вход по аккаунту


Doubling a Path Algebra, or: How to extend Indecomposable

код для вставки
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+
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
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.
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
where dij := tr(fij fji ) = dji . Thus (2) immediately follows. For a fixed partition
∆0 = ∆ +
0 ∪ ∆0 , let us define Φd ∈ k[x1 , . . . , xn ] by
О¦d =
В±di xi =
d i xi в€’
d i xi .
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.
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
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
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 .
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“
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
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.
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
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†’
and K is the classical quotient ring [19] of R0 . Consequently, there is a natural
О› в€’в†’ K вЉ•R О› =: A.
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 :
and (V, f ) the indecomposable
пЈ« пЈ¶ with пЈ«
1 0
0 0
f21 =
; f32 = пЈ­ 0 0 пЈё ; f34 = пЈ­ 1 пЈё ; f35 = пЈ­ 1 0 пЈё ; f56 =
0 1
0 1
Then the extension to a ∆-representation of type α is given by
; f65 = (О±6 ОІ)
; f43 = (в€’Оі в€’Оґ ОІ); f53 = ОіОі О±5 в€’О±
= (О±1 ОІ); f23 = О±2 в€’О±
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.
Firstly, for each vertex i ∈ ∆0 , define a linear map σi ∈ Aut kn by σi (x1 , . . . , xn ) =
(x1 , . . . , xn ), where
for j = i
пЈІ в€’xi
xj в€’ xi for j в€€ V(i)
xj =
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†’
such that
fi0 f0i = О±i В· 1Vi .
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 .
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 :
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†’
Vj в†ђв€’
в€’в†’ Vi ,
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
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€’ :
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€’ 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.
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›
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 ).
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
functor F(i,l)
to (V, f ). This gives a representation (V , f ) where (i, l) is a sink.
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
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 =
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.
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
holds in R(p) . Next we consider the exact sequence
pAd /p2 Ad в†’ Ad /p2 Ad
Ad /pAd .
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 .
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 ,
if i = n
dij =
tr p |Cn if i = n.
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 .
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}.
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›) =
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.
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.
[1] M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, Yu. I. Manin: Construction of
instantons, Phys. Lett. 65A (1978), 185-187
[2] M. F. Atiyah, R. S. Ward: Instantons and Algebraic Geometry, Commun.
Math. Phys. 55 (1977), 117-124
[3] M. Auslander: Rational singularities and almost split sequences, Trans. AMS
293 (1986), 511-531
[4] M. Auslander, I. Reiten: McKay Quivers and Extended Dynkin Diagrams,
Trans. AMS 293 (1986), 293-301
[5] D. Baer, W. Geigle, H. Lenzing: The preprojective algebra of a tame hereditary artin algebra, Comm. in Alg. 15 (1987), 425-457
[6] I. N. Bernstein, I. M. Gelfand, V. A. Ponomarev: Coxeter Functors and
Gabriel’s Theorem, Uspechi Mat. Nauk 28 (1973), 19-33 = Russ. Math. Surveys 28 (1973), 17-32
[7] V. Dlab, C. M. Ringel: The preprojective algebra of a modulated graph, in:
Springer Lecture Notes in Math. 832 (1980), 216-231
[8] P. Gabriel: Unzerlegbare Darstellungen I, Manuscripta math. 6 (1972), 71-103
[9] P. Gabriel: Christine Riedtmann and the selfinjective algebras of finite
representation-type, Proc. Conf. on Ring Theory, Antwerp 1978. New York
- Basel 1979
[10] I. M. Gelfand, V. A. Ponomarev: Model algebras and representations of
graphs, Funkc. anal. i priloˇz. 13 (1979), 1-12 = Funct. Anal. Appl. 13 (1979),
[11] P. B. Kronheimer: The construction of ALE spaces as hyper-KВЁahler quotients,
J. Diff. Geom. 29 (1989), 665-683
[12] G. Lusztig: Canonical Bases Arising from Quantized Enveloping Algebras II,
Progress of Theor. Physics Suppl. No. 102 (1990), 175-201
[13] G. Lusztig: Quivers, Perverse Sheaves, and Quantized Enveloping Algebras,
J. AMS 4 (1991), 365-421
[14] S. Mac Lane: Categories for the Working Mathematician, New York - Heidelberg - Berlin 1971
[15] J. McKay: Graphs, singularities, and finite groups, Proc. Sympos. Pure Math.
37, AMS Providence, R. I. (1980), 183-186
[16] H. Nakajima: Instantons on ALE spaces, quiver varieties, and Kac-Moody
algebras, Duke Math. J. 76 (1994), 365-416
[17] H. Nakajima: Varieties Associated with Quivers, Preprint
[18] Chr. Riedtmann: Algebren, DarstellungskВЁocher, Uberlagerungen
und zurВЁ
Comm. Math. Helv. 55 (1980), 199-224
[19] Bo StenstrВЁom: Rings of Quotients, New York - Heidelberg - Berlin 1975
Без категории
Размер файла
126 Кб
Пожаловаться на содержимое документа