " "

At

00

00

Au

010010

00000

0

00000

010000

,

00

000

,)000000

where the matrices A j , A2, A ?, A4 and Al2 stand for one of the indecomposable rep-resentations of the set 1).

Furthermore, using [14], we can list all faithful subsets of 21 of width less than4. All these have a finite number of indecomposable representations, given in [14],and with these we can construct the representations of quiver VI.

Remark. In [1] Gabriel observes that the set of quivers of finite type coincideswith the set of Dynkin diagrams without double bonds. Moreover, for each such quiv-er there is a mysterious one-to-one correspondence between the set of its indecompos-able representations and the set of positive roots of the corresponding Lie algebra.Gabriel writes that this correspondence is not accidental, but he does not see any ex-planation for this.

To the quivers considered in this present paper correspond extended Dynkin dia-grams (without double bonds), to which correspond simple (infinite-dimensional) grad-ed Lie algebras of finite height (see [17]).

Received 19/SEPT/72BIBLIOGRAPHY

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Naucn. Sem. Leningrad. Otdel. Mat. Inst. Stelclov. (LOMI) 28 (1972), 60-69 = J. Soviet Math,(to appear).

792 L. A. NAZAROVA

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Dedekind rings, and finite groups with an abelian normal divisor of index p, Izv. Akad. NaukSSSR Ser. Mat. 33 (1969), 65-89 = Math. USSR Izv. 3 (1969), 65-86. MR 41 #5479.

14. . . Kleiner, On faithful representations of partially ordered sets of finite type, Zap.Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 28 (1972), 42-59 = J. Soviet Math,(to appear).

15. I. M. Gel fand and V. A. Ponomarev, Indecomposable representations of the i-orentzgroup, Uspehi Mat. Nauk 23 (1968), no. 2 (140), 3-60 = Russian Math. Surveys 23 (1968), no.2, 1-58. MR 37 #5325.

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Translated by G. . AND ALL

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Representatio Quiver Infinite Type