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Doubling a Path Algebra, or: How to extend Indecomposable

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Doubling a Path Algebra, or: How to extend
Indecomposable Modules to Simple Modules
Wolfgang Rump
Introduction. For a field k of characteristic 0 and a representation-finite quiver
∆, it has been observed [8] that the indecomposable k-linear ∆-representations are
independent of the chosen orientation. In fact, they can be associated [6] to the
positive roots of the corresponding semisimple Lie algebra g = g(∆), where ∆
↔
denotes the unoriented graph underlying ∆. Thus if we consider the graph ∆ with
pairs of arrows in opposite direction for every edge of ∆, we may ask whether there
↔
exists a ∆-representation which induces the indecomposables for each orientation
∆ of ∆. Of course, for a reasonable solution, one should expect that the maps
representing the arrows of ∆ are connected with the opposite maps by a suitable
↔
relation, and the resulting ∆-representation should be more or less unique.
Our first aim in this paper is to show that this problem is indeed solvable (Theorem 1) and admits a unique solution depending on a fixed element О± in the root
space k∆0 which does not lie on a wall of the Weyl chambers. (For arbitrary α ∈ k∆0 ,
↔
the ∆-representation is still unique, but for some orientations ∆, the induced representation might decompose.) The mentioned relation (see (1) below) between the
maps representing ∆ and the opposite maps is well-known in the particular case
О± = 0: Then it turns into the defining relation of the preprojective algebra [10, 7] of
the graph ∆. Our construction in §2 therefore leads to a semisimple deformation A
of the preprojective algebra, which, as we shall see, can be regarded as a ”double”
of the path algebra k∆. We shall prove that the indecomposable k∆-representations
correspond to the simple A-modules.
↔
There are quite different contexts where quivers of the form ∆ and relations similar to (1) have occurred. We thank the referee for directing our attention to some
of them. McKay [15] observed that for a binary polyhedral group G, if the twodimensional irreducible representation R (over C) is tensored with all irreducibles
(a )
R1 , . . . , Rm , say R вЉ— Ri = Rj ij , the oriented graph with adjacency matrix (aij )
↔
is of the form ∆, with an extended Dynkin diagram ∆. An explanation is provided,
among others, by M. Auslander [3] (see also [4]). He establishes a one-to-one correspondence between R1 , . . . , Rm and the indecomposable projective modules over
the twisted group ring S[G], where S = C[[x, y]], and shows that the McKay quiver
↔
↔
∆ coincides with the Gabriel quiver of S[G]. Now if ∆ is regarded as a translation
↔
quiver with the identity as translation, the relations in ∆ given by the ring structure
of S[G] are just the mesh relations, that is, the relations (1) with О± = 0.
On the other hand, the inhomogenous equations (1) are connected with the resolution of singularities, and with minimum action solutions of SU2 Yang-Mills fields,
also called instantons or pseudo-particles. Atiyah and Ward [2] have shown that
self-dual instantons in R4 correspond to certain two-dimensional algebraic vector
bundles over P3 (C), and in [1] these bundles have been constructed in terms of pure
linear algebra. A quadratic equation between matrices, similar to (1), results as a
defining relation.
Every binary polyhedral group G, viewed as a subgroup of SU2 (C), gives rise
to a quotient singularity C2 /G with intersection matrix of Dynkin type. If XG denotes the 4-manifold underlying the minimal resolution of C2 /G, Peter Kronheimer
has shown [11] that XG admits the structure of an ALE hyper-KВЁahler 4-manifold,
that is, a self-dual gravitational instanton. Here, ALE (= asymptotically locally
Euclidean) signifies that at infinity, XG resembles R4 /G, and the Riemannian metric is Euclidean up to O(r −4 ). “Hyper-K¨ahler” means that XG is equipped with
three covariant-constant complex structures I, J, K connected by the relations of
quaternion units. Moreover, he proved that every ALE hyper-KВЁahler 4-manifold is
diffeomorphic to some XG . In his construction of XG , Kronheimer starts with the
manifold M = (R вЉ— EndCG)G , on which the group F of G-invariant unitary transformations of CG operates. Then XG is obtained as a hyper-KВЁahler quotient M /F ,
where M вЉ‚ M is defined by a quadratic relation similar to (1). The left-hand side
of this relation plays the rˆole of a (hyper-K¨ahler) moment map. In a similar way,
relation (1) enters into the definition of certain varieties of quivers considered by
Lusztig ([12], §8; [13], §12) and Nakajima’s quiver varieties [16, 17].
Our subsequent article will be self-contained. No use is made of the above
mentioned connections.
1. Extensions of indecomposables. Let k be a field of characteristic 0 and ∆
a finite oriented graph with vertex set ∆0 = {1, . . . , n}. A representation (V, f ) of
∆ over k is defined as a family V = (V1 , . . . , Vn ) of finite dimensional k-vector spaces
and a family f of k-linear maps fji : Vi → Vj for each arrow i → j in ∆. Let k∆
denote the path algebra of ∆, i.e. the k-algebra with paths p : i0 → i1 → . . . → ir
in ∆ as basis such that, for another path q : j0 → . . . → js in ∆, the product qp
is given by the composition i0 в†’ . . . в†’ ir в†’ j1 в†’ . . . в†’ js if ir = j0 , and qp = 0
otherwise. Then in an obvious way, each ∆-representation (V, f ) can be regarded
as a finitely generated k∆-module. The vector dim(V, f ) = (d1 , . . . , dn ) ∈ Nn with
di = dimk Vi is called the dimension vector of (V, f ).
Gabriel’s theorem [8] states that the number of (isomorphism classes of) indecomposable ∆-representations (i.e. k∆-modules) is finite if and only if the unoriented
graph ∆ underlying ∆ is a disjoint union of Dynkin diagrams. Moreover, the set of
dimension vectors of indecomposable ∆-representations coincides with the set Φ+
2
of positive roots of the semisimple Lie algebra g of type ∆. Thus for a given root
d в€€ О¦+ , there is an indecomposable representation with dimension vector d for each
orientation ∆ of ∆.
↔
Now let ∆ be a Dynkin diagram. If ∆ denotes the oriented graph with arrows
i→j and j→i for each edge i—j in ∆, the question arises for a given dimension vector
↔
d ∈ Φ+ , whether there exists a ∆-representation which induces the indecomposable
↔
∆-representations of all orientations via the natural embedding k∆ → k∆.
Our first result (Theorem 1) will give a solution to this problem, including an
↔
explicit construction of the k ∆-modules in question. Furthermore, we shall prove
↔
that these k∆-modules can be regarded as simple modules over a semisimple algebra
associated with ∆. Let us define a ∆-representation of type α = (α1 , . . . , αn ) ∈ kn
↔
as a ∆-representation (V, f ) which satisfies for each i ∈ ∆0 the relation
fij fji = О±i В· 1Vi
(1)
j
where j runs over the vicinity V(i) of i, that is, the set of vertices j adjacent to i in
∆. If d is the dimension vector of (V, f ), the following relation necessarily holds for
the О±i :
В±di О±i = 0.
(2)
в€’
Here, the + and − signs are chosen according to any fixed partition ∆0 = ∆+
0 ∪ ∆0
such that adjacent vertices belong to different signs. (Since ∆ is a tree, there are
only two such partitions!) In fact, if we take the trace on both sides of (1), we get
dij = О±i di
jв€€V(i)
where dij := tr(fij fji ) = dji . Thus (2) immediately follows. For a fixed partition
в€’
∆0 = ∆ +
0 ∪ ∆0 , let us define Φd ∈ k[x1 , . . . , xn ] by
О¦d =
В±di xi =
d i xi в€’
i∈∆+
0
d i xi .
(3)
i∈∆−
0
Theorem 1 Let ∆ be a Dynkin diagram and α ∈ kn with Φd (α) = 0. Then every
indecomposable ∆-representation (V, f ) with dimension vector d has a unique extension to a ∆-representation (V˜ , f˜) of type α. Moreover, the f˜ij depend polynomially
on О±.
As a consequence, we get the solution to the above question:
Corollary. Let (V, f ) be a ∆-representation of dimension vector d ∈ Φ+ and type
О± such that
±di αi = 0 for all d = d in Φ+ . Then the induced ∆-representations
of all orientations are indecomposable.
3
Note that this choice of О± is possible since k is infinite. The corollary follows
immediately by a proposition which we shall prove together with Theorem 1:
˜ a ∆-representation which
Proposition 1 Let ∆ be an orientation of ∆ and M
Лњ
extends some ∆-representation M . Then M has a subrepresentation with underlying
∆-representation S such that S is an indecomposable direct summand of M .
2. The double of a path algebra. Our second purpose of this paper is to
show that by Theorem 1, the decomposition g = g+ вЉ• g0 вЉ• gв€’ of a semisimple Lie
algebra g has an analogue for associative algebras. Namely, the nilpotent Lie algebra
g+ corresponds to our path algebra A+ = k∆, the negative part g− to A− = (k∆)op ,
and the abelian part g0 is replaced by some commutative algebra A0 generated by
n elements О±1 , . . . , О±n . The semisimple Lie algebra g corresponds to a semisimple
algebra A containing an order О›0 generated by A+ , Aв€’ , and A0 .
Firstly, the ∆-representations over some extension field of k can be interpreted
as modules over the k-algebra
↔
(4)
Λ = R∗∆ /I
where Rв€— is the polynomial ring k[x1 , . . . , xn ] and I the principal ideal generated by
eij eji в€’
xi e i .
(5)
Here eij denotes the path j в†’ i, and ei is the primitive idempotent (empty path)
corresponding to the vertex i ∈ ∆0 . The first sum in (5) is to be extended over the
edges i—j, the second over the vertices i in ∆. The residue class of xi modulo I will
be denoted by О±i . Thus О±1 , . . . , О±n в€€ О› generate a subring R = k[О±1 , . . . , О±n ] in the
center of О›, and О› becomes an R-algebra.
Recall that an ideal P in a ring О“ is said to be prime if for any two ideals I, J
in О“, the inclusion P вЉѓ I В· J implies P вЉѓ I or P вЉѓ J. An element r в€€ О“ is called
Лњ is said
regular if the left and right multiplication by r is injective. A subring О“ of О“
Лњ
Лњ
to be an order in О“ if each regular r в€€ О“ is invertible in О“, and every element of
Лњ is of the form r в€’1 a, and also of the form ar в€’1 , with r, a в€€ О“ and r regular. The
О“
intersection N(О“) of all prime ideals in О“ is called the prime radical ([19], chap. XV),
and О“ is said to be semiprime if N(О“) = 0.
Now we define О›0 := О›/N(О›). The natural homomorphism О›
О›0 maps R onto
a central subring R0 = k[О±
ВЇ1, . . . , О±
ВЇ n ] of О›0 , generated by the residue classes of the О±i
modulo N(О›). Let О¦ в€€ Rв€— denote the product of all О¦d , d в€€ О¦+ . By virtue of (5),
the defining relations (e2i = ei , ei eij = eij , etc.) of the eij which encode the graph
structure of ∆, are turned into a single relation Φ(α
ВЇ1, . . . , О±
ВЇ n ) = 0 in R0 encoding
the root system of ∆:
Theorem 2 The kernel of the natural map ПЃ : Rв€—
R0 is the principal ideal (О¦).
If M is any ∆-representation such that for some orientation ∆ of ∆, the underlying
k∆-module is indecomposable, then N(Λ)M = 0, i.e. M can be regarded as a Λ 0 module.
4
For each orientation ∆ of ∆, the k∆-module k∆ extends to a ∆-representation M
of type (0, . . . , 0). Thus Theorem 2 implies N(Λ)M = 0, whence k∆ ∩ N(Λ) = 0 in
О›. Therefore, the natural ring homomorphism
k∆ → Λ
О›0
(6)
is injective, i.e. k∆ and (k∆)op can be regarded as subrings of Λ0 , and Λ0 is generated
by these subrings together with R0 .
Since for each d в€€ О¦+ , there is an integral domain Rd := Rв€— /(О¦d ) isomorphic to
k[x1 , . . . , xnв€’1 ], with quotient field Kd , say, Theorem 2 yields a natural embedding
Kd =: K,
R0 в†’
(7)
dв€€О¦+
and K is the classical quotient ring [19] of R0 . Consequently, there is a natural
homomorphism
О› в€’в†’ K вЉ•R О› =: A.
(8)
Theorem 3 The ring A is semisimple, the kernel of (8) is N(О›), and thus О› 0 is an
order in A. The blocks of A correspond to the positive roots d в€€ О¦ + , and are matrix
algebras Md (Kd ) with d = d1 + . . . + dn .
This theorem leads to a fairly precise description of the structure of О›0 . In
particular, each indecomposable ∆-representation M (for some orientation ∆ of ∆)
extends to a О›0 -representation E via Theorem 1, and for fixed d = dimM and all
orientations ∆, the Λ0 -lattices E belong to a unique simple A-module Sd . (Such
lattices over an order О›0 are said to be irreducible.) The reason that M extends to
a О›0 -lattice E depends on the fact that the extension problem in Theorem 1 admits
an “integral” solution. For example, let ∆ be the following orientation of E6 :
4
вќ„
1
f12
вњІ
2
вњІ
3вњ›
5вњ›
6
and (V, f ) the indecomposable
пЈ«
∆-representation
пЈ« пЈ¶ with пЈ«
пЈ¶
1 0
1
0 0
1
1
f21 =
; f32 = пЈ­ 0 0 пЈё ; f34 = пЈ­ 1 пЈё ; f35 = пЈ­ 1 0 пЈё ; f56 =
.
0
0
0 1
1
0 1
Then the extension to a ∆-representation of type α is given by
Оґ
в€’ОІ
в€’ОІ
1
6
; f65 = (О±6 ОІ)
; f43 = (в€’Оі в€’Оґ ОІ); f53 = ОіОі О±5 в€’О±
= (О±1 ОІ); f23 = О±2 в€’О±
0
0
О±5
Оґ
О±2
where ОІ = О±3 в€’ О±2 в€’ О±5 , Оі = О±2 в€’ О±1 в€’ О±3 , and Оґ = О±5 в€’ О±3 в€’ О±6 . The entries of all
matrices fij are integral in the О±1 , . . . , О±6 . In general, this follows by means of
3. Reflection functors for ∆-representations, which, in virtue of Theorem 1, can be viewed as extensions of the functors Fi± of Bernstein, Gelfand, and
Ponomarev [6] to ∆-representations.
5
Firstly, for each vertex i ∈ ∆0 , define a linear map σi ∈ Aut kn by σi (x1 , . . . , xn ) =
(x1 , . . . , xn ), where
пЈ±
for j = i
пЈІ в€’xi
xj в€’ xi for j в€€ V(i)
xj =
(9)
пЈі
xj
otherwise.
For a given ∆-representation (V, f ) of type α, and i ∈ ∆0 , let us define a ∆representation σi+ (V, f ) of type σi (α) as follows. For j ∈ V(i), the maps fij : Vj → Vi
and fji : Vi в†’ Vj give rise to k-linear maps fi0 :
Vj в†’ Vi and f0i : Vi в†’
Vj
such that
fi0 f0i = О±i В· 1Vi .
(10)
If f0i : Vi в†’
Vj is the kernel of fi0 , then fi0 (f0i fi0 в€’ О±i В· 1L Vj ) = 0, whence there
exists a unique fi0 :
Vj в†’ Vi such that
f0i fi0 в€’ О±i В· 1L Vj = f0i fi0 .
(11)
Now the f0i and fi0 decompose into k-linear maps fji : Vi в†’ Vj and fij : Vj в†’ Vi ,
which replace the fji and fij in order to obtain Пѓi+ (V, f ). It is easily verified that (1)
is satisfied for each vertex j ∈ ∆0 if α is replaced by σi (α). The dimension vector
d is changed into Пѓi (d) = (d1 , . . . , dn ), where di = в€’di + jв€€V(i) dj and dj = dj for
j = i.
If σi+ is restricted to ∆-representations, we obtain a functor which is applicable
without any condition on ∆. If i is a sink, i.e. no arrows start at i, this functor
coincides with the “image functor” Fi+ in [6]. Dually to σi+ , we define σi− by taking
the cokernel fi0 :
Vj
Vi of f0i , etc.
Remark. Since these functors are well defined even if k is replaced by an integral
domain, they can be applied to the irreducible representations E of О›0 considered
above. However, since ПѓiВ± is universally applicable, even if i is neither a sink nor a
source, there exist “non-oriented” irreducibles of Λ0 , i.e. those irreducible Λ0 -lattices
which do not arise by extension of some indecomposable ∆-representation via Theorem 1.
Now let (V, f ) be a ∆-representation of type α and i ∈ ∆0 . If αi = 0, then (10)
and (11) state that the maps f0i , f0i and О±1i fi0 , в€’ О±1i fi0 form a biproduct diagram [14]
Vi в€’в†’
в†ђв€’
Vj в†ђв€’
в€’в†’ Vi ,
(12)
jв€€V(i)
that is, we have a natural isomorphism
Vj в€ј
= Vi вЉ• Vi . Hence Пѓi+ and Пѓiв€’ coincide
on (V, f ). In this case, we simply write Пѓi instead of ПѓiВ± . If О±i = 0, we put Пѓi = 1,
the identity functor. Via these σi , the Weyl group of ∆ operates on the category of
∆-representations.
4. Proofs. For the proof of the existence part of Theorem 1, these reflection
functors Пѓi suffice. For the uniqueness part, however, we have to make use of Пѓi+
and Пѓiв€’ :
6
Proof of Theorem 1. If (V, f ) is simple, then Φd (α) = ±αi for some i ∈ ∆0 ,
and the assertion is trivial. Otherwise, let i be a sink of ∆. Then Fi− Fi+ (V, f ) =
(V, f ), where Fi± denote the classical reflection functors [6] for ∆-representations.
As О¦d (О±) = 0 implies О¦d (О± ) = 0 for d = Пѓi (d) and О± = Пѓi (О±), we may assume by
induction that Fi+ (V, f ) is extendable to a ∆-representation (W, g) of type σi (α).
Then (V, f ) extends to the ∆-representation σi− (W, g) of type σi σi (α) = α. To
prove the uniqueness of this ∆-representation, let (V˜ , f˜) be any such extension.
Then Пѓi+ (VЛњ , fЛњ) extends Fi+ (V, f ), whence Пѓi+ (VЛњ , fЛњ) = (W, g). Therefore, we get
(VЛњ , fЛњ) = Пѓiв€’ Пѓi+ (VЛњ , fЛњ) = Пѓiв€’ (W, g). By the construction of ПѓiВ± , an inductive argument
also proves the integrality property of fЛњ.
Лњ which
Proof of Proposition 1. If i ∈ ∆0 is a source of ∆, we may apply σi− to M
в€’
+ в€’ Лњ
Лњ , whence by induction,
extends the application of Fi to M . Then Пѓi Пѓi M = M
we may assume that M has a simple direct summand S concentrated at a source i.
Лњ.
Thus О±i = 0, and S extends to a subrepresentation SЛњ of M
Next we shall focus our attention to Theorems 2 and 3. If О± is specialized to
(0, . . . , 0), then Λ turns into the preprojective algebra Π(∆) of ∆ [10, 7, 5]. For
k ∈ N, define Λk as the R-submodule of Λ generated by the paths of length ≤ k.
This gives a filtration
R вЉ‚ О› 0 вЉ‚ О›1 вЉ‚ О›2 вЉ‚ . . . вЉ‚ О›
(13)
of О›, i.e. О›i О›j вЉ‚ О›i+j for i, j в€€ N. With О›в€’1 := 0, we can form the associated graded
R-algebra (О›i /О›iв€’1 ), wherein the defining relation (5) simplifies to
eВЇij eВЇji = 0, if
eВЇij denotes the residue class of eij in О›1 /О›0 . Hence, we have a natural epimorphism
of R-algebras:
R ⊗k Π(∆)
(О›i /О›iв€’1 ).
(14)
iв€€N
Lemma 1 О› is finitely generated as an R-module.
Proof. By virtue of (14), this follows from the well-known fact that Π(∆) is finite
dimensional. For the convenience of the reader, let us give a proof (cf. [9, 18]) which
also sheds some light upon the structure of О› via (14). It suffices to show that for
large m ∈ N, any path of length ≥ m becomes zero in Π(∆). Let i ∈ ∆0 be a fixed
vertex. We choose the unique orientation ∆ of ∆ such that for each j ∈ ∆0 , there
is a path from i to j. Define Z∆ as the oriented graph with vertex set ∆0 × Z and
arrows (i, l) → (j, l) → (i, l + 1) for each arrow i → j in ∆ and l ∈ Z. (Note that,
as an abstract oriented graph, Z∆ does not depend on the orientation of ∆.) The
numbering of Z∆ is given by the embedding ∆ → Z∆ with i → (i, 0) for i ∈ ∆0 .
Now let (V, f ) be i-th projective ∆-representation, i.e. Vj = k for all j ∈ ∆0 and
fjk = 1 for each arrow k → j in ∆. There is a natural way to extend (V, f ) to
a Z∆-representation. By induction, suppose that (V, f ) is already extended to the
full subgraph Γ of Z∆, and let (i, l) be a source in Γ such that (j, m) ∈ Γ if there is
an arrow (i, l) → (j, m) in Z∆, and (i, l + 1) ∈ Γ. We apply the classical reflection
в€’
functor F(i,l)
to (V, f ). This gives a representation (V , f ) where (i, l) is a sink.
7
Define V(i,l+1) := V(i,l) , f(i,l+1)(k,l) := f(i,l)(k,l) and f(i,l+1)(j,l+1) := f(i,l)(j,l+1) for arrows
j → i → k in ∆. Thus we obtain a Z∆-representation (V, f ). If (W, g) is a nonsimple indecomposable representation of some orientation of ∆, and j is a source for
this orientation, then Wj в†’ kв€€V(j) Wk is always injective. Therefore, it is easily
seen that a path in Π(∆) starting in i vanishes if and only if the corresponding map
V(i,0) → V(j,l) in (V, f ) is zero. But since ∆ is representation-finite, V(j,m) = 0 for
sufficiently large m.
Lemma 2 Every prime ideal P of О› contains some О¦d (О±) with d в€€ О¦+ .
Proof. Since Λ is noetherian, Goldie’s first theorem ([19], chap. II, Prop. 2.6)
implies that Λ/P is an order in a simple ring B. The ideal p = P ∩ R of R is prime,
whence B contains the quotient field F of R+P/P в€ј
= R/p. By Lemma 1, О›/P and
F generate a finite dimensional F -algebra in B which therefore coincides with B.
Consequently, B gives rise to a ∆-representation over F . By Proposition 1, at least
one relation О¦d (О±) = 0 must hold in F , whence О¦d (О±) в€€ p вЉ‚ P .
Corollary. There is a positive integer s with О¦(О±)s = 0 in О›.
Proof. By Lemma 2, О¦(О±) lies in the prime radical N(О›) of О› which is a nil ideal
([19], chap. XV, Prop. 1.2).
Next we consider the natural epimorphism
ПЃ : k[x1 , . . . , xn ]
k[О±1 , . . . , О±n ] = R
(15)
with ПЃ(xi ) = О±i .
Lemma 3 The kernel of ПЃ is contained in the principal ideal (О¦).
Proof. Let d ∈ Φ+ be given. Take any orientation ∆ of ∆. By Theorem 1,
the indecomposable k∆-module with dimension vector d admits a unique extension
to a ∆-representation M of type (¯
x1 , . . . , xВЇn ) over the quotient field Kd of Rd =
в€ј
k[x1 , . . . , xn ]/(О¦d ) = k[ВЇ
x1 , . . . , xВЇn ] = k[x1 , . . . , xnв€’1 ], where xВЇi is the residue class of
xi modulo О¦d . Hence, if f в€€ KerПЃ, then fВЇM = 0 for the corresponding fВЇ в€€ Rd .
Thus fВЇ = 0, i.e. f в€€ (О¦d ). Since this holds for each d в€€ О¦+ , we obtain Ker ПЃ вЉ‚
(О¦d ) = (О¦).
For any d в€€ О¦+ , the preceding lemma implies that p := О¦d (О±) is a prime element
of R. For Ad := R(p) вЉ—R О›, we have:
Lemma 4 p В· Ad = 0.
8
Proof. Without loss of generality, we may suppose that |V(n)| ≤ 1, and
dn = 1. Let ПЂ : R в†’ R(p) be the natural homomorphism, and О±i = ПЂ(О±i ) for
i в€€ {1, ..., nв€’ 1}. The residue class field R(p) /pR(p) is isomorphic to the quotient field
Kd of Rd = R/pR which is isomorphic to the function field k(x1 , . . . , xnв€’1 ). Hence,
Kd is isomorphic to the subfield F := k(О±1 , . . . , О±nв€’1 ) of R(p) , and R(p) = F вЉ• pR(p) .
Moreover, there is a unique О±n в€€ F such that
ПЂ(О±n ) = О±n + ПЂ(p) ; О¦d (О±1 , . . . , О±n ) = 0
(16)
holds in R(p) . Next we consider the exact sequence
pAd /p2 Ad в†’ Ad /p2 Ad
Ad /pAd .
(17)
For an arbitrary orientation ∆ of ∆, let X be the indecomposable F ∆-module.
Since pAd /p2 Ad and Ad /pAd are ∆-representations of type (α1 , . . . , αn ) over F , the
corresponding F ∆-modules are isomorphic to powers of X. Hence, Ext(X, X) = 0
implies that the sequence (17) of F ∆-modules splits. Regarding Ad /p2 Ad as a ∆representation (V, f ) over the ring R(p) /p2 R(p) , the F ∆-module (V, f ) is thus of the
form (C, f ) ⊕ (pC, f ), that is, for each vertex i ∈ ∆0 , we have Vi = Ci ⊕ pCi , and
for each arrow i → j in ∆,
fji = f0ji f0
; fij = hfij f0 .
ij
ji
ij
Furthermore, the multiplication by p gives rise to an epimorphism (C, f )
(pC, f )
of F ∆-modules which splits by virtue of Ext(X, X) = 0. Therefore, p can be regarded as the natural projection in a decomposition (C, f ) = (pC, f )⊕(D, g). Thus
p induces an endomorphism p of (C, f ), and the hij induce F -linear maps hij : Cj в†’
pCi → Ci which extend the F ∆-module (C, f ) to a ∆-representation (C, f˜) of type
(0, . . . , 0, p ). For any edge i—j in ∆, let dij be the trace tr(f˜ij f˜ji ). Then dij = dji ,
and for each vertex i ∈ ∆0 ,
0
if i = n
dij =
tr p |Cn if i = n.
jв€€V(i)
But this implies tr p |Cn = 0, whence pC = 0 and thus pAd /p2 Ad = 0. By
Nakayama’s lemma, we conclude pAd = 0.
Using the notation of (7), we thus obtain that Ad is a finite dimensional Kd algebra. Since the relation О¦d (О±) = 0 in Kd holds for no positive root other than d,
Proposition 1 and a similar argument as in the preceding proof, together with the
uniqueness part of Theorem 1 imply that the ∆-representation Ad decomposes into
simple ∆-representations of a single type α. Hence, the Kd -algebra Ad is simple.
Consider the natural map
ПЂd : О› в€’в†’ Ad .
(18)
The image of πd is an order in Ad . Thus, by Goldie’s theorem ([19], chap. II, Prop.
2.6), the kernel Pd of ПЂd is a prime ideal of О›:
Pd = {a ∈ Λ | ∃r ∈ R \ RΦd (α) : ra = 0}.
9
(19)
Lemma 5 Pd ∩ R = RΦd (α).
Proof. Since Φd (α) ∈ R is prime, the inclusion “⊂” holds. Conversely, suppose
О¦d (О±) в€€ Pd . By Lemma 2, there would exist some d в€€ О¦+ with d = d and
О¦d (О±) в€€ Pd . Consider the epimorphism (15). By (19), we infer KerПЃ вЉ‚ (О¦d ), which
contradicts Lemma 3.
Lemma 6 Let M be a ∆-representation of type α ∈ F n over an extension field F
of k such that, for some orientation ∆ of ∆, the underlying k∆-module is indecomposable. If N := d∈Φ+ Pd , then N M = 0.
Proof. There is a unique k-algebra-homomorphism П„ : k[x1 , . . . , xn ] в†’ F with
П„ (xi ) = О±i . The kernel p of П„ is a prime ideal of k[x1 , . . . , xn ], and О¦d в€€ p
holds for the dimension vector d в€€ О¦+ of M . By Theorem 1, there is a
unique ∆-representation Md of type (¯
x1 , . . . , xВЇn ) over the quotient field Kd of
Rd = k[x1 , . . . , xn ]/(О¦d ), where xВЇi = xi + (О¦d ), and there is an Rd -lattice E in
Md which is a ∆-representation over the ring Rd such that M ∼
= F вЉ—Rd E. Since
Md is an Ad -module, we have N Md вЉ‚ Pd Md = 0 and thus N M = 0.
Pd and R ∩ N(Λ) = R · Φ(α).
Lemma 7 N(О›) =
dв€€О¦+
Proof. Let N be as in Lemma 6 and P a prime ideal in О›. As in the proof
of Lemma 2, Λ/P is an order in a simple F -algebra B = S d with a simple ∆representation S over F . Choose any orientation ∆ of ∆. By Proposition 1, S is
indecomposable as a k∆-module. Hence, Lemma 6 implies N (Λ/P ) ⊂ N B = 0.
Consequently, N вЉ‚ P and thus N(О›) = N . The second equation follows by Lemma
5 and Lemma 3.
в€ј
Proof of Theorem 2 and 3. Theorem 2 follows by Lemma 6 and 7. Since Ad =
Kd вЉ—R О› by Lemma 4, Theorem 3 follows immediately by (18), (19), and Lemma
7. The endomorphism ring of the simple Ad -modules is Kd since this holds for the
underlying indecomposable Kd ∆-modules.
5. An open question. We have shown that the element О¦(О±) = О¦d (О±) of О›
is nilpotent. On the other hand, Lemma 2 and (19) imply that there is a polynomial
r в€€ k[x1 , . . . , xn ] \ (О¦) with r(О±)О¦(О±) = 0. Since k[x1 , . . . , xn ] is not a principal
ideal domain for n ≥ 2, we cannot conclude Φ(α) = 0. A direct calculation in Λ,
however, shows that Φ(α) = 0 at least for ∆ = An and D4 . If Φ(α) = 0 holds, then
the distinction between R and R0 can be dropped. A further simplification would
arise if О› is semiprime: then О›0 would coincide with О›. By Lemma 7, this would
also imply О¦(О±) = 0.
Remark. The above results have been presented on a conference in Constantza,
in 1995. The first question was anwered by W. Crawley-Boevey around 1998 (cf.
his paper: Geometry of the moment map for representations of quivers, Compos.
Math.126 (2001), 257-293). His method can be extended to obtain a positive answer
also to the more general question, i. e. the above ring О› is in fact semiprime.
10
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12
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