Semi-invariants of tubular algebras1574/fulltext.pdf · The main theorem gives the generators and...

SEMI-INVARIANTS

OF TUBULAR ALGEBRAS

A dissertation presented

by

Kristin D. Webster

to

The Department of Mathematics

In partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the field of

Mathematics

Northeastern University

Boston, Massachusetts

June 2009

1

SEMI-INVARIANTS

OF TUBULAR ALGEBRAS

by

Kristin D. Webster

ABSTRACT OF DISSERTATION

Submitted in the partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Mathematics

in the Graduate School of Arts and Sciences of

Northeastern University, June 2009

2

Abstract

The focus of this thesis is on the rings of semi-invariants of the representation spaces of

tubular algebras. These rings reflect the cyclic nature of the structure of the modules

of the tubular algebras. The main theorem gives the generators and relations of the

rings of semi-invariants SI(Q/I, d) where d is a dimension vector of a module in

the tubes of an algebra where certain conditions hold. The theorem can be directly

applied to Euclidean algebras and Tubular algebras since they have separating tubular

families. The results reflect the work of Skowronski and Weyman on semi-invariants

of Euclidean Algebras. Also, in the case of tubular algebras, applications of shrinking

functors to the rings of semi-invariants are explored.

3

Acknowledgements

This work would not have been possible without the support and encouragement

of many people. First and foremost I would like to thank my advisor, Professor

Jerzy Weyman, under whose supervision I chose this topic and formulated this thesis.

His patience and knowledge has proved invaluable. I am very grateful to Professor

Gordana Todorov, she has been extremely helpful in offering mathematical support,

as well as encouragement and good advice. I am indebted to my fellow students in

the math department, from whom I learned so much. I am thankful for my family

and friends, on whose constant encouragement and love I have relied throughout my

time in graduate school. In particular, I would like to thank my husband Gideon

whose support has been my foundation. To him I dedicate this thesis.

4

Contents

Abstract 3

Acknowledgements 4

Table of Contents 6

1 Introduction 7

2 Basic Notions 9

2.1 Definition of Semi-Invariants and the Theorem of Sato and Kimura . 9

2.2 Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Quivers with Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Semi-Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Semi-invariants of Quivers with Relations . . . . . . . . . . . . . . . . 19

2.6 SI(Q, d) where RepK(Q, d) admits an open orbit . . . . . . . . . . . 20

2.7 Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 The Main Theorem 26

3.1 The Required Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Semi-invariants of Tubes . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Semi-Invariants of Euclidean Quivers 34

5 Tame Concealed Algebras 41

5.1 Semi-Invariants of Tame Concealed . . . . . . . . . . . . . . . . . . . 42

6 Tubular Algebras 51

6.1 Semi-Invariants of Tubular Algebras . . . . . . . . . . . . . . . . . . . 63

7 Shrinking Functors and Semi-Invariants 70

5

A Tame concealed to tubular algebras 75

6

1 Introduction

The three distinct classes of quiver representations are finite type, tame and wild

quivers. For the class of tame algebras, the indecomposable modules occur in each

dimension in a finite number of discrete and a finite number of one parameter families.

It is an interesting task to find geometric characterizations of the representation type

of a quiver. This leads us into the study of semi-invariants of quivers. Given a

quiver Q and a dimension vector d, to find semi-invariants we study the actions of

the corresponding products G(Q, d) of general linear groups on the affine variety

RepK(Q, d) of representations of a Q of dimension d over an algebraically closed

field K. In [36], Schofield classified semi-invariants of quivers to be determinantal

semi-invariants. In [41], Skowronski and Weyman showed that a finite connected

quiver is a Dynkin or Euclidean quiver if and only if the algebra of semi-invariants for

any given dimension vector is a complete intersection. Derksen and Weyman in [20]

expanded the idea of Schofield’s determinantal semi-invariants to apply to quivers

with relations.

Given a Euclidean quiver, we can consider tilted algebras. From a tilted algebra

in the preprojective component of the original quiver, we can use several branch

extensions to obtain a tubular algebra. These tubular algebras are quasi-titled and

have an interesting module structure. The focus of this paper is on the rings of

semi-invariants on the representation spaces of tubular algebras. These rings of semi-

invariants reflect very nicely the structure of the modules of the tubular algebras.

Theorem 1.0.1. 3.2.1 Let {Vt, E(j)i } be the simple modules in R(Λ) of tubular type

(m1, . . . ,ms). Let β be a regular dimension vector. Then the ring of regular semi-

invariants SIreg(Λ, β) has the generators and relations decribed as follows.

(a) The non-homogeneous generators are cE(j)[i+1,k] where [i, k] is a β-admissible path

on the j-th circle.

7

(b) The homogeneous generators cVt which span the space of dimension p + 1. We

fix a basis {c0, . . . , cp} of the weight space SIreg(Λ, β)〈h,−〉.

(c) There are s relations in SIreg(Λ, β) such that each relation expresses the product

of non-homogeneous semi-invariants of index zero on the j-th circle as a linear

combination of the semi-invariants c0, . . . , cp.

More generally, for any algebra whose periodic modules have the structure of a

separating tubular family, their rings of semi-invariants will also follow these patterns.

So we have the ability to apply this theorem to not only tubular algebras but also to

reformulate the results about semi-invariants for Euclidean Quivers.

8

2 Basic Notions

The preliminaries on quiver representations and semi-invariants included in this sec-

tion are classical and are included here to define notation and make this note self-

contained.

2.1 Definition of Semi-Invariants and the Theorem of Sato

and Kimura

Fix an algebraically closed field K.

Let G be an algebraic group and let V be its representation. There is an induced

action of G on the coordinate ring K[V ] of polynomial functions on V , namely for

g ∈ G, f ∈ K[V ]

(gf)(x) := f(g−1x).

For a character χ of G we define the space of semi-invariants of weight χ for the

representation V as

SI(G, V )χ = { f ∈ K[V ]|∀g ∈ G, g ◦ f = χ(g)f }.

The direct sum

SI(G, V ) := ⊕χ∈char(G)SI(G, V )χ

is a ring of semi-invariants for the representation V .

Theorem 2.1.1. (Sato-Kimura, [34]) Let G be an algebraic group and let V be its

representation. Assume that G is connected, and V has a Zariski open G-orbit. Then

a) SI(G, V ) is a polynomial ring, and the generators X1, . . . , Xd can be chosen in

the weight spaces χ1, . . . , χd.

9

b) The weights χ1, . . . , χd are linearly independent in char(G).

c) The number d and generators X1, . . . , Xd have the following interpretation. Let

O be a Zariski open G-orbit in V and let Z := V \ O. Decompose Z into

irreducible components

Z = Z1 ∪ Z2 ∪ . . . ∪ Zs

and reorder them so Z1, . . . , Zd have codimension 1 and Zd+1, . . . , Zs have codi-

mension > 1. Then Xj can be taken to be an irreducible equation of Zj.

2.2 Quivers

We can associate each finite dimensional K-algebra to a graphical structure called

a quiver. Conversely, each quiver corresponds to an associative K-algebra which is

finite dimensional under some conditions.

Definition 2.2.1. A quiver Q = (Q0, Q1, h, t : Q1 → Q0) is given by a set Q0 of

vertices {1, · · · , n} as a set of arrows Q1. An arrow a starts at t(a) and ends at h(a).

A quiver Q is said to be finite if Q0 and Q1 are finite sets. The underlying graph

Q of a quiver Q is obtained from Q by forgetting the orientation of the arrows. Q is

said to be connected if Q, the underlying graph is a connected graph, that is there

is a path between any pair of vertices.

A non trivial path in Q of length m is a sequence am . . . a1 (m ≥ 1) which satisfies

h(ai) = t(ai+1) for each i, 1 ≤ i ≤ m − 1. This path, p, starts at t(p) = t(a1) and

ends at h(am) = h(p). The set of all paths of length l (l ≥ 0) is denoted by Ql. The

path of length 1 that starts and ends at i is called the trivial path. The trivial path

is denoted by ei. A path of length l ≥ 1 with h(p) = t(p) is called a cycle.

The path algebra KQ is the K-algebra whose underlying K-vector space has as

its basis the set of all paths in Q with the product of two paths p and q being defined

10

by composition if h(q) = t(p) and as zero otherwise. We associate to each x ∈ Q0

a path ex of length zero. For x 6= y, x, y ∈ Q0 we have that exey = eyex = 0 and

exex = ex.

Definition 2.2.2. A finite dimensional representation V of Q is a set of {V (x) :

x ∈ Q0} of finite dimensional K-vector spaces together with a set of K-linear maps

{V (a) : V (ta)→ V (ha)|a ∈ Q1}.

A morphism φ : V → V ′ of two representations is a collection of K-linear maps

{φ(x) : V (x) → V ′(x); x ∈ Q0} such that for each a ∈ Q1 we have V ′(a)φ(ta) =

φ(ha)V (a) i.e. the following diagram commutes:

V (ta)V (a)

//

φ(ta)

��

V (ha)

φ(ha)

��V ′(ta)

V ′(a)// V ′(ha)

The representations of a quiver Q over K form a category denoted RepK(Q). The

dimension vector of a representation V is the function d : Q0 → ZQ0 defined by

di := dimV (i) for i ∈ Q0.

It is well known that there exists an equivalence of categories

KQ−mod ∼= RepK(Q)

Definition 2.2.3. If V and W are representation of a quiver Q, then we define the

direct sum representation V ⊕W by (V ⊕W )(x) = V (x) ⊕W (x) for every x ∈ Q0

and for each a ∈ Q1

(V ⊕W )(a) :=

V (a) 0

0 W (a)

: V (ta)⊕W (ta)→ V (ha)⊕W (ha)

11

Definition 2.2.4. A representation V is called decomposable if V is isomorphic

to a direct sum of non-zero representations. Otherwise a representation is indecom-

posable.

The category RepK(Q) of representations is hereditary therefore every represen-

tation has projective dimension ≤ 1.

Definition 2.2.5. For V,W ∈ RepK(Q) with α = dimV and β = dimW we can

define the Euler bilinear form as

〈α, β〉 :=∑x∈Q0

α(x)β(x)−∑a∈Q1

α(ta)β(ha)

Proposition 2.2.6. The spaces HomQ(V,W ) and Ext1Q(V,W ) are the kernel and

cokernel of the following linear map

dVW :⊕x∈Q0

Hom(V (x),W (x))→⊕a∈Q1

Hom(V (ta),W (ha))

where dVW is given by

{φ(x)|x ∈ Q0} 7→ {W (a)φ(ta)− φ(ha)V (a)|a ∈ Q1}.

It follows from the previous proposition that

〈α, β〉 = dimHomQ(V,W )− dimExt1Q(V,W ).

The non-isomorphic indecomposable projective modules of KQ are Pi = eiKQ

for each i ∈ Q0. The non-isomorphic indecomposable injective modules of KQ are

12

Qi = D(KQei). Every representation V ∈ RepK(Q) has a canonical resolution:

0→⊕a∈Q1

V (ta)⊗ Pha →⊕x∈Q0

V (x)⊗ Px → V → 0

Definition 2.2.7. For any K-algebra A we define the opposite algebra Aop of A to

be the K-algebra whose underlying set and vector space structure are just those of A,

but the multiplication in Aop is defined by the formula a∗b = ba. The opposite quiver

is defined to be Qop = (Q0, Q1, h′, t′) where for a ∈ Q1, t′(a) = h(a) and h′(a) = t(a).

Furthermore (KQ)op = KQop.

Definition 2.2.8. Let A be a K-algebra. The Grothendieck group of A is the

abelian group K0(A) = F/F ′ where F is the free abelian group having as its basis

the set of the isomorphism classes M of modules M is mod A and F ′ is the subgroup

of F generated by the elements M − L− N corresponding to all exact sequences

0→ L→M → N → 0

in mod A. Denote by [M ] the image of the isomorphism class M of the module M

under the canonical group epimorphism F → F/F ′.

Definition 2.2.9. For a finite dimension K-algebra A, consider the A-dual functor

(−)t = HomA(−, A) : modA→ modAop.

If we have a minimal projective presentation of a module M ∈ modA,

P1p0 // P0

p1 //M // 0

13

that is that p0, p1 are projective covers. Applying (−)t, we obtain an exact sequence

0 //M tpt0 // P t

0

pt1 // P t1

// coker pt1// 0

Denote coker ptq by TrM and call it the tranpose of M .

Definition 2.2.10. Let A be a finite dimensional K-algebra. Define the functor

D : modA→ modAop

by assigning to each right module M in modA the dual K-vector space M∗ =

HomK(M,K).

Definition 2.2.11. The Auslander-Reiten translation is defined to be the com-

position of D with Tr. In particular, we set τ = DTr and τ−1 = TrD.

Definition 2.2.12. An A-homomorphism is a section (or a retraction) whenever

it admits a left inverse (or a right inverse, respectively).

Definition 2.2.13. A homomorphism f : X → Y is said to be irreducible provided:

(a) f is neither a section nor a retraction and

(b) if f = f1f2 wither f1 is a retraction or f2 is a section

Xf //

f2 @@@@@@@ Y

Zf1

??~~~~~~~

Definition 2.2.14. Let A be a finite dimensional K-algebra, the Auslander-Reiten

quiver ΛA is defined as follows:

1. The points of Λ are the isomorphism classes [X] of indecomposable modules X

in modA.

14

2. Let [M ] and [N ] be the points in ΛA corresponding to the indecomposable

modules M , N in modA. The arrows [M ]→ [N ] are in bijective correspondence

with the vectors of a basis of the K-vectors space of irreducible homomorphism

from M to N .

Definition 2.2.15. Λi ⊂ Λ is a component if its underlying graph is connected.

Definition 2.2.16. A connected component P of ΛA is called preprojective if P is

acyclic (meaning it contains no cycles) and for any indecomposable module M in P ,

there exists t ≥ 0 and a ∈ Q0 such that M ∼= τ−tPa. An indecomposable module is

called preprojective if it belongs to the preprojective component of Λ.

Definition 2.2.17. A connected component Q of ΛA is called preinjective if Q is

acyclic and for any indecomposable module N in Q, there exists s ≥ 0 and b ∈ Q0

such that N ∼= τ sQb. An indecomposable module is called preinjective if it belongs

to the preinjective component of Λ.

Definition 2.2.18. If there exists a path from [M ] to [N ] in Λ then [M ] is a prede-

cessor of [N ] and [N ] is a successor of [M ]. If the path is of length 1, then [M ] is

a direct predecessor of [N ] and [N ] is a direct successor of [M ].

Definition 2.2.19. Let A be a finite dimensional K-algebra with a complete set

{e1, . . . , εn} of primitive orthogonal idempotents. The Cartan matrix of A is the

n× n matrix

CA =

c11 · · · c1n

.... . .

...

cn1 · · · cnn

where cji = dimK eiAej for i, j = 1, . . . , n. The Euler matrix of A is E = (C−1

A )t.

The Coxeter matrix of A is ΦA = −CtAC−1A .

15

2.3 Quivers with Relations

Definition 2.3.1. Let Q be a finite, connected quiver. The two-sided ideal of the

path algebra KQ generated by the arrows of Q is called the arrow ideal and is

denoted by RQ.

Note that

RQ = KQ1 ⊕KQ2 ⊕ · · ·

and that for each l ≥ 1,

RlQ =

⊕m≥l

KQm

where for each l ≥ 0, KQl is the subspace of KQ generated by the set Ql of all paths

of length l.

Definition 2.3.2. A two-sided ideal I of KQ is said to be admissible if there exists

m ≥ 2 such that

RmQ ⊆ I ⊆ R2

Q.

Q/I is called a quiver with relations and has associated path algebra KQ/I.

It is easily seen that when Q is a finite quiver, every admissible ideal is finitely

generated and therefore there exists a finite set of relations {ρ1, · · · , ρp} such that

I = 〈ρ1, · · · , ρp〉.

A representation V of Q satisfies the relations in I if we have that V (ρ) = 0 for

all relations ρ ∈ I. Then V ∈ RepK(Q/I).

Let A = KQ/I, then the non-isomorphic indecomposable projective modules of A

are Pa = eaA. Let V ∈ RepK(A,α), construct the module P0 = ⊕x∈Q0V (x) ⊗ Px.

Then for each V ∈ RepK(A,α) we have the presentation

0→ V(1) → P0 → V → 0

16

the kernel V(1) has the same dimension vector for each V ∈ RepK(Q,α). Define

P1 = ⊕x∈Q0V(1)(x) ⊗ Px. Continuing in this fashion we can construct the family of

projective resolutions of modules from RepK(A,α). If Q has no oriented cycles then

the global dimensions of A is finite so that

⊕i≥0

(−1)idimExtiQ(V,W )

is finite. For V,W ∈ RepK(A) with dim(V ) = α, dim(W ) = β, the Euler Form for

quivers with relations can be defined as

〈〈α, β〉〉 =⊕i≥0

(−1)idimExtiQ(V,W ).

2.4 Semi-Invariants

Given a dimension vector α, we can define the representation space of α-dimensional

representations of Q as

RepK(Q,α) = ⊕a∈Q1 HomK(Kα(ta), Kα(ha)).

If GL(α) =∏

i∈Q0GLαi(K) then GL(α) acts algebraically on Rep(Q,α) by simulta-

neous conjugation, that is, for g = (g(i))i∈Q0 ∈ GL(α) and V ∈RepK(Q,α) we define

g · V by

(g · V )(a) = g(ha)V (a)g(ta)−1,∀a ∈ Q1.

We have that RepK(Q,α) is a representation of GL(α) and the GL(α) orbits in

RepK(Q,α) are in a one-to-one correspondence with the isomorphism classes of α-

dimensional representations of Q. Since Q has no oriented cycles, it can be shown

that there is only one closed GL(α) orbit in Rep(Q,α) and therefore the invariant

17

ring K[Rep(Q,α)]GL(α) is exactly the base field K.

Consider the subgroup SL(α) ⊆GL(α) defined by SL(α) =∏

i∈Q0SLαi(K). The

action of SL(α) on Rep(Q,α) provides us with a non-trivial ring of semi-invariants.

Definition 2.4.1. Let Q be a quiver such that |Q0| = n and let f be a semi-invariant

on Rep(Q,α) and χ the character of f . A weight of the semi-invariant f is any

vector σ = (σ1, . . . , σn) for which the character χ for f can be written as

χ(g) = (det(g1))σ1 · · · (det(gn))σn .

A vector σ ∈ Zn will be called a weight if it is a weight for some semi-invariant.

Definition 2.4.2. We denote by SI(Q,α)χ the K-vector space of semi-invariants on

Rep(Q,α) with the character χ and by SI(Q,α) the graded ring

SI(Q,α) =⊕

χ∈char(GL(α))

SI(Q,α)χ

called the ring of semi-invariants for the action of GL(α) on Rep(Q,α).

Remark 2.4.3. Let V ∈ RepK(Q,α) then the codimension of the orbit O(V ) of V

is equal to dim Ext(V, V ), in particular when Ext(V, V ) = 0, O(V ) has codimension

0 therefore RepK(Q,α) has an open GL(α)-orbit.

In the case where RepK(Q,α) has an open GL(α)-orbit, the theorem of Sato

and Kimura 2.1.1 show that SI(Q,α) is a polynomial algebra. Schofield [35] has an

effective algorithm for finding the semi-invariants in this case. We call these Schofield

semi-invariants cV ’s.

Definition 2.4.4. For V ∈ RepK(Q,α), define the vector space homomorphism

(pM , V ) : Hom(P0, V ) → Hom(P1, V ) induced by the canonical presentation of M

18

given in 2.1.1. If we have V such that 〈dimV, α〉 = 0 then (pM , V ) is a square matrix,

then we can define the polynomial function cV by setting cV (M) = det(pM , V )

Theorem 2.4.5. ([15, 37]) Let Q be a quiver and α ∈ Nn. Then the ring of semi-

invariants SI(Q,α) is spanned as a K-vector space by the functions cV for represen-

tations V satisfying 〈dimV, α〉 = 0 Furthermore, the character of the semi-invariant

cV is χσ where σ = 〈dimV,−〉.

Remark 2.4.6. Schofield’s method for constructing cV ’s is as follows: For V,W ∈

Rep(Q), with dimV = α, dimW = β constructing a non-zero homomorphism from

V to W is equivalent to solving a system of homogeneous linear equations

V (a)Xha −X taW (a) = 0 for all a ∈ Q1

where X i is a αi × βi matrix of indeterminants. Since 〈〈α, β〉〉 = 0 we know that the

number of indeterminants is equal to the number of equations. If M is the matrix

of coefficients of the indeterminants, then the determinant of M is a semi-invariant.

Furthermore if (pV ,W ) is the matrix of the canonical projective resolution as above,

then we have that (pV ,W ) = M .

2.5 Semi-invariants of Quivers with Relations

The purpose of the following section is to outline the results of [20] of the extension

of the idea of Schofield’s determinantal semi-invariants to quivers with relations.

Let V ∈ RepK(Q/I). Let P1 → P0 → V → 0 be the minimal presentation of V

in RepK(Q/I).

Definition 2.5.1. Let W ∈ RepK(Q/I). Then cV can be defined as the determinant

of the matrix Hom(P0,W )→ Hom(P1,W ) whenever it is a square matrix.

19

Note that the representation space RepK(Q,α) does not have to be irreducible, we

denote the irreducible components as RepK(Q,α)j, j ∈ {1, 2, . . .}. The semi-invariant

cV is defined on the components RepK(Q/I, β) such that, for W ∈ RepK(Q/I, β),

Hom(P0,W ) = Hom(P1,W ) while Hom(V,W ) = 0 for a general W ∈ RepK(Q/I, β).

A component RepK(Q/I, β)jis called faithful over KQ/I if the ideal J = {x ∈

KQ|x|RepK(Q/I,β)j = 0} is equal to I, i.e. there are no additional relations satisfied on

RepK(Q/I, β)j.

Theorem 2.5.2. ([20]) Assume that char K = 0. For each component RepK(Q/I, β)j

the semi-invariants cV span SI(Q/I,RepK(Q/I, β)j) for KQ/I-modules of projective

dimension 1 and of dimension vectors α such that 〈〈α, β〉〉 = 0.

2.6 SI(Q, d) where RepK(Q, d) admits an open orbit

Let A = KQ/I and RepK(A, d) be the set of all representations of dimension d. The

orbits of GL(d) correspond bijectively to the isomorphism classes of representations

of dimension d. So therefore when RepK(A, d) admits an open orbit the theorem of

Sato and Kimura applies 2.1.1. In this case, SI(A, d) is a polynomial algebra. In

particular, for Q Dynkin, SI(Q, d) is a polynomial algebra. From this point on we

will only consider the cases where RepK(A, d) is not irreducible.

2.7 Tubes

The purpose of the following section is to give the main properties of algebras whose

module structure includes objects called tubes. The main theorem is concerned only

with the modules in these tubes.

Definition 2.7.1. Let Q be a connected, acyclic quiver. Define an infinite trans-

lation quiver as follows. The set of points in ZQ is (ZQ)0 = Z×Q0 = {(n, x)|n ∈

20

Z, x ∈ Q0} and for each arrow a ∈ Q1, a : x→ y there exists two arrows

(n, a) : (n, x)→ (n, y)

and

(n, a′) : (n+ 1, y)→ (n, x)

in (ZQ)1. Also, we have that τ(n, x) = (n+ 1, x).

If we have the infinite quiver A∞

1 // 2 // 3 // 4 // · · ·

from this we can have the infinite translation quiver

(1, 1)%%KKK

(0, 1)%%KKK

(−1, 1)''OOO

· · ·

· · ·##GGGG

;;wwww(1, 2)

99sss

%%KKK(0, 2)

88qqq

&&MMM(−1, 2)

99tttt

%%JJJJ

(2, 3)

&&NNNNN

99sss(1, 3)

&&NNNNN

99sss(0, 3)

77oooo

((QQQQQQQ · · ·

· · ·99ttt

88ppppp77nnnnnn

88qqqqqq

where τ(n, i) = (n + 1, i) for n ∈ Z and i ≥ 1. Therefore, we have that τ is an

automorphism and hence, so is any power τ r. For a fixed r ≥ 1, let (τ r) be the

infinite cyclic group generated by τ r and then ZA∞/(τ r) is the orbit space of of ZA∞

under the action of (τ r). Thus we are identifying (n, i) with τ r(n, i) = (n + r, i) and

each arrow a with τ ra : τ rx→ τ ry.

Definition 2.7.2. Let (T , τ) be a translation quiver. Then (T , τ) is a stable tube

of rank r if there is a isomorphism of translation quivers T ∼= ZA∞/(τ r).

Consider now the tubes formed by the indecomposable modules in an Auslander-

Reiten quiver of an algebra.

21

Definition 2.7.3. For a stable tube T , the unique τ orbit formed by the modules

having exactly one direct predecessor and exactly one direct successor is called the

mouth of T .

Definition 2.7.4. Let T be a stable tube.

1. If r = 1 we define this to be a homogeneous tube.

2. A sequence (x1, · · · , xr) of T is called a τ-cycle if τx1 = xr, τx2 = x1, · · · , τxr =

xr−1.

3. Given a point x lying on the mouth of a stable stube, a ray starting at x is

defined to be a unique infinite sectional path x = x[1]→ x[2]→ x[3]→ x[4]→

· · · in the tube T .

Let A be a finite dimensional K-algebra. A component C of mod A is said to be

a tube provided that ΛC is a tube.

Definition 2.7.5. Let I be some set, and let us consider families of Tρ, ρ ∈ I of

pairwise different tubes in mod A. Let T be the module class generated by all Tρ. T

will be called a tubular family. A tubular family is said to be stable if T does not

contain non-zero projective or injective modules.

In the cases we are interested in, the parameter set I will be the projective line

P1K.

Let A be any algebra, a brick in mod A is a module E such that EndE = K.

Two bricks E and E ′ are orthogonal if HomA(E,E ′) = 0 and HomA(E ′, E) =

0. If E1, · · · , Er is a family of pairwise orthogonal bricks in mod A and let E =

EXTA(E1, · · · , Er) denote the full subcategory of mod A (the extension category)

whose non-zero objects are all the module M such that there exists a chain of sub-

modules M = M0 ) M1 · · · ) Ml = 0 for some l ≥ 1 with Mi/Mi+1 isomorphic to

22

one of the bricks. Thus E is the smallest additive subcategory of mod A containing

the bricks Ei and closed under extensions.

Let the modules at the mouth of a tube j of rank mj be denoted, E(j)0 , . . . , E

(j)mj−1

of dimE(j)i = e

(j)i . Since the E

(j)i ’s are the simple modules at the mouth of each tube

we have the following module structure inside the tube of rank mj = k + 1:

E(j)0

""EEEEE

(j)1

##GGGGE

(j)2

##GGGG· · · E

(j)k−1

%%LLLLL E(j)k

· · · E(j)[0,1]

;;xxxx

##FFFFE

(j)[1,2]

;;xxxx

##FFFFE

(j)[2,3] · · · E

(j)[k−2,k−1]

88ppppp

&&MMMMME

(j)[k−1,k]

::vvvv· · ·

· · · E(j)[0,2]

;;xxxxE

(j)[1,3]

;;xxxx· · · E

(j)[k−2,k]

99rrrr· · ·

· · ·

%%JJJJJ · · · · · ·

E(j)[0,k]

==||||

Definition 2.7.6. For any indecomposable A module X in a tube of rank r, there

exists a unique sectional path X1 → X2 → · · · → Xm = X with X1 lying on the

mouth of the tube. We call m the regular length of X.

Definition 2.7.7. Let C be a component of the Auslander-Reiten quiver ΛA of an

algebra A. C is defined to be a standard component of ΛA if K(C) is equivalent to

the full K-subcategory of mod A whose objects are representatives of the isomorphism

classes of the indecomposable modules in C.

Theorem 2.7.8. [32] Let A be an algebra and (E1, · · · , Er) with r ≥ 1 be a τ -cycle of

pairwise orthogonal bricks in mod A such that {E1, · · · , Er} is a self-hereditary family

of mod A. Then TE is a standard stable tube of rank r and the modules E1, · · · , Er

form a complete set of modules lying on the mouth of the tube TE .

Definition 2.7.9. Define a separating tubular family of stable tubes, T , as:

23

(a) T is a family of standard stable tubes

(b) the indecomposable modules not in T fall into two classes P and Q such that

Hom(P ,Q) = Hom(Q, T ) = Hom(T ,P) = 0

(c) for any ρ ∈ I, any morphism from P to Q can be factored through Tρ.

Remark: We denote the object class of add(X∪Y ) by X∨Y . We call a separating

tubular family sincere if it contains a sincere module. A module M is sincere if

dimM is a sincere vector. A dimension vector is sincere if it contains only nonzero

components.

Theorem 2.7.10. Let T be a sincere separating tubular family of A modules sepa-

rating P from Q. Then all projective modules belong to P ∨ T , all injective modules

to T ∨Q. If X ∈ P , then proj dimX ≤ 1 if Y ∈ Q then inj dim Y ≤ 1. If in addition

T contains no non-zero injective module, then proj dimT ≤ 1 for any T ∈ T and gl

dim A ≤ 2. Similarly, if T contains no non-zero projective module then inj dimT ≤ 1

for any T ∈ T and again gl dim A ≤ 2.

Definition 2.7.11. For each tubular family T we call (n1, . . . , nm) the tubular type

where T consists of m non-homogeneous tubes and the rank of each non-homogeneous

tube is ni, (i = 1, . . . ,m).

The class of finite dimensional algebras with a separating tubular family of stable

tubes contains the tame hereditary algebras, the tame concealed algebras, the tubular

algebras, the canonical algebras and the concealed-canonical algebras.

24

Definition 2.7.12. Consider the quiver ∆(n1, . . . , nt)

a11

α12 // a1

2// · · · // a1

n1−1

α1n1

��00000000000000000

a21 α2

2

// a22

// · · · // a2n2−1

α2n AAAAAAAAA

0

α21

AA��

α11

HH��

αt1 ��;;;;;;;;...

... ω

at1 αt2

// at2// · · · // atnt−1

αtn

>>}}}}}}}}}

with relations I(x3, . . . xn)

α1m1· · ·α1

1 + xiα2m1· · ·α2

1 + αim1· · ·αi1

for i = 3, . . . , n. Then K∆(n1, . . . , nt)/I is a canonical algebra of type (n1, . . . , nt).

25

3 The Main Theorem

In this section we prove a theorem that gives abstract conditions under which the ring

of semi-invariants of certain components of a representation space has similar struc-

ture as the rings of semi-invariants of extended Dynkin quivers for regular dimension

vectors (see [SW]).

Let Λ = KQ/I be the path algebra of the quiver Q = (Q0, Q1) with the ideal I

of admissible relations.

3.1 The Required Conditions

Definition 3.1.1. We will first define the category of regular modules of index

γ. Let Λ be a finite dimensional K-algebra. For some index γ ∈ P1K, we have a

tubular family. Let the tubular type of Tγ be denoted (m1, . . . ,ms). For each Tγ, we

have a family of homogeneous indecomposable modules {Vt}, t ∈ P1K of dimension hγ

which are the homogeneous modules in Tγ. We also have a family of nonhomogeneous

modules E(j)i of dimension vectors e

(j)i , (1 ≤ j ≤ s, 0 ≤ i ≤ mj−1). The modules E

(i j)

are the modules at the mouth of the tubes. The smallest category R(Λ)γ containing

{Vt} and E(j)i closed under extensions and direct summands we will call the category

of regular modules of index γ.

Definition 3.1.2. Let Λ be of tubular type (m1, . . . ,ms). For each j = 1, . . . , s we

can associate to the tube of rank mj a graph Γ(j) with mj vertices 0, 1, . . . ,mj − 1

and edges a(j)i connecting vertices i and i + 1, with a

(j)mj−1 connecting mj − 1 and 0.

We label the vertex i with a number p(j)i . We call a directed path [i, k] on Γ(j) a

β-admissible path of index l if p(j)i = p

(j)k = l and p

(j)t > l for all vertices t inside of

[i, k].

Example 3.1.3. This example illustrates admissable paths. Consider a tube of rank

26

6, with p0 = p3 = 0, p1 = p5 = p2 = 1 and p4 = 2. Then we have the following

polygon ∆ with admissible arcs denoted by the dotted lines.

p0 p1

444444

p5

444444

p2

p4 p3

labeled 0 ME

8-

%

��

��%

-8

EM

1 O0

�000000

1

000000

��1

��

2 0

Therefore the admissible paths are [0, 3], [3, 0] of index 0 and [1, 2] of index 1.

Definition 3.1.4. The conditions on R(Λ)γ:

1. A vector

β = phγ +s∑j=1

mj−1∑i=0

p(j)i e

(j)i

(where we assume that for each j = 1, . . . , s we have min{p(j)i ; 1 ≤ i ≤ mj−1} =

0) is called a regular dimension vector. Every dimension vector in R(Λ)γ

can be written this way.

2. The dimension vectors e(j)i satisfy the relations

hγ =

mj−1∑i=0

e(j)i .

for j = 1, . . . , s, and the dimension of the space spanned by the e(j)i equals∑s

j=1mj − s+ 1,

3. The decomposition of a general vector in R(Λ)γ is given by the same formula

as for the extended Dynkin quivers [32]. In particular, for V of dimension β,

we have that

V = Vt1 ⊕ . . .⊕ Vtp ⊕⊕sj=1 ⊕[i,k] admissible E(j)[i,k].

27

4. Dimension vectors e(j)i , and the vectors e

(j)[i,k] := e

(j)i + e

(j)i+1 + . . .+ e

(j)k are Schur

roots and the generic modules E(j)[i,k] have projective dimension 1 and injective

dimension 1 over Λ.

5. The values of the Euler form are:

a) 〈〈e(j)i , e

(l)k 〉〉 = 0 if j 6= l

b) 〈〈e(j)i , e

(j)k 〉〉 =

1 i = k

−1 i = k + 1

0 otherwise

6. The general module in dimension vector R(Λ,h) is a 1-parameter family of

modules Vt (t ∈ K ∪ ∞), which are also of projective dimension 1 and of

injective dimension 1.

7. Every indecomposable module X of projective dimension ≤ 1 orthogonal to

Vt (in the sense that HomΛ(X, Vt) = Ext1Λ(X, Vt) = 0 for general t) is in the

category R(Λ).

8. The condition (6) implies by results of [20] the existence of the semi-invariant

cVu in the coordinate ring SI(Λ, β). We require that cVu(Vw) = −u+ w.

Definition 3.1.5. Let {Vt, E(j)i } be the simple modules in R(Λ) of tubular type

(m1, . . . ,ms). For each regular dimension vector β we have the ring of regular

semi-invariants

SIreg(Λ, β) := SI(GL(Q, β),R(Λ, β)).

To each admissible path [i, k] in Γ(j) we associate the semi-invariant cE(j)[i+1,k] . In

particular, each path connecting two consecutive vertices labeled by zero is admissible.

The product of all semi-invariants of index zero has weight 〈h,−〉. This includes a

28

degenerate case of the case when only one vertex of Γ(j) is labeled by zero. In such

case the whole circle gives a semi-invariant of index zero of weight 〈h,−〉.

3.2 Semi-invariants of Tubes

Theorem 3.2.1. Let {Vt, E(j)i } be the simple modules in R(Λ) of tubular type

(m1, . . . ,ms). Let β be a regular dimension vector. Then the ring of regular semi-

invariants SIreg(Λ, β) has the generators and relations decribed as follows.

(a) The non-homogeneous generators are cE(j)[i+1,k] where [i, k] is a β-admissible path

on the j-th circle,

(b) The homogeneous generators cVt which span the space of dimension p + 1. We

fix a basis {c0, . . . , cp} of the weight space SIreg(Λ, β)〈h,−〉.

(c) There are s relations in SIreg(Λ, β) such that each relation expresses the product

of non-homogeneous semi-invariants of index zero on the j-th circle as a linear

combination of the semi-invariants c0, . . . , cp.

Proof. We use the result of Derksen and Weyman [20] (and Domokos [22]) describing

the generators of the rings of semi-invariants for quiver with relations. The component

R(Λ, β) for p > 0 is faithful (otherwise change Λ to some factor algebra Λ′, and all

modules Vt and E(j)i will be also modules over Λ′) so we know that the ring of regular

semi-invariants SIreg(Λ, β) is generated by semi-invariants cW where W is a module

of projective dimension 1 orthogonal to a general module V in R(Λ)γ. A general

module in that component can be written

V = Vt1 ⊕ . . .⊕ Vtp ⊕⊕sj=1 ⊕[i,k] admissible E(j)[i,k].

The condition (6) of 3.1.4 implies that the module W has to be regular. This

allows us to repeat the reasoning from [40] and describe the generators of the rings

29

SIreg(Λ, β). We have the existence of semi-invariants c0, . . . , cp as defined by Ringel in

([32], 4.1). We know that these ci’s are linear combinations of cVt for Vt homogeneous

and orthogonal to V . Furthermore, cVt =∑p

i=0 tici.

Following [20], we check to see that for cW , W is in the orthogonal category of V .

We know based on 3.1.4.(4) and (6) that W has projective dimension 1. Clearly, for

any of the Vt because there are no maps between tubes, we have that

〈〈h, β〉〉 = 〈〈h, ph +s∑j=1

mj−1∑i=0

p(j)i e

(j)i 〉〉

= 〈〈h, ph〉〉+ 〈〈h,s∑j=1

mj−1∑i=0

p(j)i e

(j)i 〉〉 = 0

Similarly, for cE(l)[n+1,k] , we have that:

〈〈e(l)[n+1,k], β〉〉 = 〈〈e(l)

n+1 + e(l)n+2 + · · ·+ e

(l)k , β〉〉

= 〈〈e(l)n+1 + e

(l)n+2 + · · ·+ e

(l)k , ph +

s∑j=1

mj−1∑i=0

p(j)i e

(j)i 〉〉

= 〈〈e(l)n+1 + e

(l)n+2 + · · ·+ e

(l)k , ph〉〉+

s∑j=1

〈〈e(l)n+1 + e

(l)n+2 + · · ·+ e

(l)k ,

mj−1∑i=0

p(j)i e

(j)i 〉〉

Again, since there are no maps between tubes this sum simplifies to:

= 〈〈e(l)n+1 + e

(l)n+2 + · · ·+ e

(l)k ,

ml−1∑i=0

p(l)i e

(l)i 〉〉.

Considering 3.1.4.(5) we have:

= 〈〈e(l)n+1 + e

(l)n+2 + · · ·+ e

(l)k

k∑i=n

p(l)i e

(l)i 〉〉

30

and for any i ∈ n+ 1, . . . , k − 1, we have

〈〈e(l)n+1 + e

(l)n+2 + · · ·+ e

(l)k , p

(l)i e

(l)i 〉〉 = −p(l)

i + p(l)i = 0

and

〈〈e(l)n+1 + e

(l)n+2 + · · ·+ e

(l)k , p

(l)n e

(l)n 〉〉 = −p(l)

n

〈〈e(l)n+1 + e

(l)n+2 + · · ·+ e

(l)k , p

(l)k e

(l)k 〉〉 = p

(l)k

by the definition of admissible path, we have that p(l)n = p

(l)k therefore,

〈〈e(l)[n+1,k], β〉〉 = 0.

But still we need the relations and weight multiplicities of SIreg(Λ, β) to have the

full structure of this ring. We need a version of reciprocity result from [16].

Lemma 3.2.2. (Reciprocity). Let β, γ be two regular dimension vectors. Then

dim(SIreg(Λ, β)〈〈γ,−〉〉) = dim(SIreg(Λ, γ)〈〈−,−β〉〉).

Proof. Assume that the space on the left hand side has dimension m. It is spanned

by semi-invariants cW where W are regular modules of dimension γ. Let us choose

m regular modules W1, . . . ,Wm of dimension γ such that cW1 , . . . , cWm is a basis of

SIreg(Λ, β)〈〈γ,−〉〉. Then there exist m regular modules V1, . . . , Vm of dimension β such

that the determinant

det(cWi(Vj))1≤i,j≤m

is not zero. But we have by definition cWi(Vj) = cVj(Wi). Thus the functions cVj are

31

linearly independent in SIreg(Λ, γ)〈〈−,−β〉〉. This shows that

dim(SIreg(Λ, β)〈〈γ,−〉〉) ≤ dim(SIreg(Λ, γ)〈〈−,−β〉〉).

The other inequality is proved in exactly the same way.

Lemma 3.2.3. (Weight multiplicities). Let

β = ph+s∑j=1

mj−1∑i=0

p(j)i e

(j)i

(where we assume that for each j = 1, . . . , s we have min{p(j)i ; 1 ≤ i ≤ mj − 1} = 0),

and

γ = qh+s∑j=1

mj−1∑i=0

q(j)i e

(j)i

(where we assume that for each j = 1, . . . , s we have min{q(j)i ; 1 ≤ i ≤ mj − 1} = 0),

be two regular dimension vectors. Then

dim(SIreg(Λ, β)〈〈γ,−〉〉) = dim(SIreg(Λ, γ)〈〈−,−β〉〉) =

(p+ q

q

).

Moreover, the subring

⊕q≥0SIreg(Λ, β)〈〈qh,−〉〉

is a polynomial ring in p+ 1 variables, with the generators occurring in degree 1.

Proof. First we will reduce to the case where γ = qh and β = ph. Each semi-invariant

of weight 〈〈γ,−〉〉 is a product of a semi-invariant of weight 〈〈qh,−〉〉 and of a semi-

invariant of weight 〈〈γ− qh,−〉〉 because of condition (2) of 3.1.4. But the dimension

of the weight space SIreg(Λ, β)〈〈γ−qh,−〉〉 is equal to one, because of condition (3) of

3.1.4. Also, the ring SIreg(Λ, β) is a domain, so the non-zero element in the weight

space SIreg(Λ, β)〈〈γ−qh,−〉〉 is a non-zero divisor. Thus the dimensions of weight spaces

32

SIreg(Λ, β)〈〈γ,−〉〉 and SIreg(Λ, β)〈〈qh,−〉〉 are the same. Using reciprocity argument, we

are reduced to the case β = ph, γ = qh.

The general module in dimension vector β is V = Vt1⊕ . . .⊕Vtp , and in dimension

vector γ is W = Vu1 ⊕ . . . ⊕ Vuq . Therefore each semi-invariant in the weight space

SIreg(Λ, ph)〈〈qh,−〉〉 is a linear combination of monomials of degree q in semi-invariants

from the weight space SIreg(Λ, ph)〈〈h,−〉〉. Applying reciprocity and using the same

argument, plus the condition (7) of 3.1.4, which implies that dim(SIreg(Λ, h)〈〈h,−〉〉) =

2, we see that dim(SIreg(Λ, ph)〈〈h,−〉〉) ≤ p+ 1. This in turn shows that

dim(SIreg(Λ, ph)〈〈qh,−〉〉) ≤(p+ q

q

).

This reasoning shows that SIreg(Λ, ph) is naturally a factor of a polynomial ring in

≤ p+ 1 variables.

To prove the equality we observe that if any relation would occur in SIreg(Λ, ph)

(or if the number of generators would actually be smaller that p + 1), then the

dimension of the factor of Reg(Λ, ph) by the group GL(Q, ph) with the stability

condition given by h would be smaller than p. But this is impossible, because the

modules Vt1 ⊕ . . .⊕ Vtp are pairwise nonisomorphic (as dim(HomΛ(Vt, Vu)) = δt,u by

condition (7) of 3.1.4 ), and the factor in question has to include their orbits.

Now we can conclude the description of the relations in the rings SIreg(Λ, β). The

relations given in the formulation of the Theorem certainly exist. We need to show

that there are no additional relations. But condition (1) of 3.1.4 implies that the

additional relations can occur only between the semi-invariants in the weights qh.

This is impossible by 3.2.2.

We will now use this theorem to make statements about semi-invariants of Eu-

clidean Quiver and Tubular Algebras.

33

4 Semi-Invariants of Euclidean Quivers

A Euclidean graph is one of the following diagrams:

Am : ◦

PPPPPPPPPPPPPPP

nnnnnnnnnnnnnnn

◦ ◦ · · · ◦ ◦

m ≥ 1

Dm : ◦

@@@@@@@ ◦

◦ ◦ · · · ◦ ◦

~~~~~~~

@@@@@@@

◦

~~~~~~~◦

m ≥ 4

E6 : ◦

◦

◦ ◦ ◦ ◦ ◦

E7 : ◦

◦ ◦ ◦ ◦ ◦ ◦ ◦

E8 : ◦

◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

where for An we have to exclude the cyclic orientation.

Let Q be a Euclidean Quiver then A = KQ is tame hereditary. The tubular

type of Dn is (n − 2, 2, 2), of E6 is (3, 3, 2), of E7 is (4, 3, 2) and of E8 is (5, 3, 2).

The representations of Q without indecomposable preprojective or preinjective direct

summands are called regular. Denote by Dr the set of dimension vectors of all the

34

regular representations of Q. In each of these cases, for some V ∈ RepK(Q) there is

a one-dimensional subspace of D on which 〈〈dimV, dimV 〉〉 vanishes. For a minimal

dimV we let h = dimV .

If β = ph, then the general module in dimension vector β is V = Vt1 ⊕ · · · ⊕ Vtp

where the Vti are pairwise non-isomorphic, that is dim Hom(Vti , Vtj) = δi,j.

Theorem 4.0.4. ([32]) Let Q be a tame quiver of dimension d ∈ Dr with canoni-

cal decomposition d = ph +∑

i∈I ei. Then there exists semi-invariants c0, . . . , cp ∈

SI(Q,d)〈h,−〉. These c0, . . . , cp are a basis of a vector space of cV where dimV = h.

Furthermore, cVλ =∑p

i=0 ciλi

To construct these ci’s, let V,W ∈ RepK(Q,d). Notice that 〈〈d,d〉〉 = 0. There-

fore, we can use a method similar to the construction of Schofield’s cV ’s. Consider

a certain matrix with coefficients in the polynomial ring K[Rep(Q,d)][λ] obtained

from the same method as 2.4.6. Then taking the determinant of this matrix we ob-

tain a polynomial homogeneous in degree p of the form∑p

i=0 ciλi where the ci’s are

in K[Rep(Q,d)]〈h,−〉

Example 4.0.5. This example illustrates the how to find the ci’s and cV ’s for a

Euclidean quiver. Consider Q of type D4 of the subspace orientation. Let d = h =

(1, 1, 1, 1, 2). So we have V :

5

1

a1

::tttttttttttttttttttttt2

a2

GG��3

a3

WW0000000000000

4

a4

ddJJJJJJJJJJJJJJJJJJJJJJ

Let αi =

xi

yi

with coordinate functions Xi, Yi for 1 ≤ i ≤ 4. Let W ∈

35

RepK(Q,d) be the following:

K2

K

b1

99rrrrrrrrrrrrrrrrrrrrrrrK

b2

EE��K

b3

YY3333333333333

K

b4

eeLLLLLLLLLLLLLLLLLLLLLLL

with b1 =

1

0

, b2 =

0

1

, b3 =

1

1

, b4 =

λ

1

. Then the matrix of

coefficients of the equations V (a)Xha −X taW (a) = 0 for a ∈ Q1 is:

−X1 0 0 0 1 0 0 0

−Y1 0 0 0 0 1 0 0

0 −X2 0 0 0 0 1 0

0 −Y2 0 0 0 0 0 1

0 0 −X3 0 1 0 1 0

0 0 −Y3 0 0 1 0 1

0 0 0 −X4 λ 0 1 0

0 0 0 −Y4 0 λ 0 1

The determinant of this matrix is

(X1X4Y2Y3 −X1X2Y3Y4 −X3X4Y1Y2 +X2X3Y1Y4)

+(X1X2Y3Y4 −X1X3Y2Y4 −X2X4Y1Y3 +X3X4Y1Y2)λ

So we have two semi-invariants:

c0 = (X1X4Y2Y3 −X1X2Y3Y4 −X3X4Y1Y2 +X2X3Y1Y4)

36

and

c1 = (X1X2Y3Y4 −X1X3Y2Y4 −X2X4Y1Y3 +X3X4Y1Y2).

Notice for λ = 0, 1,∞ we have corresponding regular representations, let them be

V0, V1, V∞ respectively. We can then associate cV0 = c0, cV1 = c0 + c1 and cV∞ = c1.

Theorem 4.0.6. ([14]) Let Q be a Euclidean quiver. Then there are at most three

τ -orbits , ∆ = {ei, i ∈ I},∆′ = {e′i|i ∈ I ′},∆′′ = {e′′i |i ∈ I ′′} of the dimension

vectors of nonohomogeneous simple regular representations of Q. We may assume

that I = {0, 1, . . . , u−1}, I ′ = {0, 1, . . . , v−1}, I ′′ = {0, 1, . . . , w−1} and τ−Ei = Ei+1

These orbits can be represented graphically as polygons:

e0

zzzzzze1

:::::

eu−1 e2

GGGGGGG

��

ei+1 ei

∆

e′0

}}}}}}e′1

88888

e′u−1 e′2

FFFFFF

��

e′i+1 e′i

∆′

e′′0

}}}}}}e′′1

88888

e′′u−1 e′′2

FFFFFF

��

e′′i+1 e′′i

∆′′

Any dimension vector d ∈ Dr can be written uniquely in the form

d = ph +∑i∈I

piei +∑i∈I′

p′ie′i +∑i∈I′′

p′′i e′′i

for some non-negative integers p, pi, p′i, p′i with at least one coefficient in each family

{pi|i ∈ I}, {p′i|i ∈ I ′}, {p′′i |i ∈ I ′′} being zero. Label the vertices of the polygons

∆,∆′,∆′′ with the coefficients pi, p′i, p′′i . We say that the labelled arc

pi · · · pj

37

of the labelled polygon ∆ is admissible if pi = pj and pi < pk for all interior labels pk.

We denote such an admissible arc by [i, j] and define pi = pj to be the index, ind[i, j]

of [i, j]. Let the set of admissible arcs be A.

For each admissible arc, [i, j] ∈ A(d) (resp. A′(d),A′′(d)), there exists a semi-

invariant cV (resp. cV′, cV

′) with V ∼= E[i+1,j]

∼= Ei+1 ⊕ · · · ⊕ Ej (resp. V ′ ∼=

E ′[i+1,j], V′′ ∼= E ′′[i+1,j].

Example 4.0.7. This example illustrates admissable arcs. Consider a tube of rank

6, with p0 = p3 = 0, p1 = p5 = p2 = 1 and p4 = 2. Then we have the following

polygon ∆ with admissible arcs denoted by the dotted lines.

p0 p1

444444

p5

444444

p2

p4 p3

labeled 0 ME

8-

%

��

��%

-8

EM

1 O0

�000000

1

000000

��1

��

2 0

Therefore the admissible paths are [0, 3], [3, 0] of index 0 and [1, 2] of index 1. So we

would get the associated semi-invariants cE[1,3] , cE[4,0] and cE[2,2] .

Theorem 4.0.8. [40] Let Q be a Euclidean quiver and d = ph + d′ the canonical

decomposition of d ∈ Dr with p ≥ 1. Then the algebra SI(Q,d) has the presentation

SI(Q,d) ∼=

K[c0, . . . , cp, cE[i+1,j] , cE

′[r+1,s] , cE

′′[t+1,m] ]

(c0 −∏

ind[i,j]=0 cE[+1i,j] , cp −

∏ind[r,s]=0 c

E′[r+1,s] ,

∑pi=0 ci −

∏ind[t,m]=0 c

E′′[t+1,m])

with

[i, j] ∈ A(d), [r, s] ∈ A′(d), [t,m] ∈ A′′(d)

Example 4.0.9. Continuing the example from 4.0.5, we have that D4 is of tubular

38

type (2, 2, 2). Again, we have the quiver of V :

5

1

a1

::tttttttttttttttttttttt2

a2

GG��3

a3

WW0000000000000

4

a4

ddJJJJJJJJJJJJJJJJJJJJJJ

The simple regular modules at the mouth of the tube have the following dimensions:

e0 = (1, 1, 0, 0, 1)

e1 = (0, 0, 1, 1, 1)

e′0 = (1, 0, 1, 0, 1)

e′1 = (0, 1, 0, 1, 1)

e′′0 = (1, 0, 0, 1, 1)

e′′1 = (0, 1, 1, 0, 1)

Since d = h we have that all pi, p′i, p′′i = 0 so each edge in ∆,∆′,∆′′ is an admissable

arc. Therefore we have the semi-invariants that come from the simple regular modules,

namely: cE0 , cE1 , cE′0 , cE

′1 , cE

′′0 , cE

′′1 . For example, cE0(V ) is the determinant of the

matrix

−X1 0 1 0

−Y1 0 0 1

0 −X2 1 0

0 −Y2 0 1

which is X2Y1 − X1Y2 and by a similar calculation cE1(V ) = X3Y4 − X4Y3 then

cE0cE1 = c0 + c1.

39

Similarly, we can see all the relations: cV1 − cE0cE1 , cV0 − cE′0cE′1 , cV1 − cV0 − cE′′0 cE′′1

So the ring of semi-invariants is

K[cV0 , cV1 , cE0 , cE1 , cE′0 , cE

′1 , cE

′′0 , cE

′′1 ]

(cV1 − cE0cE1 , cV0 − cE′0cE′1 , cV1 − cV0 − cE′′0 cE′′1 ).

40

5 Tame Concealed Algebras

We will first discuss some properties of tilting modules in order that we can discuss

a specific family of algebras obtained from tilting modules from tame hereditary

algebras.

Definition 5.0.10. For an A module M , we denote add M the smallest additive full

subcategory of mod A containing M . An A module T is called a tilting module if the

following conditions are satisfied:

(T1) proj dim T ≤ 1

(T2) Ext1A(T, T ) = 0

(T3) For any indecomposable projective A module P there exists a short exact se-

quence

0→ P → T ′ → T ′ → 0

with T ′, T ′ in add T .

Definition 5.0.11. Algebras of the form End(T ) where T is a preprojective tilting

module over a tame hereditary algebra A are called tame concealed algebras.

Example 5.0.12. This example illustrates how tame concealed algebras are formed

from hereditary algebras. Let A = E6 with the following orientation:

· // ·

��=======

· // · // ·

· // ·

@@��

41

We can find Ti in the preprojective component of ΓA such that B = End(⊕Ti) is

a tilting module and therefore B is tame concealed:

·

��;;;;;;; ·

��:::::: ·

��666666 T2

��::::::·

��:::::: T5

��::::::·

��666666 ·

��???????

·

BB��

��<<<<<< T1

��<<<<<·

��888888

DD��·

��888888 ·

��<<<<<<

BB��·

��<<<<<< ·

��<<<<<<

BB��T4

��<<<<<·

��<<<<<<

BB��·

��<<<<<< ·

��888888

DD��·

��888888 ·

BB��

��<<<<<< T7

AAAAAA·

· · ·

@@��//

��?????? ·

AA�� // ·

AA�� //

��>>>>>> ·

CC��// ·

CC��//

��::::: ·

AA�� // ·

��>>>>>>

AA�� // ·

AA�� // ·

AA��

��>>>>>> // ·

AA�� // ·

AA��

��>>>>>> // ·

CC��// ·

��:::::

CC��// ·

AA�� // ·

!!CCCCCCC

>>}}}}}}} // · · ·

·

??��

��<<<<<< ·

BB��

��888888 ·

??��

��<<<<< ·

??��

��<<<<<< ·

??��

��<<<<< ·

��888888

BB��·

��<<<<<<

??�� ·

·

@@�� ·

AA�� ·

CC��T3

AA��·

AA��T6

AA��·

CC��·

>>}}}}}}}

Therefore QB is the quiver with relations

· // ·

��=======

· f d a _ ] Z X

@@�� //

��======= · // ·

· // ·

@@��

5.1 Semi-Invariants of Tame Concealed

Definition 5.1.1. A pair (T,F) of full subcategories of A-modules is called a torsion

pair if the following conditions are satisfied.

(a) HomA(M,N) = 0 for all M ∈ T, N ∈ F

(b) HomA(M,−)|F = 0 implies M ∈ T

(c) HomA(−, N)|T = 0 implies N ∈ F

the subcategory T is called the torsion class and its objects are called torsion

objects. The subcategory F is called the torsion-free class and its objects are

called torsion free objects.

42

Consider the full subcategory T(T ) of mod-A defined by T(T ) = {M |Ext1A(T,M) =

0}

Proposition 5.1.2. ([1], IV.3.2) Let M,N ∈ T(T ) then we have functorial isomor-

phims:

(a) HomA(M,N) ∼= HomB(HomA(T,M), HomA(T,N))

(b) Ext1A(M,N) ∼= Ext1B(HomA(T,M), HomA(T,N))

For any R ∈ T , we know that HomA(τ−R, T ) = 0, since we have a separating

tubular family and T ∈ P so therefore Ext1A(T,R) = 0. Therefore we know that all

the regular modules of A are in T(T )

Theorem 5.1.3. ([1], VI.4.3) Let A be an algebra TA a tilting module and B =

EndTA. Then the correspondence

dimM 7→ dim HomA(T,M)− dim Ext1A(T,M)

where M is an A module induces an isomorphism f : K0(A) → K0(B) of the

Groethendieck groups of A and B.

Proposition 5.1.4. ([1], VI.4.5) Let A be an algebra of finite global dimension, TA

be a tilting module, B = EndTA and f : K0(A) → K0(B) be the isomorphism as in

5.1.3. Then for any A-modules M and N we have:

〈dimM, dimN〉A = 〈f(dimM), f(dimN)〉B.

Now consider specifically, an algebra T which is a tilting module in the prepro-

jective component of A, where A is tame hereditary with B = End(T ). That is B is

tame concealed, we have the following facts: ([39], XI.3.3)

43

(a) gl.dim B ≤ 2 and pd Z ≤ 1 and id Z ≤ 1, for all but finitely many non-

isomorphic indecomposable B-modules Z that are preprojective or preinjective.

(b) The Euler form 〈−,−〉B : K0(B)×K0(B)→ Z of B is Z congruent to the Euler

form 〈−,−〉A : K0(A) × K0(A) → Z of A, that is there exists f as in 5.1.3

making the following diagram commute

K0(B)×K0(B)〈−,−〉B // Z

K0(A)×K0(A)

f×f ∼=

OO

〈−,−〉A

55lllllllllllllllll

(c) The Auslander-Reiten quiver ΛB of B has the disjoint union form

ΛB = P(B) ∪R(B) ∪Q(B)

with P(B) a preprojective component, Q(B) a preinjective component and

R(B) is a family of regular components.

(d) R(B) is non-empty and furthermore, pd Z ≤ 1 and id Z ≤ 1 for any regular

module Z ∈ R(B).

Theorem 5.1.5. ([39], XI.3.4, XI.3.7) Let B = EndTA be tame concealed algebra.

Let add R(B) be the full subcategory of mod B whose objects are all the regular B

modules.

(a) The subcategory add R(B) of mod B is serial, abelian and closed under exten-

sions.

(b) The components of R(B) form a family T B = {T Bλ }λ∈Λ of pairwise orthogonal

stable tubes. Every tube T Aλ in mod B is the image of a tube in mod A under

the functor HomA(T,−). Moreover, if rBλ denotes the rank of T Bλ and n = |Q0|,

44

then ∑λ∈Λ

(rBλ − 1) ≤ n− 2.

(c) All but finitely many of the tubes T Bλ in R(A) are homogeneous and there are

at most n− 2 non-homogeneous tubes in T Bλ .

(e) The Euler quadratic form qB : K0(B) → Z is positive semidefinite of corank

one and there exists a unique positive vector hB ∈ K0(B) such that the radical

rad qB = {x ∈ K0(B), qb(x) = 0} of qB is an infinite cyclic subgroup of K0(B)

of the form rad qB = Z · hB.

(f) If f : K0(A) → K0(B) is a group isomorphism as in 5.1.3 then the following

diagram is commutative:

K0(A)ΦA //

f ∼=��

K0(A)

f ∼=��

K0(B)ΦB // K0(B)

(g) Let add R(B) be the full subcategory of mod B whose objects are all the reg-

ular B-modules. The functor HomA(T,−) : modA → modB restricts to the

equivalences HomA(T,−) : addR(A) → addR(B) of categories such that the

following diagram is commutative:

addR(A)τ //

HomA(T,−) ∼=

��

addR(A)τ−

oo

HomA(T,−) ∼=

��addR(B)

τ // addR(B)τ−

oo

As stated in 2.6, in this case where A is tame concealed, if d is a preprojective

or preinjective dimension vector then SI(A, d) is a polynomial algebra. Therefore we

45

will focus our attention on the case where d is a regular dimension vector.

Example 5.1.6. This example illustrates how the isomorphism f : K0(A)→ K0(B)

is constructed and what it does to dimension vectors of the simple regular modules.

Again as in the example 5.0.12, we have A = E6 with the following orientation:

1 // 4

��=======

2 // 5 // 7

3 // 6

@@��

Specifically, let T1 = P2, T2 = τ−2P3, T3 = τ−2P1, T4 = τ−3P2, T5 = τ−4P3, T6 =

τ−4P1, T7 = τ−6P2

therefore:

dimT1 = (0, 1, 0, 0, 1, 0, 1)

dimT2 = (1, 1, 0, 1, 1, 1, 2)

dimT3 = (0, 1, 1, 1, 1, 1, 2)

dimT4 = (0, 1, 0, 1, 2, 1, 2)

dimT5 = (1, 1, 1, 2, 2, 1, 3)

dimT6 = (1, 1, 1, 1, 2, 2, 3)

dimT7 = (1, 2, 1, 2, 3, 2, 4)

46

where the quiver with relations of B is

2 // 5

��=======

1 g f e d b a _ ] \ Z Y X W

@@��//

��======= 4 // 7

3 // 6

@@��

Then we have f as described in 5.1.3 as the matrix:

f =

0 1 0 0 1 1 1

1 1 1 1 1 1 2

0 0 1 0 1 1 1

0 0 1 1 1 0 1

0 0 0 1 1 1 1

0 1 0 1 0 1 1

0 −1 −1 −2 −2 −2 −3

If we let S be the matrix with column i = dimTi then f = E ·S where E is the Euler

matrix of A.

Using this we can easily find the dimension vectors of the simple regular modules

and the homogeneous module as in 5.1.5.d.

47

regular module dimension in A dimension in B

H (1, 1, 1, 2, 2, 2, 3) (1, 1, 1, 1, 1, 1, 1)

E1 (1, 0, 0, 1, 1, 0, 1) (0, 0, 0, 0, 1, 0, 0)

E2 (0, 1, 0, 0, 1, 1, 1) (1, 1, 0, 1, 0, 1, 1)

E3 (0, 0, 1, 1, 0, 1, 1) (0, 0, 1, 0, 0, 0, 0)

E ′1 (1, 0, 0, 1, 0, 1, 1) (0, 1, 0, 0, 0, 0, 0)

E ′2 (0, 1, 0, 0, 1, 1, 1) (0, 0, 0, 0, 0, 1, 0)

E ′3 (0, 0, 1, 1, 0, 1, 1) (1, 0, 1, 1, 1, 0, 1)

E ′′1 (1, 1, 1, 1, 1, 1, 2) (1, 1, 1, 0, 1, 1, 1)

E ′′2 (0, 0, 0, 1, 1, 1, 1) (0, 0, 0, 1, 0, 0, 0)

A key observation of tilting theory is that the tilting module TA induces a tilting

B-module, which is the left B-module BT . Moreover, the algebra A can be recovered

from B and BT . This is seen from the tilting theorem of Brenner and Butler:

Theorem 5.1.7. ([5]) Let TA be a tilting module and B = EndTA, and (T(TA),F(TA)),

(DF(BT ), DT(BT )) be the induced torsion pairs in modA and modB respectively.

Then T has the following properites.

(a) BT is a tilting module and the canonical K-algebra homomorphism A→ End(BT )op,

given by

a 7→ (t 7→ ta)

is an isomorphism.

(b) The functors HomA(T,−) and −⊗BT induce quasi-inverse equivalences between

T(TA) and DT(BT ).

(c) The functors Ext1A(T,−) and TorB1 (−, T ) induce quasi-inverse equivalences be-

tween F(TA) and DF(TA).

48

From part (b) of this theorem more specifically, we can notice that the functors

HomA(T,−) and −⊗B T induce quasi-inverse equivalences between R(A) and R(B)

sinceR(B) ∈ DT(BT ). Similarly, we can find a tilting module TB in the preprojective

component of B such that A = End(TB). Therefore we can define a functor

ϕ′ = HomB(TB,−) : R(B)→ R(A)

which induces isomorphism of the regular modules.

Lemma 5.1.8. Let A be a tame hereditary algebra, T a tilting module from A and

B = EndT . Let ϕ = HomA(T,−) : modA→ modB and f : K0(A)→ K0(B) which

is an isomorphism of dimension vectors such that f(dimM) = dimϕ(M).

(a) If E1, . . . , Er are the simple regular modules at the mouth of a tube in A, then

ϕ(E1), · · · , ϕ(Er) are the modules at the mouth of a tube in B. Then it follows

that rank of T Aλ is r so is the rank of ϕ(T Aλ ) = T Bλ is also r. We can conclude

that the tubular type of B is the same as the tubular type of A.

(b) For any module M ∈ R(A), dimM = phA +∑s

j=1

∑mj−1i=0 p

(j)i e

(j)A,i. Then

dimϕ(M) = phB +∑s

j=1

∑mj−1i=0 p

(j)i e

(j)B,i.

(c) Let M ∈ A be a module of dimension phA +∑s

j=1

∑mj−1i=0 p

(j)i e

(j)A,i. Let V ∈ A be

orthogonal to M in the sense that HomA(M,V ) = ExtA(M,V ) = 0 then ϕ(V )

is orthogonal to ϕ(M).

Proof. (a) Since we have the equivalence of categories of regular modules from the

tilting theorem of Brenner and Butler, this is clear.

(b) By 5.1.5.d we know that hB exists. We know f(hA) = hB since this just follows

from (a) hA = eA,1 + · · ·+ eA,r and hB = f(eA,1) + · · ·+ f(eA,r).

(c) This follows from 5.1.2 since M,V ∈ T(T ). We have that HomA(M,V ) =

HomA(ϕ(M), ϕ(V )) = 0 and also that Ext1A(M,V ) = Ext1

A(ϕ(M), ϕ(V )) = 0.

49

Theorem 5.1.9. Let A be a tame hereditary algebra, T a tilting module from A and

B = EndT . Let ϕ = HomA(T,−) : modA→ modB and f : K0(A)→ K0(B) which

is an isomorphism of dimension vectors such that f(dimM) = dimϕ(M) 5.1.3. Let

d be a QA is the quiver associated to A and QB the quiver associated to B. Then we

have a natural isomorphism SI(QA, d)∼= // SI(QB, f(d)) .

Proof. The previous lemma 5.1.8 shows that the conditions (1)-(6) of 3.1.4 hold.

By 5.1.7 we have an equivalence of R(A) and R(B). From this it follows that the

orthogonal categories are the same 5.1.8 (c) and thus condition (7) holds. We know

that every module in R(B) has projective dimension 1 from 5.1.5. In general, we

have that if cV is a generating semi-invariant in SI(QA, d) then cϕ(V ) is a generating

semi-invariant in SI(Qb, f(d)). Therefore, the main theorem 3.2.1 is demonstrated in

this case.

50

6 Tubular Algebras

First, we will discuss the construction of the tubular algebras and then the module

structure. Given a finite dimensional algebra A0 and and an A0 module R we denote

byA0[R] the one-point extension of A0 by R, namely the algebra

A0[R] =

A0 R

0 K

=

a r

0 b

∣∣∣a ∈ A0, r ∈ R, b ∈ k

The quiver of A0[R] contains the quiver of A0 as a full subquiver and there is an

additional vertex ω called the extension vertex of A0[R]

The category of A0[R] modules can be described as the triples V = (V0, Vω, γV )

corresponding to the A0[R] module

V0

Vω

with

0 R

0 0

operating on it via the

map γV : R⊗k Vω → V0 adjoint to γV .

The one-point coextension [V ]A of A by V is defined by:

[V ]A = (Aop[DV ])op ≈

K DV

0 A

The vertex of the quiver of [V ]A belonging to the simple projective module

K

0

will be denoted by −ω and called the coextension vertex. The [V ]A modules can be

written as triples W = (W−ω, W0, δW ) where W−ω is a K-vectorspace, W0 an A

module and δW : DV ⊗W0 → W−ω a K-linear map.

Definition 6.0.10. A branch in b of length n is any finite full connected subquiver

of the quiver:

51

b

β+

xxppppppppppppppppp

b+ ________________β++

yysssssssb−

β−

ffNNNNNNNNNNNNNNNNN

β−+

yysssssss

b++ _______

��b+−

β+−eeKKKKKKK

��b−+ _______

��b−−

��

β−−eeKKKKKKK

b+++ ___

��b++−

\\::::

��b+−+ ___

��b+−−

]]::::

��b−++ ___

��b−+−

��

]]::::b−−+ ___

��b−−−

��

]]::::

· · ·

]];;;;· · ·

\\9999· · ·

]]<<<<<· · ·

\\9999· · ·

]]<<<<<· · ·

\\9999· · ·

^^<<<<<

whose vertices are of the form bi1...in indexed by any possible sequence i1, . . . , in ∈

{+,−} and arrows

βi1...in− : bi1...in− → bi1...in

and

βi1...in+ : bi1...in → bi1...in+

with relations βi1...in−βi1...in+.

Definition 6.0.11. Given an algebra C with quiver QC and relations σi with a vertex

b of QC then the restriction B of C to some full subquiver Q′ will be said to be a

branch of C in b of length n provided that:

a) B is a branch in b of length n.

b) There is a full subquiver Λ such that Q = Q′ ∪ Λ.

c) Q′ ∩ Λ = {b}.

d) Any relation σi has its support in either Q′ or Λ.

If we denote by {ρi} the set of all relations having suport in Q and let A = (Λ, {ρi}),

then C is said to be obtained from A by adding the branch B in b.

Let A0 be an algebra E1, . . . Et be A0 modules and B1, . . . Bt branches. Let

A0[Ei, Bi]ti=1 be inductively defined. Let A0[E1, B1] be obtained from the one-point

52

extension A0[E1] with extension vertex ω1 by adding the branch B1 in ω1, and let

A0[Ei, Bi]ti=1 = (A0[Ei, Bi]

t−1i=1)[Et, Bt].

Definition 6.0.12. The algebra A = A0[Ei, Bi]ti=1 is called a tubular extension of A0

using modules from T provided that the modules E1, . . . Et are pairwise orthogonal

ray modules from T .

Definition 6.0.13. A tubular extension A of a tame concealed algebra of extension

type (2, 2, 2, 2), (3, 3, 3), (4, 4, 2) or (6, 3, 2) is called a tubular algebra.

If the tame concealed algebra was of type (m1, . . . ,ms), an extension on a module

Ei in the tube of rank mj by a branch of length |Bi| is an algebra of extension type

(m1, · · · ,mj−1,mj + |Bi|,mj+1, · · · ,ms). An extension on a homogeneous tube by

a branch of length |Bi| is an algebra of extension type (m1, . . . ,ms, 1 + |Bi|). See

Appendix A for a list of possible (m1, . . . ,ms) on which a tubular extension can be a

tubular algebra.

Example 6.0.14. Starting with the tame concealed algebra of 5.0.12 which is of

tubular type (3,3,2), we can form tubular algebras of type (3,3,3) using a one point

extension on either E ′′1 or E ′′2 with extension vertex labeled as 1 as follows

One point extension on E ′′1 , e′′1 = (1, 1, 1, 0, 1, 1, 1):

3 // 6��===

1 //P U _ i n2j g c _ [ W T@@��

//

��=== 5 // 8

4 // 7

@@��

One point extension on E ′′2 , e′′2 = (0, 0, 0, 1, 0, 0, 0):

1

&&NNNNNNNN\ Z X V T R P

3 // 6��===

2 T W [ _ c g j

@@��//

��=== 5 // 8

4 // 7

@@��

53

Or we can form a tubular algebra of type (4,4,2) using a one point extension on

each of the tubes of rank 3. From this we get a variety of tubular algebras where the

asterisk indicates an extension vertex.

∗��>>>> Y T

N∗��>>>> G

=4· // ·��<<<

· k g c _ [ W S@@�� //

��>>>> · // ·

· // ·AA��

∗��>>>

_ SD

· // ·��<<<

∗ //8

O ^

o l j h e c a

· m h _ V Q@@�� //

��>>> · // ·· // ·

AA��

∗��>>>> N

=-· // ·��<<<

· k g c _ [ W SAA�� //

��<<< · // ·

· // ·AA��

∗@@�� p

��

∗��>>> O

=,· // ·��<<<

∗ //

p `

6A

M W _ g n

· m h d _ Z V QAA�� //

��<<< · // ·· // ·

AA��

∗��>>>>

NNNNNNNNp _ N

· // ·��<<<

· l h c _ [ V RAA�� //

��<<< · // ·

∗ N _ p

pppppppp

@@��· // ·

AA��

We can also form a tubular algebra of type (6,3,2) using a branch extension on

one of the tubes of rank 3. Let B be the branch, then |B| = 3. We obtain one

of the following quivers where the asterisks are the extension vertices and the lines

connecting vertices indicate that the arrows could be any orientation:

∗ ∗ ∗��

..

..

· // ·��>>>>

· k g c _ [ W S@@�� //

��>>>> · // ·

· // ·@@��

∗ //n _ P∗ O

?/

��@@@@// ∗

· // ·��>>>>

· k g c _ [ W S@@�� //

��>>>> · // ·

· // ·@@��

54

∗

∗ · // ·��<<<<

∗ //

hhhhhhhh

NNNN · l h c _ [ V RAA�� //

��<<< · // ·

· // ·AA��

∗�

�.

��· // ·

��<<<<

∗��

//

hhhhhhhh

NNNN · l h c _ [ V RAA�� //

��<<<< · // ·

∗ · // ·AA��

∗ ∗ ∗�� NNNN

· // ·��<<<

· k g c _ [ W S??~~~~ //

��@@@@ · // ·

· // ·AA��

∗ //n _ P∗

OOOO��

// ∗

· // ·��<<<

· k g c _ [ W S??~~~~ //

��@@@@ · // ·

· // ·AA��

Theorem 6.0.15. [33] Let A0 be an algebra with a separating tubular family T ,

separating P from Q. Let A = A0[Ei, Bi]ti=1 be a tubular extension using modules

from T . We define module classes P0, T0, Q0 in mod A as follows: Let P0 = P , let

T0 be the module class given by all indecomposable A modules M with either M |A0 (the

restriction of M to A0) non-zero and in T or else the support of M being contained in

some Bi and 〈lBx , dimM〉 < 0. Let Q0 be the module class given by all indecomposable

A modules M with either M |A0 non-zero and in Q or else the support of M being

contained in some Bi and 〈lBx , dimM〉 > 0. Then mod A = P0 ∨ T0 ∨ Q0 with T0

being a separating tubular family separating P0 from Q0.

The elements z ∈ Zn such that χ(z) = 〈z, z〉 = 0 are called the radical vectors

and they form a subgroup of Zn called radχ.

An algebra A will be said to be cotubular provided that the opposite algebra Aop is

tubular. Let A be an algebra which is both tubular and cotubular. Assume that A is

an extension of A0 and a coextension of A∞ where both A0 and A∞ are tame concealed

algebras. Let α0 be the positive radical generator of K0(A0) then 〈α0, α0〉 = 0 and

α∞ be the positive radical generator of K0(A∞) then 〈α∞, α∞〉 = 0. α0 + α∞ is

sincere and the non-sincere positive radical vectors are the positive multiples of α0

55

and α∞.

Consider dimension vector β = pα0 +qα∞, with GCD(p, q) = d, p = dp′, q = dq′,

we define the component C(p, q) to be the closure of all modules which are direct

sums of d copies of regular indecomposable modules of dimension β′ = p′α0 + q′α∞.

Given any x ∈ K0(A) we can define the index of x provided that the values

of〈α0, x〉 and 〈α∞, x〉 are not both zero. In this case, we call index (x) =〈α0, x〉〈α∞, x〉

.

For a module M we will also call the index of its dimension vector index (M).

Theorem 6.0.16. For any non-zero element in the radical of χ, its index is defined

namely index (pα0 + qα∞) =q

p= γ.

We want to define module classes Pγ, Tγ, Qγ. P0, T0, Q0 have been defined and

dually P∞, Q∞, T∞ can be defined. Q∞ is the module class of all preinjective A∞

modules. T∞ is the module class given by all indecomposable A modules M with

M |A∞ a non-zero regular A∞ module or else M |A∞ being zero and 〈α0, M〉 > 0.

P∞ is the module class given by M |A∞ a preprojectiveA∞ module or else M |A∞

being zero and 〈α0, M〉 < 0.

Note that Q0 ∩P∞ is the module class given by all indecomposable A modules M

with both 〈α0, dimM〉 > 0 and 〈α∞, dimM〉 < 0.

Theorem 6.0.17. The following module classes are pairwise disjoint:

P0, T0, Q0 ∩ P∞, T∞, Q∞

and they give the complete category of A modules.

56

M in 〈α0, M〉〈α∞, M〉 index(M)

P0 < 0 ≤ 0 negative or ∞

T0 = 0 < 0 0

Q0 ∩ P∞ > 0 < 0 positive

T∞ > 0 = 0 ∞

Q∞ ≥ 0 > 0 negative or 0

For γ =q

p, we can define Pγ, Tγ, Qγ as the module classes of all modulesM

such that 〈pα0 + qα∞, dimM〉 < 0,= 0 or > 0 respectively. We call Tγ γ-separating

provided Tγ separates Pγ from Qγ.

The next theorem determines completely the structure of all components of A.

We are particularly interested in this structure because the structure of the rings of

semi-invariants of tubular algebras is related to this module structure.

Theorem 6.0.18. ([33]) The structure of a tubular algebra A is as follows: First,

there is a preprojective component P0 which is the module classof all preprojective A0

modules. Then for any γ = q/p, there is a separating tubular P1k family Tγ, and all

Tγ are stable of type (2, 2, 2, 2), (3, 3, 3), (4, 4, 2)or (6, 3, 2) except for T0 and

T∞. Finally, there is the preinjective component Q∞ which is the module class of all

preinjective A∞ modules.

Example 6.0.19. Starting with the tame concealed algebra of 5.0.12 which is of

tubular type (3,3,2), we can form tubular algebras of type (3,3,3) using a one point

extension on E ′′1 , e′′1 = (1, 1, 1, 0, 1, 1, 1), with extension vertex labeled as 1 as follows.

3 // 6��===

1 //P U _ i n2j g c _ [ W T@@��

//

��=== 5 // 8

4 // 7

@@��

57

LetA = KQ/I, we have thatA0 is tame concealed of type E6 with h0 = (1, 1, 1, 1, 1, 1, 1)

as in example 5.0.12 with the following simple regular non-homogeneous dimension

vectors:

regular module dimension in A0

H (1, 1, 1, 1, 1, 1, 1)

E0,1 (0, 0, 0, 0, 1, 0, 0)

E0,2 (1, 1, 0, 1, 0, 1, 1)

E0,3 (0, 0, 1, 0, 0, 0, 0)

E ′0,1 (0, 1, 0, 0, 0, 0, 0)

E ′0,2 (0, 0, 0, 0, 0, 1, 0)

E ′0,3 (1, 0, 1, 1, 1, 0, 1)

E ′′0,1 (1, 1, 1, 0, 1, 1, 1)

E ′′0,2 (0, 0, 0, 1, 0, 0, 0)

A∞ is the following:

3 // 6

1 //P U _ i n2

@@��//

��=== 5

4 // 7

which is also tame concealed of type E6 with the following simple regular dimension

vectors:

58

regular module dimension in A0

H∞ (1, 1, 1, 1, 1, 1, 1)

E∞,1 (0, 1, 1, 0, 1, 0, 0)

E∞,2 (0, 1, 0, 1, 0, 0, 1)

E∞,3 (1, 1, 1, 1, 0, 1, 0)

E ′∞,1 (0, 1, 0, 1, 1, 0, 0)

E ′∞,2 (0, 1, 1, 0, 0, 1, 0)

E ′∞,3 (1, 1, 1, 1, 0, 0, 1)

E ′′∞,1 (1, 2, 1, 1, 1, 1, 1)

E ′′∞,2 (0, 1, 1, 1, 0, 0, 0)

Then we have that α0 = (0, 1, 1, 1, 1, 1, 1, 1) α∞ = (1, 3, 2, 2, 1, 1, 1, 0). In the case

where γ = 1 we have h1 = α0 + α∞ = (1, 4, 3, 3, 2, 2, 2, 1).

Definition 6.0.20. Let T be a tilting A module, T = T0⊕Tp with T0 a preprojective

tilting A0 module and Tp a projective A module, then T is called a shrinking module.

The corresponding functor ΣT = Hom(T, −) : modA→ End(T ) will be called a left

shrinking functor.

Tp is uniquely determined by A0 and A, it is just the direct sum of indecomposable

projective modules Pa with a outside A0. When A is a tubular extension of a tame

concealed algebra A0 then ΣTA = B where B is a tubular extension of B0 = End(T0).

Theorem 6.0.21. ([33]) Let A be a tubular algebra. Then there exists a left shrinking

functor Σ : A−mod→ B−mod, where B is a tubular extension of a tame concealed

canonical algebra.

Theorem 6.0.22. ([33] 5.7.1) Let A be a canonical tubular algebra. Then there exists

a proper left shrinking functor from mod A to C- mod where C is one of the algebras

C(4, λ), C(6), C(7), or C(8).

59

C(4, λ)

5 2α2

��

4

γ

^^=======

γ′��1

α1

^^=======

β1��

6 3β2

^^=======

with relations (α12 − β12)γ = 0 ,(α12 − λβ12)γ′ = 0. C(4, λ) is of type (2, 2, 2, 2) with

α0 = (0, 1, 1, 2, 1, 1) and α∞ = (1, 1, 1, 1, 0, 0).

C(6)

4α3

��2

α2oo

8 7γ′oo 6γ

oo 1

α1

^^=======

β1��

5β3

^^=======

3β2

oo

with relation (α123−β123)γ = 0. C(6) is of type (3, 3, 3) with α0 = (0, 1, 1, 2, 2, 3, 2, 1)

and α∞ = (1, 1, 1, 1, 1, 1, 0, 0).

C(7)

6α4

��4

α3oo 2α2oo

9 8γoo 1

α1

^^=======

β1��

7β4

^^=======

5β3

oo 3β2

oo

with relation (α1234−β1234)γ = 0. C(7) is of type (4, 4, 2) with α0 = (0, 1, 1, 2, 2, 3, 3, 4, 2)

and α∞ = (1, 1, 1, 1, 1, 1, 1, 1, 0).

60

C(8)

3α3

wwpppppppppppppp 2α2oo

10 9γoo 1

α1

ggNNNNNNNNNNNNNN

β1��

8β6

^^=======

7β5

oo 6β4

oo 5β3

oo 4β2

oo

with relation (α123−β123456)γ = 0. C(8) is of type (6, 3, 2) with α0 = (0, 2, 3, 1, 2, 3, 4, 5, 6, 3)

and α∞ = (1, 1, 1, 1, 1, 1, 1, 1, 1, 0).

In order to find the modules in Tγ we will first find the tubular family T1, then we

will use shrinking functors to shift T1 to any Tγ, γ ∈ Q+. These shrinking functors

are just specifically defined tilting functors.

Example 6.0.23. Continuing with example 6.0.19, we can illustrate 6.0.22. We have

A

3 // 6��===

1 //P U _ i n2j g c _ [ W T@@��

//

��=== 5 // 8

4 // 7

@@��

First we need a shrinking functor which takes A to a canonical tubular algebra.

In this case we take Tp = P1 and the following Ti:

61

P3

��========·

@@@@@@@@ T2

$$IIIIIIIIIII

P6

>>}}}}}}}

AAAAAAAAA·

��========

@@��T5

$$HHHHHHHHHHH

::vvvvvvvvvvT3 = P2

$$IIIIIIIIII·

P8

>>}}}}}}}//

AAAAAAA P5// ·

@@�� //

��========= · // T8

>>~~~~~~~//

@@@@@@@ T6

::vvvvvvvvv// ·

$$IIIIIIIIIIII

::vvvvvvvvvvvv // · · ·

P7

>>}}}}}}}}}

AAAAAAA·

@@��

��========= T7

::vvvvvvvvvvv

$$HHHHHHHHHH ·

P4

@@��·

>>~~~~~~~~T4

::vvvvvvvvvvv

That is, T2 = τ−2P3, T3 = P2, T4 = τ−2P4, T5 = τ−2P6, T6 = τ−2P5, T7 = τ−2P7

and T8 = τ−2P8.

Then B = End(Tp ⊕8i=2 Ti) is the following canonical algebra:

2 // 5

��=======

1h e b _ \ Y V

@@��//

��======= 3 // 6 // 8

4 // 7

@@��

This algebra is canonical and tubular. In the preprojective component of B, we

can find T ′i such that End(⊕T ′i ) is

· // ·

��=======

· // · // ·

· // ·

@@��

Specifically, we have that:

62

·

��?????????? ·

��???????? T ′2

��@@@@@@@@@

·

AA��

��<<<<<<<<< T ′8

��???????·

��????????

??�� ·

��?????????? T ′4

��????????

??��·

��@@@@@@@@@@ ·

·

BB��//

��99999999 ·

AA�� // ·

??�� //

��?????????? T ′7

??��// T ′6

??��//

��???????·

??�� // ·

��@@@@@@@@@@

??~~~~~~~~~~ // · · ·

·

AA��

��<<<<<<<<< ·

??��

��?????????? T ′5

??��

��???????·

·

??�� ·

??��T ′3

??~~~~~~~~~

and take T ′1 = P1 Then we have that T = End(⊕T ′i ) is of the form

4α3

��2

α2oo

8 7γ′oo 6γ

oo 1

α1

^^=======

β1��

5β3

^^=======

3β2

oo

with relation (α123 − β123)γ = 0. with Ti corresponding to vertex i. This is C(6).

6.1 Semi-Invariants of Tubular Algebras

In this section we will verify the conditions of 3.1.4 for Tubular Algebras to show the

main theorem holds in this case.

Let A = KQ/I be a tubular algebra as previously defined of tubular type T =

(m1, . . . ,ms) = (2, 2, 2, 2), (3, 3, 3), (4, 4, 2) or (6, 3, 2).

We know that the representation space Rep(A, d) for some d ∈ R(A) is irreducible

by the following theorem of Bobinski and Skowronski [7].

Theorem 6.1.1. [7] Let A be a tame concealed or tubular algebra and d ∈ R(A).

63

Then the affine variety modA(d) of all A modules of dimension vector d is irreducible,

normal, a complete intersection,

dim modA(d) = dimGL(d)− 〈〈d, d〉〉

and the maximal GL(d)-orbits in modA(d) form the open sheet consisting of nonsin-

gular points. Moreover, a module M in modA(d) is a nonsingular point if and only

if Ext2A(M,M) = 0.

We now need to show that a tubular family Tγ ∈ A, satisfies the conditions of the

main theorem 3.2.1. Primarily, we use the fact that for any γ ∈ Q∞0 , we have that

Tγ is a separating tubular PK1 family with all but T0 and T∞ being stable of tubular

type T ([32], 5.2.4).

From the properties of stable tubular families we get the following conditions are

true:

1. For each Tγ, we have a family of homogeneous indecomposable modules {Vt},

t ∈ P1K of dimension hγ which are the homogeneous modules in Tγ. We also

have a family of nonhomogeneous modules E(j)i of dimension vectors e

(j)i , (1 ≤

j ≤ s, 0 ≤ i ≤ mj − 1). In particular, for tubular algebras, we have that

h = pα0 + qα∞ as defined in section 6.

2. Let X be an indecomposable module in Tγ. If X belongs to a homogeneous

tube then dimX = ph, p ∈ Z+ where p is the regular length of X. If X is in a

non-homogeneous tube of rank mj, then dimX = ph+e(j)i + · · ·+e

(j)i +k where

the regular length is mjp+ (k+ 1). Therefore in general, if X is any module in

add T , with dimX = β we can write the dimension vector as:

β = phγ +s∑j=1

mj−1∑i=0

p(j)i e

(j)i .

64

Therefore tubular algebras satisfy condition 3.1.4.(1).

3. Conditions 3.1.4.(2) and (5) hold as a result of the properties of tubular families

([32], 3.1.3’)

4. In order to show condition 3.1.4.(3), we will need to use the idea of generic

decomposition. The following theorems of Crawley-Boevey and Schroer, aid in

our proof of this decomposition:

Theorem 6.1.2. If C is an irreducible component in moddA(k), then C =

C1 ⊕ · · · ⊕ Ct for some irreducible components Ci of module varieties moddiA (k)

with d = d1 + · · ·+ dt and with the property that the general module in each Ci

is indecomposable. Moreover, C1, . . . , Ct are uniquely determined by this, up to

reordering.

Theorem 6.1.3. If Ci ⊆ moddiA (k) are irreducible components (1 ≤ i ≤ t), and

d = d1 + · · · + dt, then C is a irreducible component of moddiA (k) if and only if

Ext1A(Ci, Cj) = 0 for all i 6= j.

We will use the notion of admissible paths as defined in 3.1.2. Consider a vector

of the form

β = phγ +s∑j=1

mj−1∑i=0

p(j)i e

(j)i .

Clearly ph decomposes into p homogeneous modules of dimension h, namely

Vt1⊕· · ·⊕Vtp . For each j, 1 ≤ j ≤ s, we can decompose d =∑mj−1

i=0 p(j)i e

(j)i using

induction on the index of the admissible path. Beginning with k = max{p(j)i },

consider the admissible paths k1, . . . kn of index k. Let E(j)ki

be the module

associated to the admissible path ki with dimension e(j)ki

. Then we can write the

dimension vector as d = d′ =∑mj−1

i=0 q(j)i e

(j)i + e

(j)k1

+ · · · + e(j)kn

where q(j)i = p

(j)i

if p(j)i 6= k and q

(j)i = k − 1 if p

(j)i = k. Next consider the admissible paths of

65

index k′ = k − 1 and then continue in this manner until the paths of index 1

are considered. Therefore we have the general decomposition:

V = Vt1 ⊕ . . .⊕ Vtp ⊕⊕sj=1 ⊕[i,k]admissible,length≥1 E(j)[i,k].

It remains to be shown that each summand has no extensions with any other.

Because there are no maps between tubes of tubular algebras, we just need

to check that the modules in the same tube have no extensions. Consider,

E(j)[i,k] and E

(j)[m,n]. Without loss of generality and by the definition of admissible

sequences, we have either 0 < m < i < k < n < mj that is [i, k] ⊂ [m,n] or

0 < i < k < m < n < mj with m− k > 1, we cannot have k = m or k = m− 1

by the construction of our decomposition. So in the case [i, k] ⊂ [m,n], we have

that

〈〈ei + · · · ek, em + · · ·+ en〉〉 = 〈〈ei + · · ·+ ek, ei−1 + · · ·+ ek〉〉

=k∑l=i

〈〈el, ei−1 + · · ·+ ek〉〉 =k∑l=i

〈〈el, el−1 + el〉〉

k∑l=i

〈〈el, el−1〉〉+k∑l=i

〈〈el, el〉〉 = −(k − i) + (k − i) = 0.

Therefore, Ext1A(E

(j)[i,k], E

(j)[m,n]) = 0. Similarly, Ext1

A(E(j)[m,n], E

(j)[i,k]) = 0. In the

case that m− k > 1, it is clear that

〈〈e[i,k], e[m,n]〉〉 = 〈〈e[m,n], e[i,k]〉〉 = 0

based on the conditions of 3.1.4.(5). So again, Ext1A(E

(j)[i,k], E

(j)[m,n]) = 0 and

Ext1A(E

(j)[m,n], E

(j)[i,k]) = 0.

66

5. Condition 3.1.4.(4): All modules X ∈ T have projective and injective dimension

1. τX is also in T , therefore Hom(D(A), τX) = 0 since T is a separating tubular

family. Similarly, Hom(τ−X,A) = 0 therefore X has injective dimension 1.

6. We can see that tubular algebras easily satisfy condition 3.1.4.(6) since we have

the condition on tubular algebras that all sincere homogeneous modules are of

dimension h = pα0 + qα∞ with p, q ∈ Z+ and furthermore, γ = q/p.

7. Condition 3.1.4.(7) holds based on the fact that Tγ is generated by modules

such that 〈dimX,h〉 = 0. Also, in the tubular case, γ = −〈dimX,α∞〉〈dimX,α0〉

.

Example 6.1.4. In the following example, we will continue example 6.0.23 applying

the main theorem to the algebra C(6). Recall that C(6) is the following algebra:

4α3

��2

α2oo

8 7γ′oo 6γ

oo 1

α1

^^=======

β1��

5β3

^^=======

3β2

oo

with relation (α123 − β123)γ = 0. For A = C(6), the tubular type of A is (3,3,3).

We have that A0 = E6 and A∞ = A5. Therefore, α0 = (0, 1, 1, 2, 2, 3, 2, 1) and

α∞ = (1, 1, 1, 1, 1, 1, 0, 0). For the purpose of this example as well as the explo-

ration of shrinking functors in the next section, we will choose h = α0 + α∞ =

(1, 2, 2, 3, 3, 4, 2, 1). Thus γ = 1 and we are dealing with the separating tubular fam-

ily T1.

We know the dimension vectors e(j)i for T1:

67

dimension

E(1)0 (0, 1, 1, 1, 1, 1, 1, 1)

E(1)1 (1, 1, 1, 1, 1, 2, 1, 0)

E(1)2 (0, 0, 0, 1, 1, 1, 0, 0)

E(2)0 (1, 1, 1, 2, 1, 2, 1, 1)

E(2)1 (0, 1, 0, 1, 1, 1, 1, 0)

E(2)2 (0, 0, 1, 0, 1, 1, 0, 0)

E(3)0 (1, 1, 1, 1, 2, 2, 1, 1)

E(3)1 (0, 0, 1, 1, 1, 1, 1, 0)

E(3)2 (0, 1, 0, 1, 0, 1, 0, 0)

Clearly we can see the illustration of 3.1.4.(2), that is that for each j, we have

that∑2

i=0 e(j)i = h. Let β = h + e

(1)1 + e

(2)0 + 2e

(2)1 , so we have p = p

(1)1 = p

(2)0 = 1 and

p(2)1 = 2. So we have the polygons:

∆1 = 0

��

=======

1 1

∆2 = 1

��

=======

2 0

∆3 = 0

��

=======

0 0

If we indicate an admissible path in ∆j by [i, k](j), we have the admissible paths:

[0, 0](1), [1, 1](2), [0, 1](3), [1, 2](3), [2, 0](3) of index 0, [1, 2](1), [0, 0](2) of index 1 and [2](2)

of index 2.

We can then find the generic decomposition of V of dimension β:

V = Vt ⊕ E(1)[1,2] ⊕ E

(2)2 ⊕ E

(2)[0,2].

68

From the main theorem 3.2.1 we have the semi-invariants from modules in the tubes:

cE(1)[0,2] , cE

(2)[1,0] , cE

(3)0 , cE

(3)1 , cE

(3)2 .

We also have c0, c1 which are linear combinations of cVi ’s. By direct calculation

the relations are c0 = cE(1)[0,2] , c1 = cE

(2)[1,0] and c0 + c1 = cE

(3)0 cE

(3)1 cE

(3)2 .

69

7 Shrinking Functors and Semi-Invariants

The main theorem in this paper gives a set of conditions under which the structure

of the ring of semi-invariants for a tubular algebra can be seen in terms of generators

and relations. Generally speaking, shrinking functors are one of the tools used with

Tubular Algebras and they can be used in the explicit calculation of the generating

cV ’s for semi-invariants. Unfortunately, although the existence of shrinking functors

is shown in [32], they are not easy to explicitly find for all cases. In the C4, C6, C7, C8

examples from theorem 6.0.22, we do have the exact calculation of the shrinking

functors. We also have theory relating the canonical tubular algebras and these C

algebras through the use of shrinking functors.

Let A be a tubular algebra. As stated earlier, if T be a tilting A module, T =

T0 ⊕ Tp with T0 a preprojective tilting A0 module and Tp a projective A module,

then T is called a shrinking module. The corresponding functor Σl = Hom(T, −) :

modA→ End(T ) will be called a left shrinking functor.

Tp is uniquely determined by A0 and A, it is just the direct sum of indecomposable

projective modules Pa with a outside A0. When A is a tubular extension of a tame

concealed algebra A0 then ΣTA = B where B is a tubular extension of B0 = End(T0)

Let σl : K0(A)→ K0(B) be the linear transformation induced by Σl.

We need the dual notion of a right shrinking functor on A which is uniquely

determined by A and A∞. Let T be a cotilting A module with T = Tq ⊕ T∞ where

T∞ is a preinjective cotilting A∞ module and Tq is an injective A module. Tq is

the direct sum of all indecomposable injective A modules Qa with a outside A∞. If

B = End(T ). The functor Σr : A−mod→ B −mod is given by:

Σr = HomA(−, T ) ·D = (TD)⊗−.

70

Σr is called the right shrinking functor.

Since the right shrinking functor is the dual notion of the left shrinking functor any

result concerning left shrinking functors, there is a corresponding result concerning

right shrinking functors. In particular, we have a linear transformation corresponding

to Σr namely σr : K0(A)→ K0(B).

Note that σl(α∞) = α0 +α∞ = σr(α0) therefore σl(pα0 + qα∞) = (p+ q)α0 + qα∞

and σr(pα0 + qα∞) = pα0 + (p+ q)α∞. Define σl (resp. σr) to be the map of indices

taking γ to γγ+1

(resp. γ + 1).

We use shrinking functors to shift T1 to Tγ, γ ∈ Q+. So first we must find the

simple modules in T1. So once we know the modules in T1, we can explicitly determine

the modules in other Tγ.

Proposition 7.0.5. For a tubular algebra A, Σr and Σl preserve the structure of the

ring of semi-invariants for any γ ∈ Q+

Proof. Following directly from 5.1.8, we know that for a tubular algebra A, Σr and Σl

preserve generic decomposition. For V ∈ R(A), Σr(V⊥) = (Σr(V ))⊥ and Σl(V

⊥) =

(Σl(V ))⊥. If 〈〈V,W 〉〉 = 0, then then by 5.1.2, we have that

〈〈Hom(T, V ),Hom(T,W )〉〉 = 0.

It follows easily from the main theorem 3.2.1, that shrinking functors preserve the

structure of the ring of semi-invariants of tubular algebras. In particular, Lemma

3.2.3 holds. If we have that for T1 the dimension of the weight space 〈〈h,−〉〉 is

2, that is the weight space is generated by a linear combination of semi-invariants

cV1,0 and cV1,1 . We can see that given a specific γ ∈ Q+, we have that the ho-

mogeneous modules in Tγ1 are of dimension h1. If we consider the weight space,

〈〈h1,−〉〉 then we have that each semi-invariant is a linear combination of monomials

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of degree 1, cVγ,0 and cVγ,1 where Hom(T, V1,i) = Vγ,i. Since dim Hom(Vγ,j, Vγ,k) =

dim Hom(Hom(T, V1,j),Hom(T, V1,k)) = dim Hom(V1,j, V1,k).

Example 7.0.6. Continuing 6.1.4, we can observe the effect of the shrinking functors

on the simple modules of T1 for A = C(6).

So we need to define the left shrinking functor and the right shrinking functor for

C(6). Let the left shrinking functor be Σl = HomC(Tl,−) with Tl = T0 ⊕ P1 where

T0 := ⊕8i=2Piτ

−2. We can identify End(Tl) with C. Therefore we also have a linear

transformation σl corresponding to Σl which is σl = ETC t where t is the matrix whose

first column is dim P1 and whose ith column is dim(Piτ−2) for 2 ≤ i ≤ 8.

t =

1 0 0 0 0 0 0 0

1 0 1 0 1 1 0 0

1 1 0 1 0 1 0 0

1 1 1 1 2 2 1 0

1 1 1 2 1 2 1 0

2 2 2 3 3 4 2 1

1 1 1 2 2 3 1 0

1 1 1 1 1 2 1 0

So then α∞σl = α0 + α∞.

Similarly, we can define the right shrinking functor for C(6). Let

Tr = (Q1 ⊕Q2 ⊕Q4)τ ⊕Q3 ⊕Q5 ⊕Q6 ⊕Q7 ⊕Q8

Notice that (Q1⊕Q2⊕Q4)τ⊕Q3⊕Q5⊕Q6 is in the preinjective component of C(6). We

72

can identify End(Tr) with C(6) and the right shrinking functor is Σr = HomC(−, Tr).

We can again define σr = −ETC t where t is the matrix whose ith column is the

dimension of the summand of Tr corresponding to the vertex i of C(6).

t =

1 1 1 1 2 2 1 1

0 1 0 1 1 1 1 1

1 1 1 1 1 2 1 1

0 0 0 1 1 1 1 1

0 0 1 0 1 1 1 1

0 0 0 0 1 1 1 1

0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 1

.

Now again, take β = h1 + e(1)1 + e

(2)0 + 2e

(2)1 as in 6.1.4 and consider a shift to T2

that is h2 = α0 + 2α∞. Let Σr((E(j)i ) = F

(j)i and σr(e

(j)i ) = f

(j)i .

Σr(V1,t ⊕ E(1)[1,2] ⊕ E

(2)2 ⊕ E

(2)[0,2]) = V2,t ⊕ F (1)

[1,2] ⊕ F(2)2 ⊕ F (2)

[0,2]

σr(β) = (4, 7, 7, 9, 10, 11, 6, 2) = h2 + f(1)1 + f

(2)0 + 2f

(2)1

Illustrating that shrinking functors preserve generic decomposition.

Furthermore, we can illustrate the isomorphism of rings of semi-invariants. No-

tice since generic decomposition is preserved, for both β and σr(β) the admissible

sequences of index 0 are the same. In particular, we have the following labelled

polygons:

73

∆1 = 0

��

=======

1 1

∆2 = 1

��

=======

2 0

∆3 = 0

��

=======

0 0

.

If we indicate an admissible path in ∆j by [i, k](j), we have the admissible paths:

[0, 0](1), [1, 1](2), [0, 1](3), [1, 2](3), [2, 0](3) of index 0, [1, 2](1), [0, 0](2) of index 1 and [2](2)

of index 2.

In the space SIreg(Λ, β) we have the semi-invariants which come from the admis-

sible paths of index zero. Therefore, we have the following generators:

cE(1)[0,2] , cE

(2)[1,0] , cE

(3)0 , cE

(3)1 , cE

(3)2 .

We also have c0, c1 which are linear combinations of cVi ’s. By direct calculation

the relations are c0 = cE(1)[0,2] , c1 = cE

(2)[1,0] and c0 + c1 = cE

(3)0 cE

(3)1 cE

(3)2 .

Consider now, the action of the functor, Σr under which we have that Σr((E(j)i ) =

F(j)i and Σr(V1,t) = V2,t. Then clearly we have the following generators:

cF(1)[0,2] , cF

(2)[1,0] , cF

(3)0 , cF

(3)1 , cF

(3)2 .

We also have c′0, c′1 which are linear combinations of cV2,i ’s. Because all the relations

come from extensions, and extensions are preserved by Σr, the relations are c0 = cF(1)[0,2] ,

c1 = cF(2)[1,0] and c0 + c1 = cF

(3)0 cF

(3)1 cF

(3)2 .

74

A Tame concealed to tubular algebras

Given that a tubular algebra is a tubular extension of a tame concealed algebra, then

there are only the following possibilities:

A0 (where T = End(TA0 )) Tubular type of T Possible tubular type of AeA3 (2, 2) anyeA4 (3, 2) (3, 3, 3)

(4, 4, 2)

(6, 3, 2)eA5 (4, 2) (4, 4, 2)

(6, 3, 2)eA5 (3, 3) (3, 3, 3)

(4, 4, 2)

(6, 3, 2)eA6 (5, 2) (6, 3, 2)eA6 (4, 3) (6, 3, 2)eA7 (4, 4) (4, 4, 2)eA7 (5, 3) (6, 3, 2)eA7 (6, 2) (6, 3, 2)eA8 (6, 3) (6, 3, 2)eD4 (2, 2, 2) anyeD5 (3, 2, 2) (3, 3, 3)

(4, 4, 2)

(6, 3, 2)eD6 (4, 2, 2) (4, 4, 2)

(6, 3, 2)eD7 (5, 2, 2) (6, 3, 2)eD8 (6, 2, 2) (6, 3, 2)eE6 (3, 3, 2) (3, 3, 3)

(4, 4, 2)

(6, 3, 2)eE7 (4, 3, 2) (4, 4, 2)

(6, 3, 2)eE8 (5, 3, 2) (6, 3, 2)

75

References

[1] I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theoryof Associative Algebras, volume 1, LMS Students Texts, No. 65 ,CambridgeUniversity Press, 2006.

[2] G. Bobinski, A. Skowronski, Geometry of Periodic Modules over Tame Concealedand Tubular Algebras. Alg. and Rep. Theory, 5, (2002), 187-200.

[3] K. Bongartz, Minimal singularities for representations of Dynkin quivers, Com-ment. Math. Helv, 69 (1994),575-611.

[4] K. Bongartz, On Degenerations and Extensions of Finite Dimensional Modules,Adv. Math. 121 (1996), 245-287.

[5] S. Brenner, M. C. R. Butler, The equivalence of certain functors occurring in therepresentation theory of algebras and species, J. London Math. Soc., 14(1976)18387.

[6] C.R. Butler, C.M. Ringel, Auslander-Reiten sequences with few middle terms andapplications to string algebras, Comm. Algebra, 15(1987), 269-290.

[7] A. Skowronski, G. Bobinski, Geometry of Period Modules over Tame Concealedand Tubular Algebras, Alg. and Rep. Theory 5 (2002), 187-200.

[8] P. Caldero, F. Chapoton, Cluster algebras as Hall algebras of quiver representa-tions, math.RT/0410187.

[9] W. Crawley-Boevey, Exceptional sequences of Representations of Quivers, Cana-dian Math. Soc. Conf. Proceedings, Vol 14 (1993) 117-124.

[10] W. Crawley-Boevey, Subrepresentations of general representations of quivers,Bull. London Math. Soc. 28 (1996),363–366.

[11] W. Crawley-Boevey, J. Scher, Irreducible Components of Varieties of Modules,Journal fr die Reine und Angewandte Mathematik (Crelle) 553 (2002), 201-220.

[12] C. DeConcini, D. Eisenbud, C. Procesi, Young diagrams and determinantal va-rieties, Inv. Math. 56 (1980), 129–165.

[13] P. Donovan, M. R. Freislich, The representation theory of finite graphs andassociated algebras, Carleton Lecture Notes 5, Ottawa, 1973.

[14] V. Dlab, C.M. Ringel, Indecomposable Representations of Graphs and Algebras,Memoirs Amer. Math. Soc. 173 (1976).

76

[15] H. Derksen, J. Weyman, Generalized quiver representations for reductive groups,Colloquium Math., 94 151-173 (2002).

[16] H. Derksen, J. Weyman, Semi-invariants of quivers and saturation forLittlewood-Richardson coefficients, J. Amer. Math. Soc. 13 (2000), no. 3, 467–479.

[17] H. Derksen, J. Weyman, On the Littlewood-Richardson polynomials, J. Algebra,255, 247-257 (2002).

[18] H. Derksen, J. Weyman, On the canonical decomposition of quiver representa-tions, Compositio Math., 133, 245-265 (2002).

[19] H. Derksen, J. Weyman, On the σ-stable decompositions for quiver representa-tions, preprint, 2001.

[20] H. Derksen, J. Weyman, Rings of semi-invariants for quivers with relations, J.Algebra, 258, 216-227 (2002).

[21] H. Derksen, A. Schofield, J. Weyman, On the number of subrepresentations ofgeneric representations of quivers, math.AG/0507393.

[22] M. Domokos, A.N. Zubkov, Semi-Invariants of Quivers as Determinants, Trans.Groups, 6, (2001), 9-24.

[23] S. Donkin, Rational Representations of Algebraic Groups, Lecture Notes in Math-ematics 1140, Springer Verlag, 1985.

[24] V. Kac, Infinite root systems, representations of graphs and invariant theory,Invent. Math. 56 (1980), 57–92.

[25] V. Kac, Infinite root systems, representations of graphs and invariant theory II,J. Algebra 78 (1982), 141–162.

[26] M.M. Kleiner, Partially ordered sets of finite type, Zap. Nauchn. Sem. Leningrad.Otdel. Mat. Inst. Steklova(LOMI) 28 (1972) 3241.

[27] I. G. MacDonald, Symmetric functions and Hall polynomials, Second edition,Oxford Mathematical Monographs, Oxford Science Publications. The ClarendonPress, Oxford University Press, New York, 1995.

[28] L. A. Nazarova, Representations of quivers of infinite type, Izv. kad. Nauk SSSR,Ser. Mat. 37(1973),752-791.

[29] L.A. Nazarova, Partially ordered sets of infinite type, Izv. Akad. Nauk SSSR 39(1975) 963991 (in Russian).

[30] L.A. Nazarova, A.V. Roiter, Representations of partially ordered sets. In Inves-tigations on Representation Theory (ed.) Fadeev , J, Soviet Math. 23 (1975).

77

[31] L.A. Nazarova, A.G. Zavadskij, Partially ordered sets of tame type, in: Ma-trix Problems, Acad. Nauk Ukr.S.S.R., Inst. Mat., Kiev, 1977, pp. 122143 (inRussian).

[32] C. M. Ringel, The rational invariants of tame quivers, Invent. Math.58(1980),217-239.

[33] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes inMath. 1099, Springer 1984.

[34] M. Sato, T. Kimura, A classification of irreducible prehomogeneous vector spacesand their relative invariants, Nagoya J. Math 65 (1977), 1-155.

[35] A. Schofield, General Representations of quivers, Proc. London Math. Soc. (3)65 (1992), 46-64.

[36] A. Schofield, Semi-invariants of quivers, J. London Math. Soc. 43 (1991), 383-395.

[37] A. Schofield, M. Van den Bergh, Semi-invariants of quivers for arbitrary dimen-sion vectors, Indag. Math. (N.S.) 12 (2001), no. 1, 125–138.

[38] D. Simson, Linear Representations of Partially Ordered Sets andVector SpaceCategories, Gordon and Breach,Switzerland, Australia, 1992.

[39] D. Simson, A. Skowronski, Elements of the Representation Theory of AssociativeAlgebras, volume 2, LMS Students Texts, No. 71 ,Cambridge University Press,2007.

[40] A. Skowronski, J. Weyman, The algebras of semi-invariants of quivers, preprint,1999.

[41] A. Skowronski, J. Weyman, The algebras of semi-invariants of canonical algebras,preprint, 1999.

[42] J. Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts inMathematics, vol. 149, Cambridge University Press, 2003.

[43] A.G. Zavadskii, Differentiation algorithm and classification of representations,Math. USSR Izvyestiya, Vol. 39 (1992), No.2, 975-1012.

[44] A.G. Zavadskii, L.A. Nazarova, Representations of Partially ordered sets of finitegrowth. Kiev (1983).

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