Some Classification of Prehomogeneous Vector Spaces Associated with Dynkin Quivers of Exceptional Type
|
|
- Melinda Anderson
- 5 years ago
- Views:
Transcription
1 International Journal of Algebra, Vol. 7, 2013, no. 7, HIKARI Ltd, Some Classification of Prehomogeneous Vector Spaces Associated with Dynkin Quivers of Exceptional Type Yukimi Ishii Institute of Mathematics University of Tsukuba Tsukuba, Ibaraki, , Japan Copyright c 2013 Yukimi Ishii. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we give a complete list of tilting Λ-modules where Λ is the tensor algebra of an oriented K-modulation (M, Ω) of a valued graph of exceptional type. Our result gives some kind of classification of prehomogeneous vector spaces associated with (M, Ω). Mathematics Subject Classification: 11S90, 16G20 Keywords: valued quiver, exceptional Dynkin quiver, tilting module, prehomogeneous vector spaces 1 Introduction Let (Γ, v) be a valued graph, i.e., Γ = {1, 2,...,n} is a finite set of vertices, and v =(v ij ) i,j Γ the set of nonnegative integers which satisfy the following condition: v ii = 0 for i Γ, and there exist strictly positive integers f i (i Γ) satisfying v ij f j = v ji f i for each i, j Γ with i j. In the case of v ij 0 which (v is equivalent to v ji 0, we use the symbol ij, v ji ) i j and we call it an edge of (Γ, v). If v ij = v ji = 1, we simply write i the exceptional types, i.e., G 2 : 1 E l (l =6, 7, 8). (1, 3) 2, F 4 : 1. In this paper, we deal with j 2 (1, 2) 3, and 4
2 306 Y. Ishii An orientation Ω of (Γ, v) is given by prescribing an ordering for every edge, (v indicated by an arrow ij, v ji ) i j or (v ij, v ji i )j. Sometimes we write i j (v instead of ij, v ji ) i j for simplicity. A valued graph (Γ, v) with an orientation Ω is called a valued quiver Q = (Γ, v; Ω). For an oriented K-modulation (M, Ω) of a valued graph (Γ, v) with M =((F i ) i Γ, ( i M j ) i,j Γ ) (see Definition 2.1) and a dimension vector d =(d i ) Z n 0, we can define a pair (G d,r d ) with G d = i Γ GL(d i; F i ) and R d = i j Hom F j (F d i i Fi im j,f d j j ) where G d acts on R d (see Definition 2.3). A point x R d is called a generic point of (G d,r d ) if dim K (G d ) x = dim K G d dim K R d where (G d ) x = {g G d ; gx = x} is the isotropy subgroup of G d at x. In our case, (G d,r d ) is a prehomogeneous vector space (abbrev. PV) in the sense that it has generic points (cf. [K]). If characteristic of K is 0, by Proposition 2.7, (G d,r d ) has a dense orbit. Moreover we show that for these type of PVs (G d,r d ), some partial tilting Λ(M, Ω)-modules correspond. For the definition of partial tilting modules, see [ASS; p. 192]. Note that although (G d,r d ) depends on the choice of the orientation and the modulation, the number of the isomorphism classes of basic tilting modules does not depend on the choice of the orientation (see from the below of Proposition 2.12) and the choice of modulation (see Remark 5.5). In this paper, we give a complete list of tilting Λ-modules of type G 2 and F 4 (see Sections 3 and 4); and these results give a classification (see Theorems 3.2 and 4.2) of PVs (G d,r d ) according to corresponding partial tilting modules. For E n -type (n =6, 7, 8), by using a computer, we count the number of isomorphism classes of (partial) tilting modules (see Sections 5,6,7). This paper consists of the following 7 sections. Section 1. Introduction Section 2. Preliminaries Section 3. The Case for exceptional type G 2 Section 4. The Case for exceptional type F 4 Section 5. The Case for exceptional type E 6 Section 6. The Case for exceptional type E 7 Section 7. The Case for exceptional type E 8 2 Preliminaries In the following, we shall always assume that a valued graph (Γ, v) is connected, i.e., for any i, j Γ, there is a sequence of vertices k 1 = i, k 2,...,k t = j such that v ksk s+1 0 for all s =1, 2,...,t 1. An orientation Ω is called ad-
3 Some classification of PVs ass. with Dynkin quivers of exceptional type 307 missible if Q =(Γ, v; Ω) has no oriented cycle. In this paper, we assume that an orientation is admissible. Definition 2.1. (An oriented K-modulation (M, Ω)) We say that M =((F i ) i Γ, ( i M j ) i,j Γ ) is a K-modulation of a valued graph (Γ, v) with v =(v ij ) when it satisfies the following conditions (see [D; p. 32]). 1. Each F i (i Γ) is a finite-dimensional division algebra over a commutative field K. 2. Each i M j is a F i -F j -bimodule on which K acts centrally, and satisfies the following conditions. (a) The dimension over F j of i M j as a right F j -module is v ij, while the dimension over F i of i M j as a left F i -module is v ji. (b) j M i = HomFi ( i M j,f i ) = Hom Fj ( i M j,f j ) as a F j -F i -bimodule. Note that if we put f i = dim K F i, then we have v ij f j = v ji f i from K ( i M j )= ( i M j ) K. For a valued quiver (Γ, v;ω), if we give a F i -F j -module i M j satisfying (v the condition (a) for each ij, v ji ) i j, then we can define j M i by the condition (b), so that we obtain a K-modulation (see [D; p. 32]). Now let (M, Ω) be a pair of a K-modulation of a valued graph (Γ, v) and an orientation Ω of (Γ, v) which we call an oriented K-modulation of a valued graph (Γ, v). Definition 2.2. (The abelian category rep(m, Ω)) A representation W =( j ϕ i,w i ) of an oriented K-modulation (M, Ω) is a pair of finite-dimensional right F i -vector spaces W i for i Γ and F j -linear mappings j ϕ i : W i Fi im j W j for each arrow i j. A morphism from W =( j ϕ i,w i ) to W =( j ϕ i,w i ) is a set α =(α i) i Γ of F i -linear mappings α i : W i W i (i Γ) such that the following diagram is commutative for every arrow i j. jϕ i W i Fi im j Wj α i 1 W i Fi im j α j jϕ i W j We denote by Hom(W, W ) the totality of all morphisms from W to W. Thus we have an abelian category rep(m, Ω) (see [D]). The category rep(m, Ω) is said to be of finite (representation) type if it possesses only a finite number of non-isomorphic indecomposable representations. It is known that rep(m, Ω) is of finite representation type if and only if (Γ, v) is a Dynkin graph (see [D; Theorem 3.1]).
4 308 Y. Ishii Definition 2.3. (A representation (G d,r d ) associated with (M, Ω)) Let (M, Ω) be an oriented K-modulation of a valued graph (Γ, v) with Card(Γ) = n and M =(F i, i M j ). For a dimension vector d =(d i ) Z n 0, the direct product G d = i Γ GL(d i; F i ) acts on the space R d = i j Hom F j (F d i i im j,f d j j ) of representations ( j x i,f d i i ) of (M, Ω) by R d x =( j x i ) i j x = ( j x i ) i j R d with j x i = g j jx i (g i 1) 1 for each i j. F d i jx i d i Fi im j F j j g i 1 g j F d i jx i Fi im i j F d j j By fixing a basis of F d i i Fi im j over F j, we can identify Hom Fj (F d i i Fi im j,f d j j ) with M(d j,v ij d i ; F j ) and hence we have R d = i j M(d j,v ij d i ; F j ). We call (G d,r d ) a representation associated with an oriented K-modulation (M, Ω). In the case where (Γ, v) is a Dynkin graph, such a representation (G d,r d ) is a finite PV, i.e., R d has only a finitely many G d -orbits. Note that dim K G d = i Γ f id 2 i and dim K R d = i j v ijf j d i d j. Definition 2.4. (The tensor algebra Λ of an oriented K-modulation (M, Ω)) We define the tensor algebra of an oriented K-modulation (M, Ω) by Λ= Λ(M, Ω) = t 0 M(t) where M (0) = F 1 F n, M (1) = i j im j, and inductively M (t+1) = M (t) M (0) M(1) where M (1) has the structure of the M (0) M (0) bimodule induced by the projections M (0) F i. Note that if j k, then we have i M j k M l =0since i M j k M l = i M j e j e k km l = i M j (e j e k ) km l =0where e j =(0,...,0, 1 Fj, 0,...,0) M (0). Hence we have M (t) = k 1 M k2 k2 M k3 kt M kt+1 where the summation runs over the paths k 1 k 2 k t+1. Therefore if the maximal length of the paths in the valued quiver Q =(Γ, v, Ω) is s, we have Λ=M (0) M (1) M (s). Multiplications in Λ is defined by the tensor product by identifying M (r) M (s) = M (r+s) and distributively. We have a category mod Λ of right Λ-module of finite length. Remark 2.5. (Equivalence of categories between rep(m, Ω) and mod Λ(M,Ω)) It is well-known (see [D; Proposition 1.2]) that there is an equivalence between the category rep(m, Ω) of representations of an oriented K-modulation (M, Ω) and the category mod Λ of right Λ-modules of finite length with Λ= Λ(M, Ω). For an object W =( j ϕ i,w i ) of rep(m, Ω), we construct a right Λ-module W as follows. Put W = i Γ W i which is naturally a Z-module. We define the right action of Λ = t 0 M(t) on W inductively. For (f i ) M (0) = F 1 F n and (w i ) i Γ W, define the right action W M (0) W by ((w i ) i Γ, (f i ) i Γ ) (w i f i ) i Γ. Fi
5 Some classification of PVs ass. with Dynkin quivers of exceptional type 309 For m ij i M j M (1) = i j im j and (w i ) i Γ W, define the right action W i M j W by ((w k ) k Γ,m ij ) (w t) t Γ where w j = j ϕ i (w i m ij ) if t = j and w t =0otherwise. Then inductively by w i(m ij m pq m qs )= s ϕ q (w i (m ij m pq ) m qs ), we obtain the right action of M (t) = M (t 1) M (0) M(1), and hence we have the right action of Λ= t 0 M(t) on W, i.e., W is a right Λ-module of finite length. For α =(αi ) i Γ : W W, the map α : W W defined by α((w i ) i )=(α i (w i )) i is clearly Λ-linear. Conversely for a right Λ-module X of finite length, i.e., an object of mod Λ, put X i = XF i which is a finite dimensional right F i -vector space for i Γ. Note that F i Λ. Then the action of i M j ( Λ) on each X i induces the F j -linear maps j ϕ i : X i Fi im j X j for each arrow i j. Thus we have an object ( j ϕ i,x i ) of the category rep(m, Ω). We define the dimension vector of a right Λ-module X by dim X = (dim Fi X i ) i Γ Z n 0 with n = Card(Γ). By this correspondence, we can regard each point x =( j x i ) i j R d = i j M(d j,v ij d i ; F j ) as a Λ-module X = F d 1 1 Fn dn on which (m ij) i j M (1) = i j im j ( Λ) acts as X x =(x 1,...,x n ) j 1 i j ( {}}{ 0,...,0, j x i (x i m ij ), 0,...,0) X. The orbit G d -orbit G d x corresponds to the isomorphism class of Λ-module X. Definition 2.6. Let, be the Ringel form of an oriented K-modulation (M, Ω) of a valued diagram (Γ, v), i.e., for vectors x =(x i ) i Γ, y =(y i ) i Γ Z n 0, x, y = xr t y, where R =(r ij ) i,j Γ with r ii = f i (i Γ),r ij = dim Ki M j (= (v v ij f j ) if ij, v ji ) i j ; and rij =0otherwise. Then we have dim X, dim Y = dim K Hom Λ (X, Y ) dim K Ext 1 Λ(X, Y ) for each Λ-modules X, Y (see [D; Proposition 2.2]). Proposition 2.7. Let K be a field of characteristic 0. Let G be a connected algebraic group defined over K acting on the vector space V with a K-structure V K. Then the K-rational points G K of G acts on V K. For a point x V, let G x = {g G; g x = x} be the isotropy subgroup of G at x. If a point x V K is a generic point, i.e., it satisfies dim K (G K ) x = dim K G K dim K V K, then we have G K x = V K with the Zariski topology in V K. Proof. Let Ω be a universal field. Since G = G K K Ω and V = V K K Ω, we have dim G x = dim G dim V, and hence we have G x = V (see [K]). By [B; p. 220], we have G K = G. Since the map f : G V defined by f(g) =g x is continuous with respect to the Zariski topology, we have G x = f(g) = f(g K ) f(g K )=G K x. Since G x = V, we have G K x = V with the Zariski topology in V. Since G K x V K and the Zariski topology in V K is weaker than the induced topology on V K from the Zariski topology in V,we obtain our result.
6 310 Y. Ishii Definition 2.8. A non-zero rational function f(x) on V K is called a relative invariant of (G K,V K ) if there exists a rational character χ of G K satisfying f(g x) = χ(g)f(x) as a rational function for all g G. By Proposition 2.7, if (G K,V K ) is a prehomogeneous vector space, i.e., there exists a generic point, then a relative invariant f(x) is uniquely determined by its corresponding character χ up to a constant multiple (cf. [K]). Let X(G K ) be the group of rational characters of G K. Proposition 2.9. Let (G K,V K ) be a prehomogeneous vector space with a generic point x 0. Let f 1,...,f l be relative invariants corresponding to characters χ 1,...,χ l respectively. Assume that these characters χ 1,...,χ l generate the subgroup X 1 (G K )={χ X(G K ); χ (GK ) x0 =1}. Then these f 1,...,f l generate all relative invariants of (G K,V K ). Proof. Let f(x) be any relative invariant of (G K,V K ) corresponding to a character χ. For any g (G K ) x0, we have f(g x 0 )=f(x 0 )=χ(g)f(x 0 ). By Proposition 2.7, we have f(x 0 ) 0,, and hence χ(g) = 1, i.e., χ X 1 (G K ). Hence we can express χ = χ m 1 1 χ m l l f(x) =cf m 1 1 f m l l with some (m 1,...,m l ) Z l, we have with some constant multiple. Lemma Let (G d,r d ) be a PV associated with an oriented K-modulation (M, Ω) of a valued graph (Γ, v), and X a Λ-module corresponding to a point x R d where Λ=Λ(M, Ω). Then x is a generic point of the PV if and only if Ext 1 Λ (X, X) =0, i.e., X is a partial tilting Λ-module. Proof. Since Λ is hereditary, we have dim K G d dim K R d = dim K Hom Λ (X, X) dim K Ext 1 Λ (X, X) = dim K(G d ) x dim K Ext 1 Λ (X, X). Hence, x is a generic point if and only if Ext 1 Λ (X, X) =0. By Proposition 2.7 and Lemma 2.10, we see that (the dense orbit of) a PV (G d,r d ) corresponds to the isomorphism class of some partial tilting Λ- module. Definition (Positive roots and indecomposable representations) Let (Γ, v) be a valued graph with Γ = {1, 2,...,n}. For each vertex k Γ, define the reflection r k : Q n Q n by x = (x i ) (y i ) where y k = x k + i Γ v ikx i and y i = x i (i k). The group W generated by the reflections is called the Weyl group of (Γ, v). An element x Q n is called arootof(γ, v) if there exists w W and i Γ satisfying x = we i where i e i =(0,...,0, ˇ1, 0,...,0). An element x =(x i ) Q n is called positive if all x i 0, and we write x 0. Note that positive roots belong to Z n 0. It is known that, in our case, the map rep(m, Ω) Z n 0 defined by W dim W induces the bijection between the isomorphism classes of indecomposable representation
7 Some classification of PVs ass. with Dynkin quivers of exceptional type 311 and the positive roots of (Γ, v) (see [DR; p. 2]). On the other hand, by Lemma 2.10, for a positive root d, the corresponding indecomposable representations x R d are generic points of (G d,r d ) since the corresponding Λ-module X of an indecomposable representation x has the property Ext 1 Λ (X, X) =0. By Theorem of Krull-Schmidt, any representation is uniquely decomposed to the direct sum of indecomposable representations. Let us decompose a representation x R d into the direct sum x = m 1 x 1 m s x s of indecomposable representations x 1,...,x s which are not isomorphic of each other and m k x k denotes the m k -copies of x k. Then x is a generic point if and only if the corresponding Λ-module m 1 X 1 m s X s is a partial tilting module. When m 1 = = m s = 1, such a module is called basic. It is known that Λ-module m 1 X 1 m s X s is a partial tilting module if and only if a basic Λ-module X 1 X s is a partial tilting modules. Hence x = x 1 x s R d is a generic point of a PV (G d,r d ) if and only if x = m 1 x 1 m s x s R d is a generic point of a PV (G d,r d ) for any m 1 1,...,m s 1. In general, for a partial tilting Λ-module X, there exists a Λ-module Y such that X Y is a tilting module which is, by definition, the number of its non-isomorphic direct summand is equal to n = Card(Γ) (see [ASS; p. 196]) where Λ = Λ(M, Ω) and (M, Ω) is a K-modulation of a valued graph (Γ, v). Therefore, to find a generic point of a PV (G d,r d ) associated with an oriented K-modulation (M, Ω) of a valued graph (Γ, v), it is sufficient to give a complete list of basic tilting Λ(M, Ω)-modules. Proposition For x =( j x i ) R d = i j Hom F j (F d i i x =( j x i) R d = i j Hom F j (F d i i and X = F d 1 1 F d n n Fi im j,f d j j ) and Fi im j,f d j j ), let X = F d 1 1 Fn dn be corresponding Λ-modules. Then a linear map α : X X with α =(α 1,...,α n ) M(d 1,d 1; F 1 ) M(d n,d n; F n ) is a Λ-homomorphism if and only if α j jx i = j x i (α i 1) for each arrow i j. Proof. Since Λ is generated by i M j, it is enough to show that α(m ij x) = m ij α( x) for m ij i M j and x =(x 1,...,x n ) X. Since α(m ij x) = (0,...,0,α j ( j x i (x i m ij )), 0,...,0) and m ij α( x) =m ij (α 1 x 1,...,α n x n )= (0,...,0, j x i ((α ix i ) m ij ), 0,...,0), we obtain our result. Now we shall consider the relation with the case of another orientation Ω. Definition Let Q =(Γ, v;ω) be a valued quiver with Γ={1, 2,...,n}. We call k Γ a sink (resp. source) if i k (resp. j k) for any arrow i j. For an orientation Ω and k Γ, we define the new orientation s k Ω by changing the direction of all arrows which contain k, and keep the arrows which do not contain k. An ordered set (k 1,...,k n ) of all vertices is called an
8 312 Y. Ishii admissible sequence of sinks if k 1 is a sink with respect to Ω and k t is a sink with respect to s kt 1 s k1 Ω for all t with 2 t n. Then the following facts hold (see [D; p. 30]). 1. s kn s kn 1 s k1 Ω=Ω 2. s k1 s k2 s kn Ω=Ω 3. Each k t is a source with respect to s kt+1 s kn Ω for 1 t n. Proposition For any sink k Γ, the reflection functor Δ + k : rep(m, Ω) rep(m,s k Ω) induces the categorical equivalence rep (k) (M, Ω) = rep (k) (M,s k Ω) where rep (k) (M, Ω) (resp. rep (k) (M,s k Ω)) is a full subcategory of rep(m, Ω) (resp. rep(m,s k Ω)) consisting of representations which do not contain, as their direct summands, the representation k e k =( j ϕ i,w i ) with the dimension vector e k =(0,...,0, ˇ1, 0,...,0), i.e., W k = F k,w i =0for all i k and all j ϕ i =0. 2. Similarly for any source k Γ, the reflection functor Δ k : rep(m, Ω) rep(m,s k Ω) is defined. For an admissible sequence of sinks (k 1,...,k n ), the covariant functor Δ : rep(m, Ω) rep(m, Ω) called the Coxeter functor defined by Δ =Δ k 1 Δ k 2 Δ k n induces the categorical equivalence rep (M, Ω) = rep (M, Ω) where rep (M, Ω) (resp. rep (M, Ω)) is a full subcategory of rep(m, Ω) consisting of representations without injective (resp. projective) direct summand. For X, X Ob(rep (M, Ω)), we have a K-linear isomorphism Hom(X, X ) = Hom(Δ X, Δ X ). Proof. For 1, see [D; p. 62]. For 2, see [D; Proposition 2.10]. Proposition ([D; Lemma 2.5, Corollary 2.7 (i)]) Let k Γ be a sink with respect to Ω. 1. If X Ob(rep (k) (M, Ω)) is indecomposable, then Δ + k X Ob(rep(k) (M,s k Ω)) is also indecomposable, and we have dim Δ + k X = r k(dim X). 2. For X, X Ob(rep (k) (M, Ω)), we have Ext 1 (X, X ) = Ext 1 (Δ + k X, Δ+ k X ). Proposition Let k Γ be a sink with respect to Ω. Then we have Ext 1 Λ(M,s k Ω) (X, e k)=0for any Λ(M,s k Ω)-module X. Proof. Since k Γ is a source of s k Ω, and it is known that there exists an admissible sequence of sinks (i 1,...,i n ) with i n = k with respect to s k Ω (see [ASS; p. 279]), we have e k Ob(rep(M,s k Ω)) is injective by [D; Lemma 2.1]. Hence we obtain our result.
9 Some classification of PVs ass. with Dynkin quivers of exceptional type 313 Proposition ([D; Corollary 2.4 (ii), p. 62]) Let k Γ be a sink with respect to Ω. LetY Ob(rep(M,s k Ω)) be a representation without e k as its direct summand. Then we have dim Fk Ext 1 Λ(M,s k Ω)(e k,y)= [r k (dim Y )] k where [(d 1,...,d n )] k = d k. Note that dim K Ext 1 Λ(M,s k Ω)(e k,y)= f k dim Fk Ext 1 Λ(M,s k Ω) (e k,y) where f k = dim K F k. Theorem If we obtain the table of (dim K Ext 1 Λ(X i,x j )) with respect to some orientation Ω, then we can obtain the similar table (dim K Ext 1 Λ (Y i,y j )) with respect to another orientation Ω from (dim K Ext 1 Λ (X i,x j )). Proof. By Propositions , we obtain our result. Hence it is essentially enough to calculate the table of (dim K Ext 1 Λ (X i,x j )) with respect to any one of orientations Ω. Our results show that the number of (partial) tilting Λ-modules does not depend on the choice of an orientation of arrows. The following proposition is well-known. Proposition In general, if End R (M) is a local ring for an R-module M, then M is indecomposable. The converse also holds if M has the composition series. 3 The Case for exceptional type G 2 In ( this) section, we consider the valued graph (Γ, v) with Γ = {1, 2} and v = 0 1, with an orientation Ω: (1, 3) 2, i.e., of type G 2. We take a K- modulation M = {F 1 = K, F 2 = L, 1M 2 = K L L } where L/K is a cubic extension obtained by adding α to K satisfying α, α 2 / K and α 3 K. We denote L as a K-L-bimodule by K L L. Then, for a dimension vector d = (d 1,d 2 ) Z 2 0, we have the representation (G d,r d ) with G d = GL(d 1 ; K) GL(d 2 ; L) and R d = M(d 2,d 1 ; L). The action of G d on R d is given by R d x BxA 1 R d for (A, B) G d. For a positive root α =(d 1,d 2 ), let x = 2 ϕ 1 : K d 1 K L L d 2 be the corresponding indecomposable representation. If we identify K d 1 K L with L d 1, we may regard x = 2 ϕ 1 as an element of M(d 2,d 1 ; L). Then the Λ-action of the corresponding Λ-module X = K d 1 L d 2 is given by x 1 y 1 x 1 l X (.,. ) ( 0., 2 ϕ 1 (. )) 0 x d1 l x d1 y d2
10 314 Y. Ishii for l L = 1 ( M 2 = ) M (1) Λ. There are 6 positive roots α k (1 k 6) of (Γ = 0 1 {1, 2}, v = ), which is listed below. We denote by x 3 0 k (resp. X k )an indecomposable representation (resp. Λ-module) corresponding to a positive root α k. Note that x k is a generic point of (G αk,r αk ) (see Definition 2.11). An indecomposable representation x k and the Λ-module X k corresponding to a positive root α k is given as follows where Λ = Λ(M, Ω) (see Definition 2.4). α 1 =(0, 1) x 1 =0:{0} L X 1 = L α 2 =(1, 0) x 2 =0:L {0} X 2 = K α 3 =(1, 1) x 3 =1:L L X 3 = K L α 4 =(2, 1) x 4 =(1α) :L 2 L X 4 = K 2 L α 5 =(3, 1) x 5 =(1αα ( 2 ):L) 3 L X 5 = K 3 L α 0 1 α 6 =(3, 2) x 6 = 0 α 2 : L 1 3 L 2 X 6 = K 3 L 2 Theorem 3.1. (The basic tilting Λ(M, Ω)-modules of type G 2 ) For the tensor algebra Λ of the oriented K-modulation (M, Ω), there exist exactly 5 isomorphism classes of basic tilting Λ-modules whose complete list is given by the following X 1 X 3 = L (K L) 2. X 2 X 5 = K (K 3 L) 3. X 3 X 6 =(K L) (K 3 L 2 ) 4. X 4 X 5 =(K 2 L) (K 3 L) 5. X 4 X 6 =(K 2 L) (K 3 L 2 ) Proof. In our case, we have r 11 = dim K K =1,r 22 = dim K L =3,r 12 = dim K ( 1 M 2 = 3, )( and ) hence the Ringel form is given by (x 1,x 2 ), (y 1,y 2 ) = 1 3 y1 (x 1,x 2 ) (see Definition 2.6). Since dim X 0 3 y k = α k (k =1,...,6), 2 we have the following matrix (3.1) of dim X i, dim X j (3.1)
11 Some classification of PVs ass. with Dynkin quivers of exceptional type 315 We can calculate dim K Hom Λ (X i,x j ) by Proposition 2.12, and its matrix is given by (3.2) Since dim K Ext 1 Λ (X i,x j ) = dim K Hom Λ (X i,x j ) dim X i, dim X j, we have the following matrix (3.3) of dim K Ext 1 Λ(X i,x j ) from (3.2) (3.3) Since X = X i X j is a tilting module if and only if dim K Ext 1 Λ (X, X) = dim K Ext 1 Λ (X i,x i )+dim K Ext 1 Λ (X i,x j )+dim K Ext 1 Λ (X j,x i )+dim K Ext 1 Λ (X j,x j )= 0, we obtain our result by (3.3). Theorem 3.2. (PVs corresponding to (partial) tilting modules of type G 2 ) For a dimension vector d =(d 1,d 2 ) Z 2 >0, a (partial) tilting module corresponding to a prehomogeneous vector space (GL(d 1 ; K) GL(d 2 ; L),M(d 2,d 1 ; L)) is given as follows. Note that if it corresponds to a 1 X i1 a 2 X i2 (resp. b j X j ), then the basic tilting module X i1 X i2 (resp. the basic partial tilting module X j ) is its invariant. 1. If d 2 >d 1, it corresponds to (d 2 d 1 )X 1 d 1 X If d 1 > 3d 2, it corresponds to (d 1 3d 2 )X 2 d 2 X If d 1 >d 2 > 2 3 d 1, it corresponds to (3d 2 2d 1 )X 3 (d 1 d 2 )X If 3d 2 >d 1 > 2d 2, it corresponds to (3d 2 d 1 )X 4 (d 1 2d 2 )X If 2d 2 >d 1 > 2 3 d 1 >d 2, it corresponds to (2d 1 3d 2 )X 4 (2d 2 d 1 )X If d 1 = d 2, it corresponds to d 1 X If d 1 =3d 2, it corresponds to d 2 X If d 1 = 3 2 d 2 (and hence d 2 is even), it corresponds to d 2 2 X 6.
12 316 Y. Ishii 9. If d 1 =2d 2, it corresponds to d 2 X 4. Proof. Since m i X i m j X j (i j) is a partial tilting module if and only if X i X j is a partial tilting module, we obtain all partial tilting modules by Theorem 3.1. For example, a tilting module m 1 X 1 m 3 X 3 corresponds to d =(d 1,d 2 )=m 1 α 1 + m 3 α 3 =(m 3,m 1 + m 3 ). Hence we have d 1 = m 3 <d 2 = m 1 + m 3 and X =(d 2 d 1 )X 1 d 1 X 3, i.e., we obtain 1. The remaining parts are obtained similarly. Remark 3.3. (The other orientation) Since the vertex 2 is the sink for the orientation Ω, we can apply Proposition 2.15 for k =2. Now we shall consider the other orientation Ω : 1 (1, 3) 2, and put Λ =Λ(M,s 2 Ω). Let Y 1,...,Y 6 be indecomposable Λ -modules corresponding to positive roots α 1,...,α 6. Since s 2 ((d 1,d 2 )) = (d 1, d 2 + d 1 ), we have the correspondence: s 2 ((1, 0)) = (1, 1), Δ + 2 X 2 = Y3 ; s 2 ((1, 1)) = (1, 0), Δ + 2 X 3 = Y 2 ; s 2 ((2, 1)) = (2, 1), Δ + 2 X 4 = Y 4 ; s 2 ((3, 1)) = (3, 2), Δ + 2 X 5 = Y 6 ; s 2 ((3, 2)) = (3, 1), Δ + 2 X 6 = Y5. Hence by Proposition 2.15, we have Ext 1 Λ (Y i,y j )(i, j 2). By Proposition 2.16, we have Ext 1 Λ (Y i,y 1 )=0(i 1). By Proposition 2.17, we have dim F2 Ext 1 Λ (Y 1,Y 2 )=[s 2 (dim Y 2 )] 2 =[s 2 (1, 0)] 2 = [(1, 1)] 2 =1. Since F 2 = L and [L : K] =3, we have dim K Ext 1 Λ (Y 1,Y 2 )=3. We can calculate similarly dim K Ext 1 Λ (Y 1,Y j )(j 3), and hence we obtain the table of dim K Ext 1 Λ(M,Ω )(Y i,y j ), which is actually given by the transposed matrix of (2.3). In particular, the number of (partial) tilting Λ-modules does not depend on the choice of an orientation of arrows. Remark 3.4. (The relative invariants) We define the injection ϕ : L M(3; K) by a (1,α,α 2 )=(1,α,α 2 )ϕ(a), p rα 3 qα 3 i.e., ϕ(a) = q p rα 3 for a = p + qα + rα 2 L = K + Kα + Kα 2. r q p Define the injective homomorphism Φ:GL(d 2 ; L) GL(3d 2 ; K) by Φ((z ij )= (ϕ(z ij )). We also define the K-isomorphism Ψ:M(d 2,d 1 ; L) = M(3d 2,d 1 ; K) by s Ψ((z ij )) = (ψ(z ij )) with ψ : L K 3 given by ψ(s + tα + uα 2 )= t. u Now let d =(d 1,d 2 ) be the one of the positive roots α 3,α 4,α 5,α 6. Then we define the pair (G K d,rk d ) by G K d = GL(d 1; K) Φ(GL(d 2 ; L)) ( GL(d 1 ; K) GL(3d 2 ; K)) and Rd K =Ψ(M(d 2,d 1 ; L)) = M(3d 2,d 1 ; K). The action of G K d on RK d is given by (g 1, Φ(g 2 )) Ψ(x) =Ψ(g 2 xg1 1 ) for (g 1,g 2 )
13 Some classification of PVs ass. with Dynkin quivers of exceptional type 317 G d and x R d. Since (G d,r d ) is a PV, by Proposition 2.7, there exists a dense G K d -orbit. If d = α i, then Ψ(x i ) is a generic point (i =3, 4, 5, 6). To obtain the all relative invariants (see Definition 2.8), it is enough to obtain the relative invariants whose corresponding characters generate the group X 1 (G K ) by Proposition 2.9. Actually we can construct them similarly to [S]. As an example, we show the case d =(3, 2). In this case, the relative invariant f(ψ(x)) with X = (x ij + y ij α + z ij α 2 ) M(2, 3; L) is given by det X 11 X 12 X 13 O O O A 11 O B 11 O C 11 O X 21 X 22 X 23 O O O O A 11 O B 11 O C 11 O O O X 11 X 12 X 13 A 12 O B 12 O C 12 O O O O X 21 X 22 X 23 O A 12 O B 12 O C 12 where X ij = ψ(x ij + y ij α + z ij α 2 ) M(3, 1), A ij = ψ(a ij + b ij α + c ij α 2 ) M(3, 1; K), B ij = ψ(α(a ij + b ij α + c i,j α 2 )) M(3, 1; K), C ij = ψ(α 2 (a ij + b ij α + c ij α 2 )) M(3, 1; K). We should choose constants so that f(ψ(x)) is not identically zero. As far as it satisfies this condition, the choice of constant gives just a constant multiple of the relative invariant ([S]). For example, we may choose as a 11 + b 11 α + c 11 α 2 =1and a 12 + b 12 α + c 12 α 2 = α. 4 The Case for exceptional type F 4 In this section, we consider the valued graph (Γ, v) with Γ = {1, 2, 3, 4} and v = with an orientation Ω : 1 2 (1, 2) 3 4, i.e., of type F 4. We take a K-modulation M = {F 1 = F 2 = K, F 3 = F 4 = L, 1 M 2 = K K K, 2 M 3 = K L L, 3 M 4 = L L L } where L/K is a quadratic extension obtained by adding a square root α of an element of K. Then, for a dimension vector d =(d 1,d 2,d 3,d 4 ) Z 4 0, we have the representation (G d,r d ) with G d = GL(d 1 ; K) GL(d 2 ; K) GL(d 3 ; L) GL(d 4 ; L) and R d = M(d 2,d 1 ; K) M(d 3,d 2 ; L) M(d 4,d 3 ; L). The action of G d on R d is given by R d x = (x, y, z) g x =(BxA 1,CyB 1,DzC 1 ) R d for g =(A, B, C, D) G d. For a positive root α =(d 1,d 2,d 3,d 4 ), the corresponding indecomposable representation x =( 2 ϕ 1, 3 ϕ 2, 4 ϕ 3 ) is given by maps 2 ϕ 1 : K d 1 K K( = K d 1 ) K d 2, 3ϕ 2 : K d 2 K L( = L d 2 ) L d 3 and 4 ϕ 3 : L d 3 L L( = L d 3 ) L d 4. We
14 318 Y. Ishii may regard x as an element of M(d 2,d 1 ; K) M(d 3,d 2 ; L) M(d 4,d 3 ; L). The Λ-action on the corresponding Λ-module X = K d 1 K d 2 L d 3 L d 4 is given by x 1 y 1 z 1 w 1 x 1 k y 1 l z 1 l (.,.,.,. ) ( 0., 2 ϕ 1., 3 ϕ 2., 4 ϕ 3. ) 0 x d1 k y d2 l z d3 l x d1 y d2 z d3 w d4 for (k, l, l ) K L L = 1 M 2 2 M 3 3 M 4 = M (1) Λ. There are 24 positive roots α k (1 k 24) of the valued graph (Γ, v) of type F 4, which is listed below. We denote by x k (resp. X k ) an indecomposable representation (resp. Λ-module) corresponding to a positive root α k. Note that x k is a generic point of a PV (G αk,r αk ). An indecomposable representation x k corresponding to a positive root α k is given as follows. α 1 =(0, 0, 0, 1) x 1 =(0, 0, 0) X 1 = {0} {0} {0} L α 2 =(0, 0, 1, 0) x 2 =(0, 0, 0) X 2 = {0} {0} L {0} α 3 =(0, 0, 1, 1) x 3 =(0, 0, 1) X 3 = {0} {0} L L α 4 =(0, 1, 0, 0) x 4 =(0, 0, 0) X 4 = {0} K {0} {0} α 5 =(0, 1, 1, 0) x 5 =(0, 1, 0) X 5 = {0} K L {0} α 6 =(0, 1, 1, 1) x 6 =(0, 1, 1) X 6 = {0} K L L α 7 =(0, 2, 1, 0) x 7 =(0, ( 1 α ), 0) X 7 = {0} K 2 L {0} α 8 =(0, 2, 1, 1) x 8 =(0, ( 1 α ), 1) X 8 = {0} K 2 L L α 9 =(0, 2, 2, 1) x 9 =(0,I 2, ( 1 α ) ) X 9 = {0} K 2 L 2 L α 10 =(1, 0, 0, 0) x 10 =(0, 0, 0) X 10 = K {0} {0} {0} α 11 =(1, 1, 0, 0) x 11 =(1, 0, 0) X 11 = K K {0} {0} α 12 =(1, 1, 1, 0) x 12 =(1, 1, 0) X 12 = K K L {0} α 13 =(1, 1, 1, 1) x 13 =(1, ( 1, ) 1) X 13 = K K L L 1 α 14 =(1, 2, 1, 0) x 14 =(, ( 1 α ), 0) X 0 14 = K K 2 L {0} ( ) 1 α 15 =(1, 2, 1, 1) x 15 =(, ( 1 α ), 1) X 0 15 = K K 2 L L ( ) 1 α 16 =(1, 2, 2, 1) x 16 =(,I 0 2, ( 1 α ) ) X 16 = K K 2 L 2 L 1 ( ) α 17 =(1, 3, 2, 1) x 17 =( 0 α 0 0,, ( 0 1 ) ) 0 1 α 1 X 17 = K K 3 L 2 L α 18 =(2, 2, 1, 0) x 18 =(I 2, ( 1 α ), 0) X 18 = K 2 K 2 L {0} α 19 =(2, 2, 1, 1) x 19 =(I 2, ( 1 α ), 1) X 19 = K 2 K 2 L L α 20 =(2, 2, 2, 1) x 20 =(I 2,I 2, ( 1 α ) ) X 20 = K 2 K 2 L 2 L 1 0 ( ) α 21 =(2, 3, 2, 1) x 21 =( 0 0 α 0 0,, ( 1 α ) ) 0 1 α 0 1
15 Some classification of PVs ass. with Dynkin quivers of exceptional type 319 X 21 = K 2 K 3 L 2 L 1 0 ( ) α 22 =(2, 4, 2, 1) x 22 =( α ,, ( 1 1 ) ) α 0 1 X 22 = K 2 K 4 L 2 L 1 0 α 23 =(2, 4, 3, 1) x 23 =( 0 0 α , 0 1 α 0, ( ) ) X 23 = K 2 K 4 L 3 L 1 0 α 24 =(2, 4, 3, 2) x 24 =( 0 0 α ( ) 0 0, 0 1 α , ) X 24 = K 2 K 4 L 3 L 2 Theorem 4.1. (The basic tilting Λ(M, Ω)-modules of type F 4 ) For the tensor algebra Λ of the oriented K-modulation (M, Ω) of type F 4, there exist exactly 66 isomorphism classes of basic tilting Λ-modules. They are given by the following list. For example, (1, 3, 6, 13) indicates the tilting module X 1 X 3 X 6 X 13 = L (L L) (K L L) (K K L L). (1) (1, 3, 6, 13) (2) (1, 3, 10, 13) (3) (1, 4, 8, 15) (4) (1, 4, 11, 19) (5) (1, 4, 15, 19) (6) (1, 6, 8, 13) (7) (1, 8, 13, 15) (8) (1, 10, 11, 19) (9) (1, 10, 13, 19) (10) (1, 13, 15, 19) (11) (2, 3, 6, 13) (12) (2, 3, 10, 13) (13) (2, 5, 9, 16) (14) (2, 5, 12, 20) (15) (2, 5, 16, 20) (16) (2, 6, 9, 13) (17) (2, 9, 13, 16) (18) (2, 10, 12, 20) (19) (2, 10, 13, 20) (20) (2, 13, 16, 20) (21) (4, 7, 8, 15) (22) (4, 7, 14, 22) (23) (4, 7, 15, 22) (24) (4, 11, 18, 19) (25) (4, 14, 18, 19) (26) (4, 14, 19, 22) (27) (4, 15, 19, 22) (28) (5, 7, 8, 17) (29) (5, 7, 12, 23) (30) (5, 7, 17, 23) (31) (5, 8, 9, 16) (32) (5, 8, 16, 24) (33) (5, 8, 17, 24) (34) (5, 12, 20, 23) (35) (5, 16, 20, 24) (36) (5, 17, 20, 23) (37) (5, 17, 20, 24) (38) (6, 8, 9, 13) (39) (7, 8, 15, 17) (40) (7, 12, 14, 22) (41) (7, 12, 21, 22) (42) (7, 12, 21, 23) (43) (7, 15, 17, 23) (44) (7, 15, 21, 22) (45) (7, 15, 21, 23) (46) (8, 9, 13, 16) (47) (8, 13, 15, 24) (48) (8, 13, 16, 24) (49) (8, 15, 17, 24) (50) (10, 11, 18, 19) (51) (10, 12, 18, 19) (52) (10, 12, 19, 20) (53) (10, 13, 19, 20) (54) (12, 14, 18, 19) (55) (12, 14, 19, 22) (56) (12, 19, 20, 21) (57) (12, 19, 21, 22) (58) (12, 20, 21, 23) (59) (13, 15, 19, 20) (60) (13, 15, 20, 24) (61) (13, 16, 20, 24) (62) (15, 17, 20, 23) (63) (15, 17, 20, 24) (64) (15, 19, 20, 21) (65) (15, 19, 21, 22) (66) (15, 20, 21, 23)
16 320 Y. Ishii Proof. To express the matrix (dim K Hom Λ (X i,x j )) simply, we introduce Y i (1 i 24) as follows (cf. 5). Y 1 = X 13,Y 2 = X 6,Y 3 = X 3,Y 4 = X 1,Y 5 = X 5,Y 6 = X 16,Y 7 = X 9,Y 8 = X 2,Y 9 = X 15,Y 10 = X 17,Y 11 = X 24,Y 12 = X 8,Y 13 = X 12,Y 14 = X 21,Y 15 = X 23,Y 16 = X 20,Y 17 = X 4,Y 18 = X 14,Y 19 = X 22,Y 20 = X 7,Y 21 = X 10,Y 22 = X 11,Y 23 = X 18,Y 24 = X 19. Then we have the following table (4.1) of dim K Hom Λ (Y i,y j ). A 1 A 2 A 3 A 4 A 5 A 6 0 A 1 A 2 A 3 A 4 A A 1 A 2 A 3 A A 1 A 2 A A 1 A A 1 (4.1) and each 4 4 matrices A 1...A 6 is given by ) ) A 1 =, A 2 =, A 3 = A 5 = ( ( ), A 6 = ( ( ). ( ), A 4 = ( ) , 0222 The table of the Ringel form Y i,y j is given by (4.2). A 1 A 2 A 3 A 4 A 5 A 6 t A 1 A 1 A 2 A 3 A 4 A 5 t A 2 t A 1 A 1 A 2 A 3 A 4 t A 3 t A 2 t A 1 A 1 A 2 A 3 t A 4 t A 3 t A 2 t A 1 A 1 A 2 t A 5 t A 4 t A 3 t A 2 t A 1 A 1 (4.2) Since dim K Ext 1 Λ (Y i,y j ) = dim K Hom Λ (Y i,y j ) dim Y i, dim Y j, we have the following table (4.3) of dim K Ext 1 Λ(Y i,y j ) from (4.1), (4.2) t A t A t 2 A t A t 3 A t 2 A (4.3) t A t 4 A t 3 A t 2 A t A t 5 A t 4 A t 3 A t 2 A 1 0 Thus we obtain our result.
17 Some classification of PVs ass. with Dynkin quivers of exceptional type 321 From Theorem 4.1, we have immediately a complete list of PVs associated (1, 2) with the F 4 -type quiver. Although (G d,r d )isa finite PV for any dimension d and also a trivial PV in the sense of [K], our list can be regarded as a finer classification of such PVs. Theorem 4.2. (PVs corresponding to tilting modules of type F 4 ) For a dimension vector d =(d 1,d 2,d 3,d 4 ) Z 4 >0, a tilting module corresponding to a prehomogeneous vector space (GL(d 1 ; K) GL(d 2 ; K) GL(d 3 ; L) GL(d 4 ; L),M(d 2,d 1 ; K) M(d 3,d 2 ; L) M(d 4,d 3 ; L)) is given as follows. Note that if it corresponds to a 1 X i1 a 2 X i2 a 3 X i3 a 4 X i4, then the basic tilting module X i1 X i2 X i3 X i4 is its invariant. 1. If d 1 <d 2 <d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 3 d 2 )X 3 (d 2 d 1 )X 6 d 1 X If d 2 <d 1,d 2 <d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 3 d 2 )X 3 (d 1 d 2 )X 10 d 2 X If 2d 3 <d 2,d 1 <d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 2 2d 3 )X 4 (d 3 d 1 )X 8 d 1 X If 2d 3 <d 1 <d 2,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 2 d 1 )X 4 (d 1 2d 3 )X 11 d 3 X If d 3 <d 1 < 2d 3 <d 2,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 2 2d 3 )X 4 (2d 3 d 1 )X 15 (d 1 d 3 )X If d 3 <d 2,d 1 + d 2 < 2d 3,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (2d 3 d 1 d 2 )X 6 (d 2 d 3 )X 8 d 1 X If d 2 < 2d 3 <d 1 + d 2,d 1 <d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 3 d 1 )X 8 (2d 3 d 2 )X 13 (d 1 + d 2 2d 3 )X If 2d 3 <d 2 <d 1,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 1 d 2 )X 10 (d 2 2d 3 )X 11 d 3 X If d 2 <d 1,d 3 <d 2 < 2d 3,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (d 1 d 2 )X 10 (2d 3 d 2 )X 13 (d 2 d 3 )X If d 3 <d 1 <d 2 < 2d 3,d 3 <d 4, it corresponds to (d 4 d 3 )X 1 (2d 3 d 2 )X 13 (d 2 d 1 )X 15 (d 1 d 3 )X If d 1 <d 2 <d 4 <d 3, it corresponds to (d 3 d 4 )X 2 (d 4 d 2 )X 3 (d 2 d 1 )X 6 d 1 X If d 2 <d 1,d 2 <d 4 <d 3, it corresponds to (d 3 d 4 )X 2 (d 4 d 2 )X 3 (d 1 d 2 )X 10 d 2 X 13.
18 322 Y. Ishii 13. If d 1 <d 4, 2d 4 <d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 2 2d 4 )X 5 (d 4 d 1 )X 9 d 1 X If 2d 4 <d 1 <d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 2 d 1 )X 5 (d 1 2d 4 )X 12 d 4 X If d 4 <d 1 < 2d 4 <d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 2 2d 4 )X 5 (2d 4 d 1 )X 16 (d 1 d 4 )X If d 1 + d 2 < 2d 4,d 4 <d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (2d 4 d 1 d 2 )X 6 (d 2 d 4 )X 9 d 1 X If d 1 <d 4,d 2 < 2d 4 <d 1 + d 2,d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 4 d 1 )X 9 (2d 4 d 2 )X 13 (d 1 + d 2 2d 4 )X If 2d 4 <d 2 <d 1,d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 1 d 2 )X 10 (d 2 2d 4 )X 12 d 4 X If d 4 <d 2 < 2d 4,d 2 <d 1,d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (d 1 d 2 )X 10 (2d 4 d 2 )X 13 (d 2 d 4 )X If d 4 <d 1 <d 2 < 2d 4,d 2 <d 3, it corresponds to (d 3 d 2 )X 2 (2d 4 d 2 )X 13 (d 2 d 1 )X 16 (d 1 d 4 )X If d 1 <d 4 <d 3, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (d 3 d 4 )X 7 (d 4 d 1 )X 8 d 1 X If 2d 4 <d 1 <d 3, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (d 3 d 1 )X 7 (d 1 2d 4 )X 14 d 4 X If d 4 <d 1 < 2d 4,d 1 <d 3, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (d 3 d 1 )X 7 (2d 4 d 1 )X 14 (d 1 d 4 )X If d 4 <d 3, 2d 3 <d 1 <d 2, it corresponds to (d 2 d 1 )X 4 (d 1 2d 3 )X 11 (d 3 d 4 )X 18 d 4 X If d 3 + d 4 <d 1 < 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (2d 3 d 1 )X 14 (d 1 d 3 d 4 )X 18 d 4 X If d 3 <d 1, 2d 4 <d 1 <d 3 + d 4, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (d 1 2d 4 )X 14 (d 1 d 3 )X 19 (d 3 + d 4 d 1 )X If d 4 <d 3 <d 1 < 2d 4, 2d 3 <d 2, it corresponds to (d 2 2d 3 )X 4 (2d 4 d 1 )X 15 (d 1 d 3 )X 19 (d 3 d 4 )X If d 1 <d 4,d 3 + d 4 <d 2,d 1 + d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 2 d 3 d 4 )X 7 (d 4 d 1 )X 8 d 1 X 17.
19 Some classification of PVs ass. with Dynkin quivers of exceptional type If 2d 4 <d 1,d 3 + d 4 <d 2,d 1 + d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 2 d 3 d 4 )X 7 (d 1 2d 4 )X 12 d 4 X If d 4 <d 1 < 2d 4,d 3 + d 4 <d 2,d 1 + d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 2 d 3 d 4 )X 7 (2d 4 d 1 )X 17 (d 1 d 4 )X If 2d 4 <d 2,d 3 <d 2,d 1 + d 2 <d 3 + d 4, it corresponds to (d 2 2d 4 )X 5 (d 2 d 3 )X 8 (d 3 + d 4 d 1 d 2 )X 9 d 1 X If d 1 <d 4, 2d 4 <d 2,d 1 +2d 2 < 2d 3 +2d 4,d 3 +d 4 <d 1 +d 2, it corresponds to (d 2 2d 4 )X 5 (d 4 d 1 )X 8 (2d 3 +2d 4 d 1 2d 2 )X 16 (d 1 + d 2 d 3 d 4 )X If d 1 <d 4, 2d 3 +2d 4 <d 1 +2d 2,d 2 <d 3 +d 4,d 1 +d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 4 d 1 )X 8 (d 1 +2d 2 2d 3 2d 4 )X 17 (d 3 + d 4 d 2 )X If 2d 4 <d 1,d 3 <d 2 <d 3 + d 4,d 1 + d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (d 1 2d 4 )X 12 (d 3 + d 4 d 2 )X 20 (d 2 d 3 )X If d 1 <d 4, 2d 4 <d 2,d 3 <d 2,d 1 +2d 2 < 2d 3 +2d 4, it corresponds to (d 2 2d 4 )X 5 (2d 3 +2d 4 d 1 2d 2 )X 16 (d 1 d 4 )X 20 (d 2 d 3 )X If d 1 < 2d 4,d 1 +d 2 <d 3 +2d 4,d 2 <d 3 +d 4,d 1 +d 2 < 2d 3, it corresponds to (2d 3 d 1 d 2 )X 5 (2d 4 d 1 )X 17 (d 3 +d 4 d 2 )X 20 (d 1 +d 2 d 3 2d4)X If d 4 <d 1,d 1 + d 2 <d 3 +2d 4,d 1 +2d 2 < 2d 3 +2d 4,d 1 + d 2 < 2d 3,it corresponds to (2d 3 d 1 d 2 )X 5 (d 1 +2d 2 2d 3 2d 4 )X 17 (d 1 d 4 )X 20 (d 3 +2d4 d 1 d 2 )X If d 4 <d 3 <d 2,d 1 + d 2 < 2d 4, it corresponds to (2d 4 d 1 d 2 )X 6 (d 2 d 3 )X 8 (d 3 d 4 )X 9 d 1 X If d 1 <d 4,d 2 < 2d 3 <d 1 + d 2,d 3 + d 4 <d 2, it corresponds to (d 2 d 3 d 4 )X 7 (d 4 d 1 )X 8 (d 1 + d 2 2d 3 )X 15 (2d 3 d 2 )X If d 2 < 2d 3, 2d 3 +2d 4 <d 1 + d 2,d 1 <d 3, it corresponds to (d 3 d 1 )X 7 (2d 3 d 2 )X 12 (d 1 + d 2 2d 3 2d 4 )X 14 d 4 X If 2d 3 +d 4 <d 1 +d 2, 2d 4 <d 1,d 1 +d 2 < 2d 3 +2d 4,d 1 <d 3, it corresponds to (d 3 d 1 )X 7 (d 1 2d 4 )X 12 (2d 3 +2d 4 d 1 d 2 )X 21 (d 1 + d 2 2d 3 d 4 )X If d 3 +d 4 <d 2, 2d 3 <d 1 +d 2, 2d 4 <d 1,d 1 +d 2 < 2d 3 +d 4, it corresponds to (d 2 d 3 d 4 )X 7 (d 1 2d 4 )X 12 (d 1 +d 2 2d 3 )X 21 (2d 3 +d 4 d 1 d 2 )X 23.
20 324 Y. Ishii 43. If d 3 +d 4 <d 2, 2d 3 <d 1 +d 2,d 4 <d 1, 2d 1 +d 2 < 2d 3 +2d 4, it corresponds to (d 2 d 3 d 4 )X 7 (d 1 + d 2 2d 3 )X 15 (2d 3 +2d 4 2d 1 d 2 )X 17 (d 1 d 4 )X If d 2 < 2d 3,d 1 < 2d 4, 2d 3 + d 4 <d 1 + d 2,d 1 <d 3, it corresponds to (d 3 d 1 )X 7 (2d 4 d 1 )X 15 (2d 3 d 2 )X 21 (d 1 + d 2 2d 3 d 4 )X If d 1 + d 2 < 2d 3 + d 4,d 3 + d 4 <d 2,d 1 < 2d 4, 2d 3 +2d 4 < 2d 1 + d 2,it corresponds to (d 2 d 3 d 4 )X 7 (2d 4 d 1 )X 15 (2d 1 + d 2 2d 3 2d 4 )X 21 (2d 3 + d 4 d 1 d 2 )X If d 3 <d 2,d 1 + d 2 <d 3 + d 4,d 2 < 2d 4 <d 1 + d 2, it corresponds to (d 2 d 3 )X 8 (d 3 + d 4 d 1 d 2 )X 9 (2d 4 d 2 )X 13 (d 1 + d 2 2d 4 )X If d 1 <d 4 <d 3,d 2 < 2d 4, 2d 3 <d 1 + d 2, it corresponds to (d 4 d 1 )X 8 (2d 4 d 2 )X 13 (d 1 + d 2 2d 3 )X 15 (d 3 d 4 )X If d 1 <d 4,d 2 < 2d 4,d 3 + d 4 <d 1 + d 2 < 2d 3, it corresponds to (d 4 d 1 )X 8 (2d 4 d 2 )X 13 (2d 3 d 1 d 2 )X 16 (d 1 + d 2 d 3 d 4 )X If d 1 <d 4, 2d 4 <d 2 <d 3 + d 4, 2d 3 <d 1 + d 2, it corresponds to (d 4 d 1 )X 8 (d 1 + d 2 2d 3 )X 15 (d 2 2d 4 )X 17 (d 3 + d 4 d 2 )X If d 4 <d 3, 2d 3 <d 2 <d 1, it corresponds to (d 1 d 2 )X 10 (d 2 2d 3 )X 11 (d 3 d 4 )X 18 d 4 X If d 2 < 2d 3,d 3 + d 4 <d 2 <d 1, it corresponds to (d 1 d 2 )X 10 (2d 3 d 2 )X 12 (d 2 d 3 d 4 )X 18 d 4 X If d 3 <d 2 <d 3 + d 4, 2d 4 <d 2 <d 1, it corresponds to (d 1 d 2 )X 10 (d 2 2d 4 )X 12 (d 2 d 3 )X 19 (d 3 + d 4 d 2 )X If d 4 <d 3 <d 2 <d 1,d 2 < 2d 4, it corresponds to (d 1 d 2 )X 10 (2d 4 d 2 )X 13 (d 2 d 3 )X 19 (d 3 d 4 )X If d 3 + d 4 <d 1 <d 2 < 2d 3, it corresponds to (2d 3 d 2 )X 12 (d 2 d 1 )X 14 (d 1 d 3 d 4 )X 18 d 4 X If 2d 3 +2d 4 <d 1 + d 2,d 3 <d 1 <d 3 + d 4,d 2 < 2d 3, it corresponds to (2d 3 d 2 )X 12 (d 1 +d 2 2d 3 2d 4 )X 14 (d 1 d 3 )X 19 (d 3 +d 4 d 1 )X If 2d 4 <d 1 <d 2 <d 3 + d 4,d 3 <d 1, it corresponds to (d 1 2d 4 )X 12 (d 1 d 3 )X 19 (d 3 + d 4 d 2 )X 20 (d 2 d 1 )X If 2d 4 <d 1,d 3 <d 1,d 3 + d 4 <d 2,d 1 + d 2 < 2d 3 +2d 4, it corresponds to (d 1 2d 4 )X 12 (d 1 d 3 )X 19 (2d 3 +2d 4 d 1 d 2 )X 21 (d 2 d 3 d 4 )X 22.
21 Some classification of PVs ass. with Dynkin quivers of exceptional type If 2d 4 < d 1,d 1 < d 3,d 2 < d 3 + d 4, 2d 3 < d 1 + d 2, it corresponds to (d 1 2d 4 )X 12 (d 3 + d 4 d 2 )X 20 (d 1 + d 2 2d 3 )X 21 (d 3 d 1 )X If d 4 <d 3 <d 1 <d 2 < 2d 4, it corresponds to (2d 4 d 2 )X 13 (d 2 d 1 )X 15 (d 1 d 3 )X 19 (d 3 d 4 )X If 2d 3 <d 1 +d 2,d 4 <d 1 <d 3,d 2 < 2d 4, it corresponds to (2d 4 d 2 )X 13 (d 1 + d 2 2d 3 )X 15 (d 1 d 4 )X 20 (d 3 d 1 )X If d 1 +d 2 < 2d 3,d 4 <d 1,d 3 <d 2 < 2d 4, it corresponds to (2d 4 d 2 )X 13 (2d 3 d 1 d 2 )X 16 (d 1 d 4 )X 20 (d 2 d 3 )X If 2d 1 + d 2 < 2d 3 +2d 4,d 2 <d 3 + d 4,d 3 +2d 4 <d 1 + d 2, 2d 3 <d 1 + d 2, it corresponds to (d 1 + d 2 2d 3 )X 15 (2d 3 +2d 4 2d 1 d 2 )X 17 (d 3 + d 4 d 2 )X 20 (d 1 + d 2 d 3 2d 4 )X If 2d 3 <d 1 + d 2 <d 3 +2d 4,d 4 <d 1, 2d 4 <d 2, it corresponds to (d 1 + d 2 2d 3 )X 15 (d 2 2d 4 )X 17 (d 1 d 4 )X 20 (d 3 +2d 4 d 1 d 2 )X If d 3 <d 1 < 2d 4 <d 2 <d 3 + d 4, it corresponds to (2d 4 d 1 )X 15 (d 1 d 3 )X 19 (d 3 + d 4 d 2 )X 20 (d 2 2d 4 )X If d 3 <d 1 < 2d 4,d 3 + d 4 <d 2 < 2d 3, it corresponds to (2d 4 d 1 )X 15 (d 1 d 3 )X 19 (2d 3 d 2 )X 21 (d 2 d 3 d 4 )X If d 1 <d 3,d 1 < 2d 4,d 2 <d 3 + d 4, 2d 3 +2d 4 < 2d 1 + d 2, it corresponds to (2d 4 d 1 )X 15 (d 3 +d 4 d 2 )X 20 (2d 1 +d 2 2d 3 2d 4 )X 21 (d 3 d 1 )X 23. Proof. We can obtain the results similarly to Theorem 3.2. Remark 4.3. PVs corresponding to a partial tilting module whose number of direct summands is less than 4 appears as the boundary of the above list. For example, the case of the boundary d 1 <d 2 <d 3 = d 4 (resp. d 1 <d 2 = d 3 = d 4 ) of 1. corresponds to a partial tilting module (d 3 d 2 )X 3 (d 2 d 1 )X 6 d 1 X 13 (resp. (d 2 d 1 )X 6 d 1 X 13 ). Since we assume that d Z 4 >0, the partial tilting module X k alone does not appear for k =1,...,12, 14, 18. Remark 4.4. Define ( the injective ) homomorphism Φ d : GL(d; L) GL(2d; K) pij q by (p ij + q ij α) ij ij α 2 (i, j =1,...,d) and the K-isomorphism q ij p ij ij ( ) Ψ d,d : M(d, d ; L) M(2d, d pst ; K) by (p st +q st α) st (s =1,...,d; t = 1,...,d ). Then define the pair (G K d,rk d ) by q st st
22 326 Y. Ishii G K d = GL(d 1; K) GL(d 2 ; K) Φ d3 (GL(d 3 ; L)) Φ d4 (GL(d 4 ; L)) R K d = M(d 2,d 1 ; K) Ψ d3 d 2 (M(d 3,d 2 ; L)) Ψ d4 d 3 (M(d 4,d 3 ; L)). Similarly to the case of G 2, we can construct the relative invariants of type F 4 of the above PV (G K d,rk d ). 5 The Case for exceptional type E 6 In this section, we considerthe valued graph (Γ, v) with Γ = {1, 2, 3, 4, 5, 6} and v = with an orientation Ω : , i.e., of type E 6. There exist 36 positive roots α 1,...,α 36 given by α 1 =(0, 0, 0, 0, 0, 1), α 2 =(0, 0, 0, 0, 1, 0), α 3 =(0, 0, 0, 1, 0, 0), α 4 =(0, 0, 0, 1, 1, 0), α 5 =(0, 0, 1, 0, 0, 0), α 6 =(0, 0, 1, 0, 0, 1), α 7 =(0, 0, 1, 1, 0, 0), α 8 =(0, 0, 1, 1, 0, 1), α 9 =(0, 0, 1, 1, 1, 0), α 10 =(0, 0, 1, 1, 1, 1), α 11 =(0, 1, 0, 0, 0, 0), α 12 =(0, 1, 1, 0, 0, 0), α 13 =(0, 1, 1, 0, 0, 1), α 14 =(0, 1, 1, 1, 0, 0), α 15 =(0, 1, 1, 1, 0, 1), α 16 =(0, 1, 1, 1, 1, 0), α 17 =(0, 1, 1, 1, 1, 1), α 18 =(0, 1, 2, 1, 0, 1), α 19 =(0, 1, 2, 1, 1, 1), α 20 =(0, 1, 2, 2, 1, 1), α 21 =(1, 0, 0, 0, 0, 0), α 22 =(1, 1, 0, 0, 0, 0), α 23 =(1, 1, 1, 0, 0, 0), α 24 =(1, 1, 1, 0, 0, 1), α 25 =(1, 1, 1, 1, 0, 0), α 26 =(1, 1, 1, 1, 0, 1), α 27 =(1, 1, 1, 1, 1, 0), α 28 =(1, 1, 1, 1, 1, 1), α 29 =(1, 1, 2, 1, 0, 1), α 30 =(1, 1, 2, 1, 1, 1), α 31 =(1, 1, 2, 2, 1, 1), α 32 =(1, 2, 2, 1, 0, 1), α 33 =(1, 2, 2, 1, 1, 1), α 34 =(1, 2, 2, 2, 1, 1), α 35 =(1, 2, 3, 2, 1, 1), α 36 =(1, 2, 3, 2, 1, 2). We take a K-modulation M = {F 1 = = F 6 = K, 1 M 2 = K K K, 3 M 2 = KK K, 3M 4 = K K K, 5M 4 = K K K, 3M 6 = K K K } where K K K is a commutative field K as a K-K-bimodule. Take an admissible sequence of sinks (k 1,...,k 6 )=(6, 4, 5, 2, 1, 3) with respect to Ω (see Definition 2.13). For any t satisfying 1 t 6, let e kt Ob(rep(M,s kt s k6 Ω)) be the representation k t with the dimension vector e kt =(0,...,0, ˇ1, 0,..., 0). Then define the representation P kt Ob(rep(M, Ω)) by Δ k 1 Δ k t 1 e kt. Let Δ =Δ k 1 Δ k 6 = Δ 6 Δ 4 Δ 5 Δ 2 Δ 1 Δ 3 : rep(m, Ω) rep(m, Ω) be the Coxeter functor. Take the Coxeter element δ = r k6 r k1 of the Weyl group (see Definition 2.11 and see [D, p. 44]).
23 Some classification of PVs ass. with Dynkin quivers of exceptional type 327 Proposition 5.1. The 36 representations Δ s P t (1 t 6 and 0 s 5) are the complete representatives of isomorphism classes of indecomposable representations in rep(m, Ω), and the 6 representations Δ 5 P t (1 t 6) are the complete representatives of isomorphism classes of indecomposable injective representations. Moreover we have dim Δ s P t = δ s (dim P t )(1 t 6 and 0 s 5) and dim P kt = r k1 r kt 1 (e kt ). Proof. For any indecomposable representation X Ob(rep(M, Ω)), there exist t (1 t 6) and a non-negative integer s satisfying X = Δ s P t (see [D; p. 71 and p. 79 Theorem 2.19 (v)]). For any indecomposable representation Y Ob(rep(M, Ω)), the following (a), (b) hold. (a) If Δ Y 0, then Δ Y is indecomposable and dim Δ Y = δ 1 (dim Y ) (see [D; p. 69, Lemma 2.9]). (b) Δ Y =0 Y is injective δ 1 (dim Y ) /Z 6 0 (see [D; p. 73, Proposition 2.12]) By [D; Lemma 2.5], we have dim P kt = r k1 r kt 1 (e kt ), and hence dim P 1 = α 22, dim P 2 = α 11, dim P 3 = α 15, dim P 4 = α 3, dim P 5 = α 4, dim P 6 = α 1.For example, dim Δ 1 P 1 = δ 1 α 22 = α 8, dim Δ 2 P 1 = δ 1 α 8 = α 16, dim Δ 3 P 1 = δ 1 α 16 = α 24, dim Δ 4 P 1 = δ 1 α 24 = α 7, dim Δ 5 P 1 = δ 1 α 7 = α 2, and δ 1 α 2 =(0, 0, 0, 1, 1, 0) / Z 6 0. By similar direct calculation, we can show that the set of dim Δ s P t (1 t 6 and 0 s 5) coincides the set of all positive roots and δ 1 (dim Δ 5 P t ) / Z 6 0 (1 t 6). Hence we obtain our result. Now we shall calculate dim K Hom(Δ s 1 P t1, Δ s 2 P t2 )(1 t 1,t 2 6 and 0 s 1,s 2 5). Proposition 5.2. The following assertions 1, 2, 3 hold. 1. For 1 t 1,t 2 6 and 0 s 5, we have dim K Hom(P t1,p t2 ) = dim K Hom(Δ s P t1, Δ s P t2 ). 2. For 1 s 2 4, 1 s 5 s 2 and 1 t 1,t 2 6, we have dim K Hom(P t1, Δ s 2 P t2 ) = dim K Hom(Δ s P t1, Δ s 2 s P t2 ). 3. For 1 s 1 4, 1 s 5 s 1 and 1 t 1,t 2 6, we have dim K Hom(Δ s 1 P t1,p t2 ) = dim K Hom(Δ s 1 s P t1, Δ s P t2 ).
24 328 Y. Ishii Proof. By 2 of Proposition 2.14, we have our result. Note that {Δ 5 P t 1 t 6} is the set of complete representatives of isomorphism classes of indecomposable injective representations (see [D; Proposition 2.8]). Proposition 5.3. For 1 s 1 4 and 1 t 1,t 2 6, we have dim K Hom(Δ s 1 P t1,p t2 )=0. Proof. Since Δ s 1 P t1 is a non-projective indecomposable representation, and P t2 is an indecomposable projective representation, by [D; p. 75, Lemma 2.13], we obtain our result. Proposition 5.4. For 1 t 6 and 0 s 5, we have (dim K Hom(P 1, Δ s P t ), dim K Hom(P 2, Δ s P t ),...,dim K Hom(P 6, Δ s P t )) = δ s (dim P t ). Proof. Since the maximal length of paths is 1 in our orientation Ω, we have Λ=Λ(M, Ω) = M (0) M (1) =(F 1 F 6 ) ( 1 M 2 3 M 2 3 M 4 3 M 6 5 M 4 ) (see Definition 2.4). Put e t =(0,...,0, 1 Ft, 0,...,0) F 1 F 6 Λ. Then we have 1 Λ = e e 6, e 2 t = e t (1 t 6) and e t1 e t2 = e t2 e t1 = 0(1 t 1 t 2 6), and hence we obtain Λ = e 1 Λ+ + e 6 Λ. By [ASS; p. 19, Lemma 4.2], we have End Λ (e t Λ) = e t Λe t as K-algebras. Since e t Λe t = (0,...,0,F t, 0,...,0) = F t, the endomorphism ring End Λ (e t Λ) is a field, and hence by Proposition 2.19, each e t Λ is indecomposable. By Remark 2.5, the dimension vector of the indecomposable representation corresponding to e t Λ Ob( mod Λ )(1 t 6) is (dim K e t Λe 1, dim K e t Λe 2,...,dim K e t Λe 6 ). This coincides with dim P t. For example, we have e 3 Λ=F 3 ( 3 M 2 3 M 4 3 M 6 ) and hence e 3 Λe k =0(k =1, 5), e 3 Λe k = 3 M k (k =2, 4, 6), e 3 Λe 3 = F 3. This implies that (dim K e 3 Λe 1,...,dim K e 3 Λe 6 )=(0, 1, 1, 1, 0, 1) = α 15 = dim P 3. Hence the indecomposable representation corresponding to e t Λ(1 t 6) coincides with P t up to isomorphisms. The right Λ-module corresponding the representation Δ s P t =( j ϕ (s,t) i,w (s,t) i )(1 t 6and 0 s 5) is given by M (s,t) = i Γ W (s,t) i (see Remark 1.5). By [ASS; p. 19, Lemma 4.2], we have dim K Hom Λ (e t Λ,M (s,t) ) = dim K M (s,t) e t (1 t, t 6 and 0 s 5). Since dim K M (s,t) e t = dim K W (s,t) t = [dim Δ s P t ] t where [dim Δ s P t ] t stands for the t -th component of dim Δ s P t, we have (dim K Hom Λ (e 1 Λ,M (s,t) ),...,dim K Hom Λ (e 6 Λ,M (s,t) )) = dim Δ s P t = δ s (dim P t ). Since e t Λ (resp. M (s,t) ) corresponds to P t (resp. Δ s P t ), we obtain (dim K Hom(P 1, Δ s P t ), dim K Hom(P 2, Δ s P t ),...,dim K Hom(P 6, Δ s P t )) = δ s (dim P t ) for 1 t 6 and 0 s 5.
5 Quiver Representations
5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )
More informationWhen is the Ring of 2x2 Matrices over a Ring Galois?
International Journal of Algebra, Vol. 7, 2013, no. 9, 439-444 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.3445 When is the Ring of 2x2 Matrices over a Ring Galois? Audrey Nelson Department
More informationThe preprojective algebra revisited
The preprojective algebra revisited Helmut Lenzing Universität Paderborn Auslander Conference Woodshole 2015 H. Lenzing Preprojective algebra 1 / 1 Aim of the talk Aim of the talk My talk is going to review
More information1. Quivers and their representations: Basic definitions and examples.
1 Quivers and their representations: Basic definitions and examples 11 Quivers A quiver Q (sometimes also called a directed graph) consists of vertices and oriented edges (arrows): loops and multiple arrows
More informationThe real root modules for some quivers.
SS 2006 Selected Topics CMR The real root modules for some quivers Claus Michael Ringel Let Q be a finite quiver with veretx set I and let Λ = kq be its path algebra The quivers we are interested in will
More informationLECTURE 16: REPRESENTATIONS OF QUIVERS
LECTURE 6: REPRESENTATIONS OF QUIVERS IVAN LOSEV Introduction Now we proceed to study representations of quivers. We start by recalling some basic definitions and constructions such as the path algebra
More informationSTABILITY OF FROBENIUS ALGEBRAS WITH POSITIVE GALOIS COVERINGS 1. Kunio Yamagata 2
STABILITY OF FROBENIUS ALGEBRAS WITH POSITIVE GALOIS COVERINGS 1 Kunio Yamagata 2 Abstract. A finite dimensional self-injective algebra will be determined when it is stably equivalent to a positive self-injective
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to
More information12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.
12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End
More informationNoetherian property of infinite EI categories
Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result
More informationSIGNED EXCEPTIONAL SEQUENCES AND THE CLUSTER MORPHISM CATEGORY
SIGNED EXCEPTIONAL SEQUENCES AND THE CLUSTER MORPHISM CATEGORY KIYOSHI IGUSA AND GORDANA TODOROV Abstract. We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism
More informationRepresentations of quivers
Representations of quivers Michel Brion Lectures given at the summer school Geometric methods in representation theory (Grenoble, June 16 July 4, 2008) Introduction Quivers are very simple mathematical
More informationREPRESENTATION THEORY. WEEKS 10 11
REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here Crawley-Boevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.
More informationThus we get. ρj. Nρj i = δ D(i),j.
1.51. The distinguished invertible object. Let C be a finite tensor category with classes of simple objects labeled by a set I. Since duals to projective objects are projective, we can define a map D :
More information6. Dynkin quivers, Euclidean quivers, wild quivers.
6 Dynkin quivers, Euclidean quivers, wild quivers This last section is more sketchy, its aim is, on the one hand, to provide a short survey concerning the difference between the Dynkin quivers, the Euclidean
More informationOn real root representations of quivers
On real root representations of quivers Marcel Wiedemann Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Pure Mathematics July
More information4.1. Paths. For definitions see section 2.1 (In particular: path; head, tail, length of a path; concatenation;
4 The path algebra of a quiver 41 Paths For definitions see section 21 (In particular: path; head, tail, length of a path; concatenation; oriented cycle) Lemma Let Q be a quiver If there is a path of length
More informationCHAPTER 1. AFFINE ALGEBRAIC VARIETIES
CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationA note on standard equivalences
Bull. London Math. Soc. 48 (2016) 797 801 C 2016 London Mathematical Society doi:10.1112/blms/bdw038 A note on standard equivalences Xiao-Wu Chen Abstract We prove that any derived equivalence between
More informationIndecomposable Quiver Representations
Indecomposable Quiver Representations Summer Project 2015 Laura Vetter September 2, 2016 Introduction The aim of my summer project was to gain some familiarity with the representation theory of finite-dimensional
More informationLevel 5M Project: Representations of Quivers & Gabriel s Theorem
Level 5M Project: Representations of Quivers & Gabriel s Theorem Emma Cummin (0606574) March 24, 2011 Abstract Gabriel s theorem, first proved by Peter Gabriel in 1972 [1], comes in two parts. Part (i)
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationRepresentations of quivers
Representations of quivers Gwyn Bellamy October 13, 215 1 Quivers Let k be a field. Recall that a k-algebra is a k-vector space A with a bilinear map A A A making A into a unital, associative ring. Notice
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GROUP ACTIONS ON POLYNOMIAL AND POWER SERIES RINGS Peter Symonds Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 GROUP ACTIONS ON POLYNOMIAL
More informationOn minimal disjoint degenerations for preprojective representations of quivers
On minimal disjoint degenerations for preprojective representations of quivers Klaus Bongartz and Thomas Fritzsche FB Mathematik BUGH Wuppertal Gaußstraße 20 42119 Wuppertal e-mail: Klaus.Bongartz@math.uni-wuppertal.de
More informationENUMERATING m-clusters USING EXCEPTIONAL SEQUENCES
ENUMERATING m-clusters USING EXCEPTIONAL SEQUENCES KIYOSHI IGUSA Abstract. The number of clusters of Dynkin type has been computed case-by-case and is given by a simple product formula which has been extended
More informationTopics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationThe sections deal with modules in general. Here, we start with a ring R, all modules are R-modules.
5 Extensions GivenrepresentationsM,N ofaquiver,wewanttointroduceavectorspaceext (M,N) which measures the possible extensions Here, by an extension of R-modules (where R is a ring, for example the path
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationDerived Canonical Algebras as One-Point Extensions
Contemporary Mathematics Derived Canonical Algebras as One-Point Extensions Michael Barot and Helmut Lenzing Abstract. Canonical algebras have been intensively studied, see for example [12], [3] and [11]
More informationWIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES
WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,
More informationCategories of noncrossing partitions
Categories of noncrossing partitions Kiyoshi Igusa, Brandeis University KIAS, Dec 15, 214 Topology of categories The classifying space of a small category C is a union of simplices k : BC = X X 1 X k k
More informationValued Graphs and the Representation Theory of Lie Algebras
Axioms 2012, 1, 111-148; doi:10.3390/axioms1020111 Article OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Valued Graphs and the Representation Theory of Lie Algebras Joel Lemay Department
More informationAuslander Algebras of Self-Injective Nakayama Algebras
Pure Mathematical Sciences, Vol. 2, 2013, no. 2, 89-108 HIKARI Ltd, www.m-hikari.com Auslander Algebras of Self-Injective Nakayama Algebras Ronghua Tan Department of Mathematics Hubei University for Nationalities
More informationThe Diamond Category of a Locally Discrete Ordered Set.
The Diamond Category of a Locally Discrete Ordered Set Claus Michael Ringel Let k be a field Let I be a ordered set (what we call an ordered set is sometimes also said to be a totally ordered set or a
More informationne varieties (continued)
Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we
More informationA NOTE ON GENERALIZED PATH ALGEBRAS
A NOTE ON GENERALIZED PATH ALGEBRAS ROSA M. IBÁÑEZ COBOS, GABRIEL NAVARRO and JAVIER LÓPEZ PEÑA We develop the theory of generalized path algebras as defined by Coelho and Xiu [4]. In particular, we focus
More informationRepresentation type, boxes, and Schur algebras
10.03.2015 Notation k algebraically closed field char k = p 0 A finite dimensional k-algebra mod A category of finite dimensional (left) A-modules M mod A [M], the isomorphism class of M ind A = {[M] M
More informationQuiver Representations and Gabriel s Theorem
Quiver Representations and Gabriel s Theorem Kristin Webster May 16, 2005 1 Introduction This talk is based on the 1973 article by Berstein, Gelfand and Ponomarev entitled Coxeter Functors and Gabriel
More informationSEMI-STABLE SUBCATEGORIES FOR EUCLIDEAN QUIVERS
SEMI-STABLE SUBCATEGORIES FOR EUCLIDEAN QUIVERS COLIN INGALLS, CHARLES PAQUETTE, AND HUGH THOMAS Dedicated to the memory of Dieter Happel Abstract. In this paper, we study the semi-stable subcategories
More informationQUIVERS, REPRESENTATIONS, ROOTS AND LIE ALGEBRAS. Volodymyr Mazorchuk. (Uppsala University)
QUIVERS, REPRESENTATIONS, ROOTS AND LIE ALGEBRAS Volodymyr Mazorchuk (Uppsala University) . QUIVERS Definition: A quiver is a quadruple Q = (V, A, t, h), where V is a non-empty set; A is a set; t and h
More informationAUSLANDER-REITEN THEORY FOR FINITE DIMENSIONAL ALGEBRAS. Piotr Malicki
AUSLANDER-REITEN THEORY FOR FINITE DIMENSIONAL ALGEBRAS Piotr Malicki CIMPA, Mar del Plata, March 2016 3. Irreducible morphisms and almost split sequences A algebra, L, M, N modules in mod A A homomorphism
More informationAlgebraic Geometry
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationDECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY C.M. RINGEL)
DECOMPOSITION OF TENSOR PRODUCTS OF MODULAR IRREDUCIBLE REPRESENTATIONS FOR SL 3 (WITH AN APPENDIX BY CM RINGEL) C BOWMAN, SR DOTY, AND S MARTIN Abstract We give an algorithm for working out the indecomposable
More informationAlgebraic Geometry: Limits and Colimits
Algebraic Geometry: Limits and Coits Limits Definition.. Let I be a small category, C be any category, and F : I C be a functor. If for each object i I and morphism m ij Mor I (i, j) there is an associated
More informationRepresentation Theory
Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim
More informationA visual introduction to Tilting
A visual introduction to Tilting Jorge Vitória University of Verona http://profs.sci.univr.it/ jvitoria/ Padova, May 21, 2014 Jorge Vitória (University of Verona) A visual introduction to Tilting Padova,
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationOn the modular curve X 0 (23)
On the modular curve X 0 (23) René Schoof Abstract. The Jacobian J 0(23) of the modular curve X 0(23) is a semi-stable abelian variety over Q with good reduction outside 23. It is simple. We prove that
More informationResearch Article r-costar Pair of Contravariant Functors
International Mathematics and Mathematical Sciences Volume 2012, Article ID 481909, 8 pages doi:10.1155/2012/481909 Research Article r-costar Pair of Contravariant Functors S. Al-Nofayee Department of
More informationTOWERS OF SEMISIMPLE ALGEBRAS, THEIR GRAPHS AND JONES INDEX
SARAJEVO JOURNAL OF MATHEMATICS Vol12 (25), No2, (2016), 335 348, Suppl DOI: 105644/SJM12307 TOWERS OF SEMISIMPLE ALGEBRAS, THEIR GRAPHS AND JONES INDEX VLASTIMIL DLAB Dedicated to the memory of Professor
More informationPULLBACK MODULI SPACES
PULLBACK MODULI SPACES FRAUKE M. BLEHER AND TED CHINBURG Abstract. Geometric invariant theory can be used to construct moduli spaces associated to representations of finite dimensional algebras. One difficulty
More informationRELATIVE HOMOLOGY. M. Auslander Ø. Solberg
RELATIVE HOMOLOGY M. Auslander Ø. Solberg Department of Mathematics Institutt for matematikk og statistikk Brandeis University Universitetet i Trondheim, AVH Waltham, Mass. 02254 9110 N 7055 Dragvoll USA
More informationExamples of Semi-Invariants of Quivers
Examples of Semi-Invariants of Quivers June, 00 K is an algebraically closed field. Types of Quivers Quivers with finitely many isomorphism classes of indecomposable representations are of finite representation
More informationThe Endomorphism Ring of a Galois Azumaya Extension
International Journal of Algebra, Vol. 7, 2013, no. 11, 527-532 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2013.29110 The Endomorphism Ring of a Galois Azumaya Extension Xiaolong Jiang
More informationOn p-monomial Modules over Local Domains
On p-monomial Modules over Local Domains Robert Boltje and Adam Glesser Department of Mathematics University of California Santa Cruz, CA 95064 U.S.A. boltje@math.ucsc.edu and aglesser@math.ucsc.edu December
More informationNOTES ON CHAIN COMPLEXES
NOTES ON CHAIN COMPLEXES ANDEW BAKE These notes are intended as a very basic introduction to (co)chain complexes and their algebra, the intention being to point the beginner at some of the main ideas which
More informationNOTES ON SPLITTING FIELDS
NOTES ON SPLITTING FIELDS CİHAN BAHRAN I will try to define the notion of a splitting field of an algebra over a field using my words, to understand it better. The sources I use are Peter Webb s and T.Y
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationA Remark on Certain Filtrations on the Inner Automorphism Groups of Central Division Algebras over Local Number Fields
International Journal of lgebra, Vol. 10, 2016, no. 2, 71-79 HIKRI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.612 Remark on Certain Filtrations on the Inner utomorphism Groups of Central
More informationNOTES ON BASIC HOMOLOGICAL ALGEBRA 0 L M N 0
NOTES ON BASIC HOMOLOGICAL ALGEBRA ANDREW BAKER 1. Chain complexes and their homology Let R be a ring and Mod R the category of right R-modules; a very similar discussion can be had for the category of
More informationCoils for Vectorspace Categories
Coils for Vectorspace Categories Bin Zhu Department of mathematical science, Tsinghua University, Beijing 100084, P. R. China e-mail: bzhu@math.tsinghua.edu.cn Abstract. Coils as components of Auslander-Reiten
More informationDimension of the mesh algebra of a finite Auslander-Reiten quiver. Ragnar-Olaf Buchweitz and Shiping Liu
Dimension of the mesh algebra of a finite Auslander-Reiten quiver Ragnar-Olaf Buchweitz and Shiping Liu Abstract. We show that the dimension of the mesh algebra of a finite Auslander-Reiten quiver over
More informationQuiver Representations
Quiver Representations Molly Logue August 28, 2012 Abstract After giving a general introduction and overview to the subject of Quivers and Quiver Representations, we will explore the counting and classification
More informationCross Connection of Boolean Lattice
International Journal of Algebra, Vol. 11, 2017, no. 4, 171-179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7419 Cross Connection of Boolean Lattice P. G. Romeo P. R. Sreejamol Dept.
More informationLectures on Algebraic Theory of D-Modules
Lectures on Algebraic Theory of D-Modules Dragan Miličić Contents Chapter I. Modules over rings of differential operators with polynomial coefficients 1 1. Hilbert polynomials 1 2. Dimension of modules
More informationALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS
ALGEBRA QUALIFYING EXAM, FALL 2017: SOLUTIONS Your Name: Conventions: all rings and algebras are assumed to be unital. Part I. True or false? If true provide a brief explanation, if false provide a counterexample
More informationHilbert function, Betti numbers. Daniel Gromada
Hilbert function, Betti numbers 1 Daniel Gromada References 2 David Eisenbud: Commutative Algebra with a View Toward Algebraic Geometry 19, 110 David Eisenbud: The Geometry of Syzygies 1A, 1B My own notes
More informationAlgebras of Finite Representation Type
Algebras of Finite Representation Type Marte Oldervoll Master of Science Submission date: December 2014 Supervisor: Petter Andreas Bergh, MATH Norwegian University of Science and Technology Department
More informationQUIVERS AND LATTICES.
QUIVERS AND LATTICES. KEVIN MCGERTY We will discuss two classification results in quite different areas which turn out to have the same answer. This note is an slightly expanded version of the talk given
More informationEssays on representations of p-adic groups. Smooth representations. π(d)v = ϕ(x)π(x) dx. π(d 1 )π(d 2 )v = ϕ 1 (x)π(x) dx ϕ 2 (y)π(y)v dy
10:29 a.m. September 23, 2006 Essays on representations of p-adic groups Smooth representations Bill Casselman University of British Columbia cass@math.ubc.ca In this chapter I ll define admissible representations
More informationWinter School on Galois Theory Luxembourg, February INTRODUCTION TO PROFINITE GROUPS Luis Ribes Carleton University, Ottawa, Canada
Winter School on Galois Theory Luxembourg, 15-24 February 2012 INTRODUCTION TO PROFINITE GROUPS Luis Ribes Carleton University, Ottawa, Canada LECTURE 3 3.1 G-MODULES 3.2 THE COMPLETE GROUP ALGEBRA 3.3
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationSTANDARD COMPONENTS OF A KRULL-SCHMIDT CATEGORY
STANDARD COMPONENTS OF A KRULL-SCHMIDT CATEGORY SHIPING LIU AND CHARLES PAQUETTE Abstract. First, for a general Krull-Schmidt category, we provide criteria for an Auslander-Reiten component with sections
More informationReview of Linear Algebra
Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space
More informationHigher dimensional homological algebra
Higher dimensional homological algebra Peter Jørgensen Contents 1 Preface 3 2 Notation and Terminology 5 3 d-cluster tilting subcategories 6 4 Higher Auslander Reiten translations 10 5 d-abelian categories
More informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationInterval Decomposition of Infinite Zigzag Persistence Modules
Interval Decomposition of Infinite Zigzag Persistence Modules Magnus Bakke Botnan arxiv:1507.01899v1 [math.rt] 7 Jul 2015 Abstract We show that every infinite zigzag persistence module decomposes into
More informationAisles in derived categories
Aisles in derived categories B. Keller D. Vossieck Bull. Soc. Math. Belg. 40 (1988), 239-253. Summary The aim of the present paper is to demonstrate the usefulness of aisles for studying the tilting theory
More informationSELF-EQUIVALENCES OF THE DERIVED CATEGORY OF BRAUER TREE ALGEBRAS WITH EXCEPTIONAL VERTEX
An. Şt. Univ. Ovidius Constanţa Vol. 9(1), 2001, 139 148 SELF-EQUIVALENCES OF THE DERIVED CATEGORY OF BRAUER TREE ALGEBRAS WITH EXCEPTIONAL VERTEX Alexander Zimmermann Abstract Let k be a field and A be
More informationA generalized Koszul theory and its applications in representation theory
A generalized Koszul theory and its applications in representation theory A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Liping Li IN PARTIAL FULFILLMENT
More informationLECTURE 11: SOERGEL BIMODULES
LECTURE 11: SOERGEL BIMODULES IVAN LOSEV Introduction In this lecture we continue to study the category O 0 and explain some ideas towards the proof of the Kazhdan-Lusztig conjecture. We start by introducing
More informationStructures of AS-regular Algebras
Structures of AS-regular Algebras Hiroyuki Minamoto and Izuru Mori Abstract In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras
More informationVector spaces, duals and endomorphisms
Vector spaces, duals and endomorphisms A real vector space V is a set equipped with an additive operation which is commutative and associative, has a zero element 0 and has an additive inverse v for any
More informationEXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY
EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1. Categories 1.1. Generalities. I ve tried to be as consistent as possible. In particular, throughout the text below, categories will be denoted by capital
More informationGroups of Prime Power Order with Derived Subgroup of Prime Order
Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*
More informationSTABLE MODULE THEORY WITH KERNELS
Math. J. Okayama Univ. 43(21), 31 41 STABLE MODULE THEORY WITH KERNELS Kiriko KATO 1. Introduction Auslander and Bridger introduced the notion of projective stabilization mod R of a category of finite
More informationLie Algebra Cohomology
Lie Algebra Cohomology Carsten Liese 1 Chain Complexes Definition 1.1. A chain complex (C, d) of R-modules is a family {C n } n Z of R-modules, together with R-modul maps d n : C n C n 1 such that d d
More informationDedicated to Helmut Lenzing for his 60th birthday
C O L L O Q U I U M M A T H E M A T I C U M VOL. 8 999 NO. FULL EMBEDDINGS OF ALMOST SPLIT SEQUENCES OVER SPLIT-BY-NILPOTENT EXTENSIONS BY IBRAHIM A S S E M (SHERBROOKE, QUE.) AND DAN Z A C H A R I A (SYRACUSE,
More informationSTRONGNESS OF COMPANION BASES FOR CLUSTER-TILTED ALGEBRAS OF FINITE TYPE
STRONGNESS OF COMPANION BASES FOR CLUSTER-TILTED ALGEBRAS OF FINITE TYPE KARIN BAUR AND ALIREZA NASR-ISFAHANI Abstract. For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion
More informationφ(a + b) = φ(a) + φ(b) φ(a b) = φ(a) φ(b),
16. Ring Homomorphisms and Ideals efinition 16.1. Let φ: R S be a function between two rings. We say that φ is a ring homomorphism if for every a and b R, and in addition φ(1) = 1. φ(a + b) = φ(a) + φ(b)
More informationGraded modules over generalized Weyl algebras
Graded modules over generalized Weyl algebras Advancement to Candidacy Robert Won Advised by: Dan Rogalski December 4, 2014 1 / 41 Overview 1 Preliminaries Graded rings and modules Noncommutative things
More informationCluster-Concealed Algebras
Cluster-Concealed Algebras Claus Michael Ringel (Bielefeld/Germany) Nanjing, November 2010 Cluster-tilted algebras Let k be an algebraically closed field Let B be a tilted algebra (the endomorphism ring
More informationDecompositions of Modules and Comodules
Decompositions of Modules and Comodules Robert Wisbauer University of Düsseldorf, Germany Abstract It is well-known that any semiperfect A ring has a decomposition as a direct sum (product) of indecomposable
More informationPartial orders related to the Hom-order and degenerations.
São Paulo Journal of Mathematical Sciences 4, 3 (2010), 473 478 Partial orders related to the Hom-order and degenerations. Nils Nornes Norwegian University of Science and Technology, Department of Mathematical
More informationInjective Modules and Matlis Duality
Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following
More informationHILBERT FUNCTIONS. 1. Introduction
HILBERT FUCTIOS JORDA SCHETTLER 1. Introduction A Hilbert function (so far as we will discuss) is a map from the nonnegative integers to themselves which records the lengths of composition series of each
More informationIntroduction to the representation theory of quivers Second Part
Introduction to the representation theory of quivers Second Part Lidia Angeleri Università di Verona Master Program Mathematics 2014/15 (updated on January 22, 2015) Warning: In this notes (that will be
More information