Indecomposables Of Quiver Representations In The Of Abelian Groups

Ainsley Pullen Cl´ement Maria University of Queensland

1 1 Introduction Persistent Homology In the simplest sense, a quiver representation is the algebraic object obtained by identifying vector spaces and the maps between them with the nodes and arrows of a or multi-graph. These quiver representations have applications in persistent homology, which studies the life-span of features such as holes within evolving shapes. Figure 1 shows a shape evolving with the associated homology groups forming an An type quiver. This demonstrates how An type quivers are partic- ularly useful because they are easily used to represent changes over time. Figure 1 also shows the

Figure 1: Example from persistent homology barcode which tracks the lifespan of holes. As the shape evolves we can differentiate between the long-lived features, such as the red hole, which are the relevant features and the short lived features which tend to be noise in the data. This information is encoded into the quiver representations and can be applied through persistent homology in diverse areas such as computer vision and data analysis (Edelsbrunner and Harer, 2010)

Previous Results Gabriel’s Theorem is a fundamental result in quiver theory, which provides us with a classification of finite type quivers. Finite type quivers are those with only finitely many isomorphism classes of indecomposable representations. It was first proved by Gabriel in 1972 that finite type quivers correspond to quivers whose underlying graph is one of the Dynkin diagrams (see Figure 2).

Figure 2: Dynkin diagrams

2 An alternative approach to this result was provided by Bernˇste˘ın,Gel’fand, Ponomarev who presented a constructive method. Their proof relies on reflection which send an indecom- posable object to either another indecomposable object or to zero (Bernˇste˘ın,Gel’fand, Ponomarev, 1973). Crawley-Boevey proved the related property that any quiver representations consisting of only finite dimensional vector spaces can be decomposed into the direct sum of indecomposable representations (Crawley-Boevey, 2015).These indecomposable representations are the interval rep- resentations introduced in section 2.2. The proof proceeds by producing a descending chain condi- tion on the images and kernels. A more modern treatment is presented in forthcoming monograph Persistence Theory: From Quiver Representations to Data Analysis (Oudot, 2015). This paper follows the approach given there.

2 Background 2.1 Quiver Representations

Definition 2.1 A quiver consists of two sets Q0, Q1 and two maps h, t : Q1 → Q0

• Q0 is the set of vertices, which for us will be finite and indexed on N

• Q1 is the set of arrows, which we will also consider to be finite

The head map h assigns a head ha to every arrow a ∈ Q0 and the tail map, t, assigns a tail ta to every arrow a ∈ Q1

In particular we are interested in An type quivers which are linear shaped quivers of finite length (shown Figure 3). The direction of the arrows is arbitrary denoted by the headless arrows.

•1 •2 ... •n−1 •n

Figure 3: An type quiver

Definition 2.2 A representation of Q over a field k is a pair V = (Vi, va)

• Vi is a set of k-vector spaces {Vi|i ∈ Q0}

• va is the set of k-linear maps, {va : Vsa → Vta |a ∈ Q1}

Choice of the vector spaces and linear maps can be arbitrary and no composition law is enforced.

Definition 2.3 The direct sum of any two representations V, W is denoted V ⊕ W with

• Vi ⊕ Wi as the vector spaces for all i ∈ Q0   va 0 • va ⊕ wa = as the maps for all a ∈ Q1 0 wa

3 The direct sum is also known as the co-product. Additionally, there exists a zero object, called the trivial representation, in which all the vector spaces and maps are equal to 0. Any nontriv- ial representation that is isomorphic to the direct sum of two nontrivial representations is called decomposable. Otherwise it is an indecomposable representation.

Definition 2.4 A φ between two representations V, W of Q is the set of k-linear maps φ : Vi → Wi, such that the following diagram commutes for every a ∈ Q1 :

va Vsa / Vta

φsa φta   Ws / Wt a wa a

The morphism, φ, is called a monomorphism if every φi is injective, an epimorphism ∼ if every φi is surjective, and an isomorphism (denoted =) if every φi is bijective. There exists an identity morphism 1V : V → V defined by (1V)i = 1Vi for all i ∈ Q0. Furthermore, are composed pointwise by composing the linear maps at each vertex i independently. Consequently, representations of quivers over a field, k, form an abelian category, denoted Rep(Q, k).

Definition 2.5 A category, C is an algebraic object comprised of

• a class of objects, denoted Obj(C),

• a class of morphisms between objects, denoted HomC (X,Y ) for X,Y ∈ Obj(C) and

• a binary operation for the composition of morphisms, denoted HomC (X,Y ) × HomC (Y,Z) → HomC (X,Z) for X,Y,Z ∈ Obj(C) such that a identify morphism exists and associativity holds.

For a category to be additive it must satisfy three further properties (Gel’fand and Manin, 2003). The category must have a zero object (i.e. additive identity) and co-products. Furthermore HomC (X,Y ) must be an abelian group and the composition of morphisms must be bi-additive meaning it satisfies f(x + y, z) = f(x, z) + f(y, z) and f(x, y + z) = f(x, y) + f(x, z)

Furthermore for a category to be abelian it must satisfy four additional axioms (Gel’fand and Manin, 2003). Firstly every morphism must have a kernel and cokernel. A morphism f : X → Y has a kernel g : G → Y such that g ◦ f = 0 and a cokernel, Y/im(f), where both are defined pointwise. Additionally, every monomorphism is the kernel of its cokernel, every epimorphism is is the cokernel of its kernel and every morphism can be decomposed into the composition of an epimorphism and a monomorphism.

It is important to note that although quiver representations are comprised of objects and maps between them, Rep(Q, k) refers to the category where the objects are representations themselves and the morphisms are as described in Def 2.4.

4 Definition 2.6 A given representation V has

−−→ > • a dimension vector, dimV = (dim V1, ..., dim Vn) ; and −−→ Pn • a dimension, dim V = ||dimV|| = i=1 dim Vi

Defining dimension in this way produces a Remak decomposition so we have that V can be decomposed into the direct sum of finitely many indecomposables. We can then further show that this decomposition is unique up to isomorphism and re-ordering of elements in the sum. This is known as the Krull-Schmidt Property.

Theorem 2.7 (Krull-Remak-Schmidt) For any finite-dimensional V there are indecomposable ∼ representations V1,..., Vr such that V = V1 ⊕ ... ⊕ Vr. And for any indecomposable W1,..., Ws if ∼ V = W1 ⊕ ... ⊕ Ws then r = s and there exists a permutation σ such that Vi = Wσ(i) for 1 ≤ i ≤ r

We also have a quadratic form, qQ which associates a scalar with the dimension vector. For Dynkin n quivers qQ is positive definite i.e. qQ(x) > 0 for any nonzero vector x ∈ Z . Dynkin quivers are in fact the only positive definite quivers. A positive root of qQ is a vector x such that qQ(x) = 1. > For An type quivers the positive roots are vectors of the form (0,..., 0, 1,..., 1, 0,..., 0) with the first and last 1s occurring at some position b ≤ d ∈ [1, n]. In particular (0,..., 0, 1, 0,..., 0)> is a positive root. For a more in depth treatment of the quadratic form, including proofs of these properties, refer to (Oudot, 2015).

2.2 Indecomposables

For An type quivers these indecomposables are known as interval representations. They are an An type quiver which has a isomorphic to k at every node i ∈ [b, d] where b ≤ d ∈ [1, n]. These spaces are connected by identity isomorphisms. All other nodes have the trivial zero vector space which are mapped to or from with the zero map. Interval representations are denoted I[b, d] and an example can be seen in the following diagram The vector space is said to be born at node

0 0 kb ... kd 0

Figure 4: Interval Representation b and die and node d. This representation has the corresponding dimension vector of the form (0,..., 0, 1,..., 1, 0,..., 0)> where the 1s span the interval [b, d] which we saw earlier was a positive root of qQ.

2.3 Reflections + − Reflections, denoted si , si are applied to a node i in the quiver and act by reflecting the arrows incident with that node (see figure 5). The node must either be a source, alternatively - accessible, meaning all incident arrows are outgoing or a sink, alternatively + accessible meaning all incident arrows are incoming.

5 Q : •1 o •2 o •3 / •4 o •5 + s4 Q : •1 o •2 o •3 o •4 / •5

Figure 5: Example of a reflection

2.4 Reflection Functors In contrast to reflections, reflection functors, denoted R+, R−, are applied to the quiver represen- tation. This produces the following diagram when we reflect at node i, which is a sink for the purposes of this example.

V < i b

V1 ... Vi−1 Vi+1 ... Vn ]e 9A

∗ Vi

Figure 6: Reflection at node i

Before and after we reach the node i, the quiver stays the same. Reading the diamond along the top is the original quiver whilst along the bottom is the new quiver with reflected arrows and ∗ a new vector space, Vi . This vector space is precisely defined to have the properties we require (Oudot, 2015). For any quiver representation which admits such a decomposition into indecompos- ∼ Lr able interval representations, we have V = j=1 IQ[bj, dj]. Furthermore each reflection affects the interval representation based on the following set of rules:

0 if i = b = d ;  j j  IQ[i + 1, dj] if i = bj < dj;  + IQ[i, dj] if i + 1 = bj ≤ dj; Ri IQ[bj, dj] = [b , i − 1] if b < d = i; IQ j j j  IQ[bj, i] if bj < dj = i − 1;  IQ[bj, dj] otherwise These rules are easy to implement from an algorithmic approach, which is one reason that we prefer Bernˇste˘ınet al’s approach. This reflection either preserves the dimension or sends it to zero, which is captured in the following corollary.

−−→ + Corollary 2.8 For any sink or source, i, in a indecomposable representation V either qQ(dimRi V) = −−→ + 0 or qQ(dimRi V) = qQ(V)

6 2.5 Bernˇste˘ınet. al’s Proof of Gabriel’s Theorem Corollary 2.8 shows that reflecting either preserves the dimension or sends it to zero. Consequently, there is a series of reflections that reduces an An type quiver the trivial representation, with trivial vector spaces joined by zero maps. Without loss of generality assume that this is the minimum sequence, so R+ ... R+ 6= 0. Thus, R+ ... R+ ∼ where is a simple representation with is−1 i1 is−1 i1 = Sr Sr the space k at some arbitrary node r and the zero vector space otherwise. This is intuitive because you can understand a simple representation as ‘one step’ from triviality. Furthermore R+ ... R+ ij i1 is indecomposable for every j < s. Consequently we have −−→ −−→ −−→ −−→ q (dim ) = q (dimR+ ) = ... = q (dimR+ ... R+ ) = q (dim ) = 1 Q V Q i1 V Q is−1 i1 V Q Sr −−→ Due to properties of the quadratic form this implies that dimV = (0,..., 0, 1,..., 1, 0,..., 0)> with the first and last 1s occurring at some positions b ≤ d ∈ [1, n] (Bernˇste˘ın,Gel’fand, Pono- marev, 1973). So V is isomorphic to the interval representations I[b, d] introduced in section 2.2. We can further show there is in fact an isomorphism between the set of isomorphism classes of indecomposable representations and the set of interval representations (Bernˇste˘ın,Gel’fand, Pono- marev, 1973). First we show that different interval representations can not be isomorphic. This is due to different interval representations having different dimension vectors because span [b, d] is unique. Secondly, there exists an indirect proof that every interval representation is indecomposable.

This proof that An type quivers have only finitely many isomorphism classes of indecomposable representations extends to all Dynkin quivers. However this requires a bit more work because the connection to interval representations is not immediate. We can further prove, by enumeration, that Dynkin quivers are the only quivers which satisfy the property of having only finitely many isomorphism classes of indecomposables (Oudot, 2015).

3 Generalisation to Abelian Groups 3.1 Motivation Bernˇste˘ınet al’s reflection functors approach provides us with a constructive method for decompos- ing quiver representations. This is beneficial when using quiver representations in an algorithmic setting. Algorithmic approaches to quiver representations are particularly useful when applied to persistent homology. However, the homology groups we wish to model in persistent homology are often produced with coefficients from Z. This is because integer homology groups provide us with complete information. If we instead restrict ourselves to homology groups with coefficients from a field we may get inaccurate outputs. Consider a torus and a Klein bottle (see Figure 7).

(a) Torus (b) Klein bottle

Figure 7: Two distinct topological objects

7 ∼ The homology groups of both of these objects when computed over the field Z2 are H0(Z2) = Z2, ∼ ∼ H1(Z2) = Z2 ⊕ Z2 and H2(Z2) = Z2. These are relatively simple objects and so we would like to be able to differentiate them. When the homology groups are computed over Z then we have the ∼ ∼ ∼ ∼ torus identified as H0(Z) = Z, H1(Z) = Z ⊕ Z and H2(Z) = Z and the Klein bottle as H0(Z) = Z, ∼ ∼ H1(Z) = Z ⊕ Z2 and H2(Z) = 0. So, Z2 fails to account for the higher dimensional torsion of the Klein bottle but we can distinguish between these two objects with integer homology groups. Furthermore, the Universal Coefficient Theorem states that the integer homologies contains any other case as a by-product (Hatcher, 2002). However, as a corollary of this theorem homology groups with coefficients from Z form abelian groups of the following form: β H(F) =∼ F vector space ∼ β H(Z) = Z ⊕ Z/d1Z ⊕ ... ⊕ Z/dnZ abelian group | {z } torsion Whilst this would allow us to model ‘persisting torsion’ as in the Klein bottle case, quiver repre- sentations as we have considered them thus far require vector spaces. Therefore, it is of interest to investigate the properties of quiver representations over the category of abelian groups. In addition to the applications this would have for persistent homology, generalising quiver representations to different categories is also of interest to theoretical physicists who can use them to model Higgs Bundles (Gothen and King, 2005). Ideally we would be able to extend the results which I have outlined for quiver representations over vector spaces to the case of abelian group. Towards this aim we can prove that the category of quiver representations over abelian groups is abelian. This result shows that quiver representations over abelian groups have unique decompositions.

3.2 Proof The proof proceeds by two parts. First we prove if the category C is additive then the category of representation over C, denoted Rep(Q, C) or R, is also additive. In order to do this we show it satisfies the properties from Def 2.5. This is a useful result because in additive categories we have the intuitive notion of indecomposability as defined in Def 2.3. This is clear from the use of direct sums in corollary 3.3. However, most additive categories that we are interested in are abelian categories.

Secondly, we show that if C is abelian then so is Rep(Q, C). In addition to additivity we are required to show four additional axioms outlined in Def 2.5. Corollary 3.1 Rep(Q, C) has a zero object.

C is an additive category so it contains a zero object. We take this object as every element of Q1. It also contains a trivial morphism which we take as every element of Q0. This is the zero object of Rep(Q, C).

Corollary 3.2 If I, J, K ∈ Obj(R) then HomR(I, J) is an abelian group and HomR(I, J) × HomR(J, K) → HomR(I, K) is bi-additive.

Let I = (U, φ), J = (V, χ), K = (W, ψ) be objects of R = Rep(Q, C). As HomC (Ui,Vi) is an abelian group for all i ∈ Q0, HomR(I, J) is also an abelian group. It follows that HomR(I, J)× HomR(J, K) → HomR(I, K) is bi-additive because for each i ∈ Q0 the paring HomC (Ui,Vi)× HomC (Vi,Wi) → HomC (Ui,Wi) is bi-additive.

8 Corollary 3.3 Rep(Q, C) has coproducts.

We want to define the direct sum of (V, φ) and (W, ψ) ∈ Obj(R) . As C is additive, for all i ∈ Q0, Ui = Vi ⊕ Wi is an object of C with the maps li : Vi → Ui and hi : Wi → Ui. Consider the following diagram, where ca = lh(a) ◦ φa and da = hh(a) ◦ ψa.

φa Vt(a) / Vh(a)

ca lt(a) lh(a)  #  Ut(a) / Uh(a) O ; O da ht(a) hh(a)

ψa Wt(a) / Wh(a)

By property of co-products, there exists a unique morphism χa : Ut(a) → Uh(a) such that ca = χalt(a) and da = χaht(a). Thus lh(a)φa = χalt(a) and hh(a)φa = χaht(a). Let (U, χ) be the representation Ui = Vi ⊕ Wi for i ∈ Q0 and for each a ∈ Q1, χa : Ut(a) → Uh(a) as defined above. Then (U, χ) is the direct sum of (V, φ) and (W, ψ) with the morphisms

L = {li}i∈Q0 :(V, φ) → (U, χ) and H = {hi}i∈Q0 :(W, ψ) → (U, χ) as required.

Definition 3.4 A category, A, is additive if HomC (X,Y ) is an abelian group, the composition of morphisms is bi-additive and it has co-products and a zero object. By Corollary 3.1-3.3, we have proved that Rep(Q, C) is an additive category where C is an additive category.

We now proceed to the second part of the proof where we assume C is abelian. Corollary 3.5 Every morphism has kernel and cokernel.

We want to define ker f where f :(V, φ) → (W, ψ) is a morphism. As C is abelian, for each i ∈ Q0, 0 0 fi : Vi → Wi is a morphism then it has a kernel (Vi , µi) where µi : Vi → Vi such that fiµi = 0 for all i ∈ Q0. 0 0 0 For each a ∈ Q0 we have fh(a)(φaµt(a)) = 0 then there is an unique morphism φa : Vt(a) → Vh(a) 0 such that φaµt(a) = µh(a)φa. 0 0 If we set µ = {µi}i∈Q0 :(V , φ ) → (V, φ) then the following diagram commutes.

φ0 0 a 0 Vt(a) / Vh(a)

µt(a) µh(a)

 φa  Vt(a) / Vh(a)

ft(a) fh(a)

 ψa  Wt(a) / Wh(a)

Thus f ◦ µ = 0 and we can see that ker f = ((V 0, φ0), µ) is the kernel of f. Similarly one can show that f has a co-kernel.

9 Corollary 3.6 Every monomorphism is the kernel of its cokernel.

Let f :(V, φ) → (W, ψ) be a monomorphism. Then fi : Vi → Wi is a monomorphism for each 0 i ∈ Q0. As C is abelian if (V , ki) = coker fi we have (Vi, fi = ker ci, for all i ∈ Q0. As 0 0 0 ch(a)ψa : Wt(a) → Wh(a) is such that (ch(a)ψa) = 0 there is a unique ψa : Wt(a) → Wh(a) for all a ∈ Q1 such that the following diagram commutes.

φa Vt(a) / Vh(a)

ft(a) fh(a)

 ψa  Wt(a) / Wh(a)

ct(a) ch(a)

ψ0 0 a 0 Wt(a) / Wh(a)

0 0 Then ((W , ψ ), c) = coker f. So we have (Vi, fi) = ker ci and therefore ((V, φ), f) = ker c as required.

Corollary 3.7 Every epimorphism is the cokernel of its kernel.

Let f :(V, φ) → (W, ψ) be an epimorphism. Then fi : Vi → Wi is an epimorphism for each i ∈ Q0. 0 0 As C is abelian if (V , ki) = ker fi we have fiki = 0, for all i ∈ Q0. As φakt(a) : Vt(a) → Vh(a) is such 0 0 0 that fh(a)(φakt(a)) = 0 there is a unique φa : Vt(a) → Vh(a) for all a ∈ Q1 such that the following diagram commutes. φ0 0 a 0 Vt(a) / Vh(a)

kt(a) kh(a)

 φa  Vt(a) / Vh(a)

ft(a) fh(a)

 ψa  Wt(a) / Wh(a)

0 0 Then ((V , φ ), k) = ker f. So we have (Wi, fi) = coker ki and therefore ((W, ψ), f) = coker k as required.

Corollary 3.8 Every morphism can be written as composition of an epimorphism and a monomor- phism.

Let f :(V, φ) → (W, ψ) be a morphism. Since C is abelian we know that for each i ∈ Q0 0 0 fi : Vi → Wi can be written as fi = higi, where gi : Vi → Wi is an epimorphism and hi : Wi → Wi 0 is a monomorphism with Wi ∈ Obj(C).

10 Because of the commutative properties of morphisms, given in Def. 2.3. then for each a ∈ Q1 0 0 0 there is a unique ψa : Wt(a) → Wh(a) such that the following diagram commutes.

φa Vt(a) / Vh(a)

gt(a) gh(a)

ψ0 0 a 0 Wt(a) / Wh(a)

ht(a) hh(a)

 ψa  Wt(a) / Wh(a)

0 0 0 0 That is ψagt(a) = gh(a)φa and ψaht(a) = hh(a)ψa. Then if we take the representation (W , φ ) 0 0 and the morphisms h = {hi}i∈Q0 :(W , ψ ) → (W, ψ) and g = {gi}i∈Q0 :(V, φ) → (W, ψ) it follows that g is an epimorphism and h is a monomorphism and f = h ◦ g.

Definition 3.9 A category, A, is abelian if it is additive and every morphism has a kernel and cokernel, every monomorphism is the kernel of its cokernel, every epimorphism is is the cokernel of its kernel and every morphism can be decomposed into the composition of an epimorphism and a monomorphism.

By Corollary 3.5-3.8 , we have proved Rep(Q, C) is an abelian category where C is an abelian category.

Lemma 3.10 Rep(Q, Ab) is an abelian category, where Ab is the category of abelian groups.

4 Conclusion

We have shown that the category of quiver representations over abelian groups is abelian. This means that quiver representations over abelian groups admit unique decomposition as in the Krull- Remak-Schmidt theorem (Theorem 2.7). Carlsson and Zomordian show that it possible to compute individual homology groups over principle ideal domains. This means we have part of the persistent homology information in the integer case that we are motivated by (Carlsson and Zomordian, 2015). However, it is still difficult to utilise this information in a computational setting without further details about how these groups are connected in their decomposition. Thus, it is of interest to characterise what these indecomposables look like as we were able to do with interval representations (Section 2.2). However, this requires further work and early investigations indicate that this is a very difficult problem, even for small finitely generated abelian groups.

5 Acknowledgments

I would like to express my sincere gratitude to my supervisor, Cl´ement Maria and the Australian Mathematical Sciences Institute for providing the funding for this work.

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