Centra of Quiver Algebras

Centra of Quiver Algebras

Elin Gawell

Centra of Quiver Algebras

Elin Gawell

©Elin Gawell, Stockholm 2014 ISBN 978-91-7447-960-7

Printed in Sweden by US-AB, Stockholm 2014

Distributor: Department of Mathematics, Stockholm University

Abstract

A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by mono- mials and (anti-)commutativity relations. We give a combinatorial descrip- tion of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiver algebra and state necessary and suffi- cient conditions for the center to be finitely genterated as a K -algebra. Ex- amples are provided of partly (anti-)commutative quiver algebras that are Koszul algebras. Necessary and sufficient conditions for finite generation of the Hochschild cohomology ring modulo nilpotent elements for a partly (anti-)commutative Koszul quiver algebra are given.

Sammanfattning

En delvis (anti-)kommutativ kogeralgebra är en kogeralgebra kvotad med ett (anti-)kommutativitetsideal, dvs ett kvadratiskt ideal genererat av monom och (anti-)kommutativitetsrelationer. Vi ger en kombinatorisk beskrivning av dessa ideal och deras associerade generatorgrafer med hjälp av vilken man snabbt kan avgöra om idealet är admissibelt eller inte. Vi beskriver centrum av en delvis (anti-)kommutativ kogeralgebra och formulerar nöd- vändiga och tillräckliga villkor för att centrum ska vara ändligt genererat som en K -algebra. Exempel ges på delvis (anti-)kommutativa kogeralgebror som är Koszulalgebror. Tillräckliga och nödvändiga villkor ges för ändlig generering av Hochschild-kohomologiringen modulo nilpotenta element för delvis (anti-)kommutativa Koszulkogeralgebror.

Contents

Abstract v

Sammanfattning vii

Acknowledgements xi

1 Introduction 13

1.1 Centra of quiver algebras and finite generation of the Hochschild cohomology ring . . . 13 1.2 Applications in the theory of support varieties . . . 16

2 Preliminaries 17

2.1 Quivers and path algebras . . . 17 3 Commutativity ideals and centra of quiver algebras 21 3.1 Admissible commutativity and anti-commutativity ideals . . . 21 3.2 Centra of partly (anti-)commutative quiver algebras . . . 32 3.3 The ring structure of the center . . . 37 3.4 Finitely generated centra . . . 42

4 Projective resolutions and Koszul algebras 45

4.1 Projective resolutions of partly commutative and partly anti- commutative quiver algebras . . . 45 4.2 Introduction to Koszul quiver algebras . . . 53 4.3 Koszulity of some partly (anti-)commutative quiver algebras . 54 5 The graded center and finite generation of the Hochschild coho-

mology ring 57

5.1 Graded centra of partly (anti-)commutative quiver algebras . . 57 5.2 Introduction to the Hochschild cohomology ring . . . 59 5.3 Finite generation of the Hochschild cohomology ring . . . 61

Acknowledgements

First, I would like to thank my advisor Christian Gottlieb for providing help- ful comments, finding unclarities and for lots of support and encourage- ment. My gratitude goes to my second advisor Qimh Xantcha, Uppsala Uni- versity, for generously sharing his ideas, teaching me how to do mathemati- cal research and supporting me in every way.

I want to acknowledge Sarah Witherspoon, Texas A&M, for giving me the article with the example that was the starting point for this work and intro- ducing me to the topic of support varieties of finite-dimensional algebras.

I also want to thank Wojciech Chachólski, KTH, for introducing me to the facinating topic of quiver algebras and for encouragement.

Helpful comments were provided by my opponent, Jan Snellman, Lin- köping University.

I’m grateful to the late Torsten Ekedahl for accepting me as his PhD- student and even though this thesis turned out to concern an area of math- ematics quite different from the mathematical research I started with I’m sure that I gained a lot of mathematical insights from working with him.

Without the support and encouragement from my family, especially my husband Niklas, and all my friends this would not have been possible. I’m especially thankful towards my friends Scarlett Szpryngiel, Elin Ottergren and Linda Joelsson who encouraged me to finish this project, even when I felt like I wanted to burn all my notes and smash my computer.

And last, but not least, I would like to thank all my colleagues and former colleagues at the departments of mathematics at SU and KTH.

1. Introduction

1.1 Centra of quiver algebras and finite generation of the Hochschild cohomology ring

In this thesis we study two types of finite-dimensional quiver algebras, quiver algebras bound by a commutativity ideal and quiver algebras bound by an anti-commutativity ideal. A commutativity ideal is a quadratic ideal gener- ated by monomials and relations of the form ab − ba, analogously an anti- commutativity ideal is generated by monomials and relations of the form ab+ba. We call such algebras partly (anti-)commutative algebras. The start- ing point for our interest in such algebras is the following example:

Example 1.1.1. [Sna08] Let K be a field and letΛ = KQ/I where Q is the quiver

◦

b

DD

a

 c //◦

and I = 〈a², b², ab − ba, ac〉. Then the Hochschild cohomology ring modulo nilpotent elements

HH^∗(Λ)/N ^∼₌

½ K ⊕ K [a,b]b if charK = 2 K ⊕ K [a², b²]b² if charK 6= 2

This very simple quiver algebra provided a counterexample to a conjec- ture made by Snashall and Solberg in [SS04], that the Hochschild cohomol- ogy ring modulo nilpotent elements, HH^∗(Λ)/N, of a finite-dimensional K -algebraΛ is always finitely generated as a K -algebra. The finite gener- ation of HH^∗(Λ)/N is true for quiver algebras bound by monomial ideals (see [GSS04]). Quiver algebras bound by commutativity ideals are therefore a logical next step to investigate.

The algebra in Example 1.1.1 is a partly commutative quiver algebra.

Readers unfamiliar with quiver algebras can find all the basic defintions and results needed to understand this thesis in Chapter 2.

In Chapter 3 we introduce commutativity ideals and anti-commutativity ideals; quadratic ideals generated by only monomials and binomial com- mutativity or anti-commutativity relations. We describe these ideals and in- troduce the associated generator graphs, a combinatorial description used to determine if the ideal is admissible or not, i.e. if the quiver algebra bound by the ideal is finite-dimensional or not.

The main result in Section 3.1 is Theorem 3.1.25, which gives a short and easy way to determine if the ideal is admissible.

Theorem. A commutativity (or anti-commutativity) ideal, I , is admissible if and only if the generator graph corresponding to the orthogonal ideal I^⊥, does not contain any directed cycle.

The Hochschild cohomology ring modulo nilpotence in Example 1.1.1 has a connection to the center of the algebra since for Koszul quiver algebras we have that HH^∗(Λ)/N ^∼_{= Zgr}(Λ^!) (see Theorem 5.2.3 proven in [BGSS08]).

This is the reason for our investigation of the centra of partly (anti-)commuta- tive quiver algebras. The centra of such algebras is explicitly given in Theo- rem 3.2.7 and Theorem 3.2.9.

Theorem. Let I be a square-free commutativity ideal. The positively graded part of the center of K Q/I has a basis given by all non-zero products a1a2. . . ak, of loops with the same basepoint, such that

1. All ai commute non-trivially modulo I : aiaj= ajai6= 0 for all i and j . 2. For all arrows b in the quiver, one of the following two options holds:

• b commutes with all ai.

• There exist i and j such that a_ib = 0 = baj.

Theorem. Let I be a square-free anti-commutativity ideal. The positively graded part of the center of K Q/I has a basis given by all non-zero products a₁a₂. . . a_k such that

1. If k is even:

• The monomial contains an even number of each arrow, ai.

• For all arrows a_i, a_jin the monomial a_ia_j= −aja_i6= 0.

• For all arrows b in the quiver, one of the following two options holds:

– b anti-commutes non-trivially with all ai, i.e. bai= −aib 6=

0.

– There exist i and j such that aib = 0 = baj. 2. If k is odd:

• The monomial contains an odd number of each arrow, ai.

• For all arrows ai, ajin the monomial aiaj= −ajai6= 0.

• For all other arrows b, there exist i and j such that aib = 0 = baj. The main results in Section 3.4 are necessary and sufficient conditions for the center to be finitely generated as a K -algebra, see Theorem 3.4.4 and Theorem 3.4.8. Here Qxdenotes the subquiver of Q consisting of the vertex x and all arrows that either starts or ends in x and I_xthe intersection of the ideal I and the algebra K Qx.

Theorem. Suppose I is an (anti-)commutativity ideal such that I^⊥is admis- sible. Then Z (K Q/I ) is finitely generated as a K -algebra if and only if for all x in Q0either Z (K Qx/Ix) is trivial or

Ix⊇ 〈ab ± ba, ca, ad〉 a, b loops with basepoint x,

c, d arrows such that o(c) 6= t(c) = x,t(d) 6= o(d) = x

.

The connection between the Hochschild cohomology ring and the cen- ter requires that the algebra is a Koszul algebra, i.e. that it has a linear, mini- mal, projective resolution. In Chapter 4 we introduce the relation graph and use it to construct linear, projective resolutions for some special types of al- gebras. We also give examples of finite-dimensional partly (anti-)commuta- tive quiver algebras that are Koszul algebras.

In Chapter 5 we state a result for finite generation of the Hochschild co- homology ring for a partly (anti-)commutative Koszul quiver algebra. For Koszul quiver algebras, K Q/I , the Koszul dual is given by K Q^op/I_o^⊥(see The- orem 4.2.6 proven in [GMV98]) and the Hochschild cohomology ring mod- ulo nilpotent elements has a particularly nice structure, since HH^∗(Λ)/N ^∼₌ Zgr(Λ^!). We first investigate the graded center of K Q^op/I_o^⊥and then find nec- essary and sufficient conditions for finite generation of the Hochschild co- homology ring modulo nilpotence. The main theorems in the chapter are Theorem 5.3.2 and Theorem 5.3.3.

Theorem. Let I be an admissible (anti-)commutativity ideal andΛ = KQ/I a Koszul algebra. Then the Hochschild cohomology ring modulo nilpotence, HH^∗(Λ)/N, is finitely generated if and only if for all x ∈ Q0either Z ((Λ^!)x) is trivial or for any pair of loops a, b with basepoint x we have that ab −ba ∈ Ix

(or ab + ba ∈ Ix) and the only monomial in Ixcontaining a loop a is a².

1.2 Applications in the theory of support varieties

The theory of support varieties of finitely generated modules over a finite- dimensional K -algebraΛ using Hochschild cohomology was introduced by Snashall and Solberg in [SS04]. One essential property needed to apply their theory is that the Hochschild cohomology ring modulo nilpotent elements, HH^∗(Λ)/N, is finitely generated as a K -algebra. This was known to be true for:

• Finite-dimensional selfinjective algebras of finite representation type over an algebraically closed field [GSS03]

• Finite-dimensional monomial algebras [GSS04].

• Finite-dimensional algebras of finite global dimension [Hap89].

• Any block of a group ring of a finite group [Eve61].

• Any block of a finite dimensional cocommutative Hopf algebra [FS97].

• Λ = KQ/I is a finite-dimensional Nakayama algebra over a field K bound by an admissible ideal I generated by a single relation [SS04].

Snashall and Solberg conjectured that HH^∗(Λ)/N is always finitely gener- ated as a K -algebra whenΛ is a finite dimensional algebra over a field K . Xu found a counterexample to this conjecture when the field K has character- istic 2 [Xu08], which was later generalized to all characteristics by Snashall [Sna08] (see example 1.1.1). Several people have been working on finding the necessary and sufficient conditions to make HH^∗(Λ)/N a finitely gener- ated algebra over HH^∗(Λ), for example Parker and Snashall [PS11] and from this work also more classes of counterexamples have been found [XZ11].

We provide examples of finite-dimensional algebras where HH^∗(Λ)/N is finitely generated and also algebras where it is not.

2. Preliminaries

2.1 Quivers and path algebras

This section is aimed at those who are not familiar with quivers and path algebras. It hopefully contains all the basic defintions one needs to be able to understand Chapter 3 and provides a ground for the whole thesis. In this thesis K is always assumed to be an algebraically closed field.

Definition 2.1.1. A quiver Q is a directed graph. It consists of two sets: Q₀ (called vertices or points) and Q1(called arrows), and two mapso_,t_{: Q1→ Q0} which associates each arrow a ∈ Q1with its origino_{(a) ∈ Q0}and its target t(a).

A priori there are no restrictions at all on a quiver Q, it can be infinite, contain multiple edges, loops, be disconnected and/or look in any other way.

Example 2.1.2.

::◦ ((

hh ◦

◦_dd

$$ 

DD

◦ ^//◦ ^//◦ ^//◦

A quiver is finite if Q0and Q1are finite sets. In the rest of this thesis the quiver Q is always assumed to be finite. The underlying graph Q is obtained by forgetting the orientation of the arrows in Q. A quiver is connected if its underlying graph is a connected graph.

Definition 2.1.3. A path of length n ≥ 1 in a quiver Q is a sequence a1a2. . . an

where ai∈ Q1andt(a_{i −1}) =o(ai). The length of a path p = a1a2. . . anequals the number of arrows in the path, i.e.l(p) = n. Each x ∈ Q0is a path of length 0. We let Qldenote the set of all paths of length l in Q. We defineo_{(p) =}o(a1) andt_{(p) =}t(an).

A path p is called a cycle ifo_{(p) =}t(p). A loop is a cycle of length 1, i.e.

an arrow a such thato_{(a) =}t(a). The origin and target,o_{(a) =}t(a), of a loop are sometimes called the basepoint of the loop a. We say that a quiver is acyclic if it does not contain any cycles. If p = a1a2. . . an is a path, we say that aia_{i +1}. . . a_k is a subpath of p if 1 ≤ i ≤ k ≤ n.

Definition 2.1.4. The path algebra K Q of a quiver Q is the K -algebra whose underlying vector space has a basis consisting of all paths of length ≥ 0 and multiplication defined by

a1a2. . . an∗ b1b2. . . bk= δt(an)o(b₁)a1a2. . . anb1b2. . . bk,

whereδt(an)o(b1) denotes the Kronecker delta. Hence the product of two paths, p1∗ p2= 0 ift(p1) 6=o(p2) and p1∗ p2= p1p2ift(p1) =o(p2). If xi

and x_jare two paths of length 0 then x_i∗ xj= δxixjx_i.

The path algebra is an associative algebra, in general non-commutative.

The elements in Q0 are pairwise orthogonal idempotents. If Q0 is finite, then K Q has an identity element,P

x∈Q0x. The path algebra K Q is finite- dimensional if and only if Q is finite and acyclic. If the quiver Q is not con- nected, then the path algebra splits into a direct sum K Q =Ln

i =1K Qⁱ, where Qⁱare the connected components of Q. Hence we can without loss of gen- erality assume that Q is always connected.

Lemma 2.1.5. K Q is a graded algebra.

Proof. By the property of the basis elements we have a direct sum decom- position of K Q = KQ0⊕KQ1⊕KQ2⊕· · ·⊕KQl⊕. . . of the K -vector space KQ, where K Q_l is the subspace of K Q generated by Q_l, the set of paths of length l . That (K Q_m) ·(KQn) ⊆ KQ_m+nfor all m, n ≥ 0 is easy to see, since the prod- uct of a path of length m and a path of length n is either 0 or a path of length m + n.

Definition 2.1.6. The two-sided ideal of the path algebra K Q generated by the arrows of Q is called the arrow ideal and denotedJ.

There is an obvious direct sum decomposition ofJ_{= KQ1⊕ KQ2⊕ · · · ⊕} K Q_l⊕ . . . ofJas a K -vector space. HenceJ^l₌^L_m≥lK Q_mfor each l ≥ 1.

Definition 2.1.7. A two-sided ideal I of K Q is said to be admissible if there exists m ≥ 2 such that

J^m_{⊆ I ⊆}J²

If I is an admissible ideal of K Q, then the quotient algebra K Q/I is called a bound quiver algebra.

For any admissible ideal, I , the algebra K Q/I is finite dimensional and associative. We can also note that the Jacobson radical of K Q/I is the arrow ideal modulo the ideal,J/I . This is seen by realizing that the only maximal right ideals (which are the same as the left maximal ideals) are of the form m_{i= K x1⊕ K x2}⊕ · · · ⊕ K xi −1⊕ K xi +1⊕ K xi +2⊕ · · · ⊕ K xn⊕L

j ≥1K Qj, where Q₀= {x1, . . . , xn}. The intersection of these ideals is clearlyJ/I . The algebra (K Q/I )/rad(K Q/I ) is isomorphic to K Q/J. In the rest of this thesis we will denoteJ/I byr.

Example 2.1.8. (i) Consider the following quiver

◦^{zz a2}

a1 $$

The ideal 〈a³₁, a²₂, a1a2− a2a1〉 is clearly admissible, but if we remove any of the three generators, the ideal fails to be admissible.

(ii) For any finite quiver Q and any m ≥ 2 the idealJ^mis of course admis- sible.

(iii) If Q is finite and acyclic, thenJ^m= 0 for m big enough, and hence the zero ideal is admissible in this case.

(iv) Let Q be the quiver below

◦ ^{a2 //}◦

a3



◦

a1

__???????

a4

DD

The ideal 〈a1a2a3− a4〉 is not admissible, since it is not contained in J². The ideal 〈a1a₂〉 is not admissible, since it does not contain a₄ⁿfor any n, and hence there exist no m ≥ 2 such thatJ^m_{⊆ 〈a1}a2〉. The ideal

〈a1a2, a₄²〉 is an example of an admissible ideal of KQ.

Definition 2.1.9. A relation, ρ, in Q with coefficients in K is a K -linear combination of paths, pi, of length at least 2 with the same origin and tar- get. Thus, ρ = P^m_{i =1}λip_i such thato_{(pi) =}o_{(pj) and}t_{(pi) =}t_{(pj) for all} 0 ≤ i , j ≤ m, λi∈ K (not all zero).

If m = 1 the relation is called a monomial relation.

Definition 2.1.10. Let xK Q y be the K -vector space consisting of all paths starting in x ∈ Q0and ending in y ∈ Q0. Two relationsρ1andρ2are said to be unrelated ifρ1∈ xKQ y and ρ2∈ wKQz where either x 6= w or y 6= z (or both).

This gives a decomposition of K Q as a vector space, since K Q = M

x,y∈Q0

xK Q y.

Hence any element in K Q can be written as a unique sum of unrelated rela- tions.

Lemma 2.1.11. If I is an admissible ideal then I is finitely generated as a K Q-module.

Proof. Let m ≥ 2 be an integer such thatJ^m⊆ I . We then have a short exact sequence 0 →J^m_{→ I → I /}J^m → 0 of KQ-modules. Hence, ifJ^mand I /J^m are finitely generated, then so is I . We have thatJ^mis generated by all paths of length m, and since Q is finite so is the set of paths of length m. Thus I /J^m is an ideal of a finite-dimensional algebra K Q/J^m and hence it’s a finite- dimensional K -vector space and a finitely generated K Q-module.

Lemma 2.1.12. Letσ ∈ KQ, with σ = ρ1+ · · · + ρn, unrelated relationsρi. We then have 〈σ〉 = 〈ρ1, . . . ,ρn〉.

Proof. Ifσ is not a relation (and hence does not lie in one of the xKQ y), then xσy is either 0 or a relation ρi. Sinceσ = Px,y∈Q0xσy, the non-zero ele- ments in the set {xσy | x, y ∈ Q0} form a finite set of relations, {ρ1,ρ2, . . . ,ρn}, that generates 〈σ〉.

It follows from the lemma above that for any ideal I generated in degree 2 or higher there exists a finite set of relations such that I = 〈ρ1, . . . ,ρn〉.

Corollary 2.1.13. Let pi, pj∈ KQ. If pi, pj∉ I but pi− pj∈ I then pi− pjis a relation.

Proof. If pi− pj is not a relation, then pi, pj are unrelated and, by Lemma 2.1.12, we have that 〈pi, p_j〉 = 〈pi− pj〉 and hence pi, p_j∈ I .

It follows from Corollary 2.1.13 that for any non-zero monomial p ∈ KQ/I origin of p,o(p), and target of p,t(p), are well-defined.

Corollary 2.1.14. Let ai, aj∈ Q1. Assume aiaj∉ I . If aiaj− ajai∈ I , then ai

and ajare loops.

Proof. Since aiaj− ajaiis a relation we have thato(ai) =o(aj) andt(ai) = t_{(aj). Since ai}a_j 6= 0 we have that o_{(ai) =}t_(aj) and hence ai and aj are loops.

When I is a homogeneous ideal, then K Q/I is a graded algebra.

3. Commutativity ideals and centra of quiver algebras

3.1 Admissible commutativity and anti-commutativity ideals

We always assume that Q is a finite, connected quiver and K is an alge- braically closed field. If nothing else is written ai denote an arrow. When we use the terms monomial we always refer to a path with coefficient 1. A binomial is a sum or difference of two monomials.

Definition 3.1.1. A commutativity ideal is an ideal generated by quadratic monomials aiajand relations of the form akal−alak. An anti-commutativity ideal is an ideal generated by quadratic monomials a_ia_jand relations of the form akal+ alak.

If charK = 2 we have that every anti-commutativity ideal is also a com- mutativity ideal. When working in characteristic 2 one may consider all such ideals to be commutativity ideals.

Definition 3.1.2. A minimal generating set of an (anti-)commutativity ideal is a set I2= {aia_j, aka_l−ala_k}i , j ,k,l(or I2= {aia_j, aka_l+ala_k}i , j ,k,l) such that akal− alak∈ I2(or akal+ alak∈ I2) implies that akal∉ I2and alak∉ I2.

The generators of the form aiajare called the monomial generators and the generators of the form akal− alak and akal+ alak are called the bino- mial generators.

Any x ∈ I can be written as x =X

i , j

pi jaiajqi j+X

k,l

p_kl(a_ka_l− ala_k)q_kl

where pi j, qi j, pkland qklare paths in Q.

Definition 3.1.3. Assume aiaj ∉ I . If aiaj− ajai ∈ I2 or aiaj+ ajai ∈ I2

we say that the transposition (aia_j) is an allowed transposition. Two paths, p, q, in K Q are equivalent p ∼ q, if p can be obtained from q by allowed transpositions.

It is easy to see that ∼ is an equivalence relation.

Lemma 3.1.4. Let I be a commutativity ideal with minimal generating set

〈aiaj, akal−alak〉i , j ,k,l. Any monomial m ∈ I is equivalent to a monomial of the form paiajq, where aiaj∈ I and p, q paths. Conversely, all monomials of these types lie in I .

Proof. Assume m ∈ I . Then m =X

i , j

pi jaiajqi j+X

k,l

p_kl(a_ka_l− ala_k)q_kl

for some paths pi j, qi j, pkl and qkl. Since m is a monomial either m = pa_ia_jq, where p, q paths, or we have cancellations in the expression. As- sume that m 6= paiajq for each aiaj∈ I , then m = m1− (m1− m2) − (m2− m3) − (m3− m4) − ...(m_{i −1}− mi), where mi= m and mj− mj +1= p(akal− a_la_k)q for some p, q ∈ KQ and aka_l− ala_k ∈ I2. By definition m_j ∼ m_{j +1} and hence m1∼ m2∼ m3∼ · · · ∼ mi= m. By assumption m1is of the form paiajq, and hence m is equivalent to a monomial of the form paiajq.

If m ∼ paiajq and aiaj∈ I , then it is obvious that m ∈ I .

Lemma 3.1.5. Let I be an anti-commutativity ideal with minimal generating set 〈aiaj, akal+alak〉i , j ,k,l. Any monomial m ∈ I is equivalent to a monomial of the form paiajq, where aiaj∈ I and p, q paths. Conversely, all monomials of these types lie in I .

Proof. Analogous to the proof of Lemma 3.1.4.

Corollary 3.1.6. If a₁²a2a3a4. . . an∈ I and a1a2. . . an∉ I then a²₁∈ I . Proof. This follows immediately from 3.1.4 and 3.1.5.

Lemma 3.1.7. Let I be a commutativity ideal. Let b and c be monomials.

Any binomial in I is either the sum of two monomials in I or of the form b −c where b ∼ c.

Proof. Assume b − c ∈ I with b,c ∉ I . Then b − c =P

k,lpkl(akal− alak)qkl, where a_ka_l− ala_k are the binomial generators of I . We have that b and c are monomials, and hence b − c = m1− m2+ m2− m3+ . . . m_{i −1}− mi, where b = m1, c = mi and each mj− mj +1= p(akal− alak)q for some generator (a_ka_l− ala_k) and some paths p, q. By definition mj ∼ mj +1for all j , and hence b ∼ c.

If b ∼ c then it is obvious that b − c ∈ I .

Corollary 3.1.8. Let I be a commutativity ideal and let p and q be non-zero paths in K Q not contained in I . Assume that ap ∉ I and aq ∉ I . If ap−aq ∈ I , then p − q ∈ I .

Proof. Assume ap − aq ∈ I , then ap ∼ aq. This implies p ∼ q, and hence p − q ∈ I by Lemma 3.1.7.

If I is an anti-commutativity ideal we get a slight modification of the result in Lemma 3.1.7. A binomial is a sum or difference of two paths, i.e. if b and c are monomials, then b + c and b − c are binomials.

Lemma 3.1.9. Let I be an anti-commutativity ideal. Then any binomial in I is either

(i) a sum of two monomials in I or

(ii) of the form m₁+mnwhere m₁∼ mnare monomials and the number of transpositions used to transform m1to mnis odd or

(iii) of the form m1−mnwhere m1∼ mnare monomials and the number of transpositions used to transform m₁to m_nis even.

Proof. SupposeP

k,lpkl(akal+ alak)qklis a binomial. ThenP

k,lpkl(akal+ a_la_k)q_kl= m1+ m2− (m2+ m3) + m3+ · · · + (−1)ⁿ(m_n−1+ mn), where m_j∼ m_{j +1}for all j and any pair mj+ m_{j +1}correspond to exactly 1 transposition.

HenceP

k,lpkl(akal+ alak)qkl= m1+(−1)ⁿmn, where mnis obtained from m₁by n − 1 transpositions.

Definition 3.1.10. Let I be an (anti-)commutativity ideal. The algebra K Q/I is then said to be a partly (anti-)commutative algebra.

Definition 3.1.11. To every commutativity or anti-commutativity ideal I we associate a directed graphΓI, called the generator graph, in the following way:

• To every arrow a ∈ Q1there is a vertex which we also call a.

• For every monomial ab ∈ I2we associate a directed edge from a to b (the monomial a²will give rise to a loop in the graph).

• For every (anti-)commutativity relation ab −ba ∈ I2or ab +ba ∈ I2we associate an undirected edge between a and b.

We will abuse the notation and let ab denote not only the path of length 2 in Q and the corresponding monomial ab ∈ KQ/I , but also the edge be- tween a and b inΓI. If a clarification is needed we will denote the edge ab with either an arrow (if it is directed) a → b or a — b (if it’s undirected).

If there exists a sequence of directed edges a1 → a2→ · · · → an → a1

from a1to a1we say that the generator graph contains a directed cycle.

Example 3.1.12. (i) Let I = 〈a₁², a²₂, a²₃, a1a2− a2a1, a1a3, a3a1〉. The gen- erator graphΓIfor this ideal is

a1

((

 a2 ss

a3

hh EE

(ii) Let I = 〈a1a₂〉. Assume that the quiver also contains an arrow a3that isn’t a part of any generator of I (i.e. appears in the generator graph as an isolated vertex). The generator graphΓIfor this ideal is

a1 //a2

a₃

(iii) Let I = 〈a₁², a₂², a₃², a₁a₂− a2a₁, a₂a₃− a3a₂, a₁a₃, a₃a₁, a₂a₄〉. The gen- erator graphΓIis

a1



a2ss

||||||||

a3

VVEE a4

The generator graph does not contain all the information about the un- derlying quiver and hence not all the information needed to determine K Q/I . The following example points out how the same generator graph can be as- sociated to several different quiver algebras.

Example 3.1.13. Let I = 〈a1a2, a2a3〉. Then the generator graph ΓIis simply a1→ a2→ a3.

This ideal can be an ideal in several different quiver algebras, for example any of the quivers depicted below.

◦

a2

DD

a₁

 a3 //◦

◦

a3 $$ ^a1 ((

◦

a₂

hh

◦ ^{a1 //}◦ ^{a2 //}◦ ^{a3 //}◦

In some cases we can determine facts about the underlying quiver from the generator graph of the ideal I , for example if a1a2is an undirected edge we know that a1and a2are loops at the same vertex.

We will use the generator graph to detemine when a commutativity or anti-commutativity ideal is admissible. But for this we need to define the orthogonal ideal. K Q2can be viewed as a vector space spanned by all paths in Q of length 2. This vector space is equipped with a scalar product 〈x, y〉 =

〈(x1, x2, x3, . . . , xn), (y1, y2, . . . , yn)〉 =Pn i =1xiyi.

Definition 3.1.14. Let I be a quadratic ideal with minimal generating set I2. The orthogonal ideal I^⊥is defined by

I^⊥= 〈q ∈ KQ2|〈p, q〉 = 0 ∀p ∈ I2〉

Lemma 3.1.15. Let V be a vector space with basis e₁, e₂, . . . en, and let W be a subspace with basis

e1− e2, e3− e4, . . . , e_2k−1− e2k, e_2k+1, e_2k+2, . . . , em. Then

e1+ e2, e3+ e4, . . . e_2k−1+ e2k, e_m+1, e_m+2, . . . , en

is a basis of W^⊥.

Proof. The orthogonal complement W^⊥can be written

{x ∈ V |〈x, y〉 = 0 for all y in the generator set of W }.

That W^⊥is a subspace of V is obvious. A simple calculation shows that e1+ e2, e3+ e4, . . . e_2k−1+ e2k, e_m+1, e_m+2, . . . , en∈ W^⊥.

Assume that dimW = m − k, then

A =





































1 −1 0 0 . . . 0 0 0 0 . . . 0 0 0 . . . 0 0 0 1 −1 . . . 0 0 0 0 . . . 0 0 0 . . . 0 ... ... ... ... . .. ... ... ... ... ... ... ... ... ... ...

0 0 0 0 . . . 1 −1 0 0 . . . 0 0 0 . . . 0 0 0 0 0 . . . 0 0 1 0 . . . 0 0 0 . . . 0 0 0 0 0 . . . 0 0 0 1 . . . 0 0 0 . . . 0 ... ... ... ... . .. ... ... ... ... ... ... ... ... ... ...

0 0 0 0 . . . 0 0 0 0 . . . 1 0 0 . . . 0 0 0 0 0 . . . 0 0 0 0 . . . 0 1 0 . . . 0





































is a n × (m − k)-matrix that defines a linear mapping ϕ where

ϕ(y) = ¡〈e1− e2, y〉,〈e3− e4, y〉,...,〈e2k−1− e2k, y〉,〈e2k+1, y〉,...,〈en, y〉¢T

. Hence W^⊥= kerϕ and dimV = dim(Imϕ)+dimW^⊥. We have that Imϕ is the column space of A and since the dimension of the column space equals the dimension of the row space we have that dimV = dimW + dimW^⊥. Since e1+ e2, e3+ e4, . . . e_2k−1+ e2k, e_m+1, e_m+2, . . . , en∈ W^⊥and these n − (m − k) vectors are linearly independent we get that the set is a basis of W^⊥

In characteristic 2 we have that W and W^⊥can have a non-empty inter- section. This is not a problem and the lemma above holds anyway.

Example 3.1.16. (i) Let Q be the quiver

◦

a₂

qq

a₁



a3

DD

and I = 〈a₁², a₂², a₃², a1a2− a2a1, a1a3, a3a1〉. Then I^⊥= 〈a1a2+ a2a1, a2a3, a3a2〉.

(ii) Let Q be the quiver

◦

a₁

((◦

a2

hh ^a3 //◦

and let I = 〈a1a2〉. Then I^⊥= 〈a2a1, a1a3〉.

(iii) Let Q be the quiver

◦

a2

qq

a1



a3

DD a₄ //◦

and let I = 〈a₁², a₂², a₃², a1a2− a2a1, a2a3− a3a2, a1a3, a3a1, a2a4〉. Then I^⊥= 〈a1a₂+ a2a₁, a₂a₃+ a3a₂, a₁a₄, a₃a₄〉.

Note that if I is a commutativity ideal, then I^⊥is an anti-commutativity ideal. With information of both I and I^⊥we can recover the quiver Q, since the minimal generating set of I and I^⊥ together list all possible paths of length 2 in Q.

Proposition 3.1.17. Any non-zero path ab of length 2 in K Q is represented in the generator graphsΓIorΓI^⊥ by exactly one of the following:

(i) Two undirected edges a — b, one inΓIand one inΓI^⊥

(ii) A directed edge a → b in ΓI

(iii) A directed edge a → b in ΓI^⊥.

Proof. If ab is a non-zero path, then, since ab is a basis element in K Q2. By Lemma 3.1.15 we have that either ab ∈ I or ab ∈ I^⊥or there exist (anti-) commutativity relations in the ideals I and I^⊥. The definition ofΓI tells us how to represent these three cases and we get the result in the proposition.

Lemma 3.1.18. If I is an admissible, commutativity ideal, then I contains all non-zero squares in K Q and I^⊥is a square-free anti-commutativity ideal.

Proof. Assume that I is admissible, but does not contain a²6= 0. Then, by Corollary 3.1.6, it will not contain aⁿ for any n and hence we get a contra- diction to the assumption that I was admissible. Hence an admissible ideal I always has to contain all non-zero squares and hence I^⊥will be square- free.

Lemma 3.1.19. Let I be a commutativity ideal.

(i) If a1a2. . . an∈ I we have that aia_{i +1}∉ I^⊥for some i .

(ii) If a1a2. . . an∉ I and aia_{i +1}∉ I^⊥, then a1a2. . . a_{i −1}a_{i +1}aia_{i +2}. . . an∉ I . (iii) If a1a2. . . an∉ I , then for any 1 ≤ i ≤ n − 1 there exists an edge (directed

or undirected) a_ia_{i +1}inΓI^⊥.

Proof. (i) By Lemma 3.1.4 we have that a1a2. . . an ∼ pakalq for some a_ka_l ∈ I and some paths p and q. Either a1a2. . . an = paka_lq and then there exist a pair aia_{i +1}= akal ∈ I which by Lemma 3.1.15 im- plies aia_{i +1}∉ I^⊥, or there exist an allowed transposition (aia_{i +1}). An allowed transposition corresponds to a commutativity relation in I , and aia_{i +1}− a_{i +1}a_i∈ I gives aia_{i +1}+ a_{i +1}a_i∈ I^⊥, which implies that aia_{i +1}∉ I^⊥.

(ii) Assume a₁a₂. . . an∉ I , then for all 1 ≤ i ≤ n − 1 we have that aia_{i +1}∉ I . We also assume aia_{i +1}6= 0, and hence, by Proposition 3.1.17 we have that aia_{i +1}correspond to either a directed edge ai→ ai +1inΓI^⊥, which would imply that aia_{i +1}∈ I^⊥, or undirected edges in bothΓI

andΓI^⊥. By assumption aia_{i +1}∉ I^⊥, hence aia_{i +1}− a_{i +1}ai∈ I , i.e. we have an allowed transposition (aia_{i +1}) and

a1a2. . . an∼ a1a2. . . a_{i −1}a_{i +1}aia_{i +2}. . . an. Hence a1a2. . . a_{i −1}a_{i +1}aia_{i +2}. . . an∉ I .

(iii) If a1a₂. . . an∉ I we have that aia_{i +1}∉ I for all 1 ≤ i ≤ n − 1. By Propo- sition 3.1.17 this implies that we either have a directed edge ai→ a_{i +1} or an undirected edge ai— a_{i +1}inΓI^⊥.

Lemma 3.1.20. If aiaj is represented by an undirected edge in ΓI, then ai

and a_j are loops at the same vertex. For any pair of loops a_i and a_j at the same vertex both aiaj and ajaihave to be represented by edges inΓI and/or ΓI^⊥

Proof. If aiaj is represented by an undirected edge we have that aiaj− a_ja_i∈ I and by Corollary 2.1.14 we have that aiand ajare loops at the same vertex.

If ai and aj are loops at the same vertex, then aiaj and ajai are never trivially 0 and hence, by Proposition 3.1.17, we have that aia_jis represented by a directed edge in eitherΓIorΓI^⊥or undirected edges in both.

Lemma 3.1.21. Let I be an admissible commutativity ideal. Assume that a1a2. . . ana1 is a path not contained in I such that ai6= aj for i 6= j . Then there exists a path a1aia_{i +1}. . . aka1∉ I such that aia_{i +1}. . . ak is a subpath of a₁a₂. . . ana₁, a₁ai∈ I^⊥and a_ka₁∈ I^⊥.

Proof. Assume a₁a_j ∉ I^⊥ for all 1 ≤ j ≤ n. Let pn = a1a₂. . . ana₁. Then, by Lemma 3.1.19(iii), we would have an allowed transposition (a1a2) and hence pn is equivalent to a2a1a3a4. . . ana1. By Lemma 3.1.19(iii) we then

have an edge a1a3and since we assume that there is no directed edge from a1 we get that (a1a3) is an allowed transposition. Inductively we get that (a1ai) are allowed transpositions for all 2 ≤ i ≤ n and hence pn∼ a2a3. . . ana²₁. By Lemma 3.1.18 we have that a²₁∈ I and hence pn∈ I and we have a con- tradiction. Hence there exist an aisuch that a1ai ∈ I^⊥. Assume that i is as small as possible, i.e. a1a_m∉ I^⊥for m < i . Then

a1a2. . . ak∼ a2a3. . . a_{i −1}a1aia_{i +1}. . . ana1, which implies that a₁aia_{i +1}. . . ana₁∉ I .

Now assume that aja1∉ I^⊥for every 1 ≤ j ≤ n. By the same inductive argument as above, get that (aja1) are allowed transpositions all 1 < j < n.

This gives a contradiction since a₁ai ∈ I^⊥ and by Lemma 3.1.15 this im- plies that a1ai− aia1∉ I2. Hence there exist an ak such that aka1 ∈ I^⊥. By assumption above (a1am) are allowed transpositions for m < i , hence k ≥ i . Assume that k is as big as possible, i.e. aja1 ∉ I^⊥ for j > k. Then a1aia_{i +1}. . . ana1∼ a1aia_{i +1}. . . aka1a_k+1a_k+2. . . an and hence we have that a1aia_{i +1}. . . aka1∉ I .

Proposition 3.1.22. Let a1a2. . . an∉ I . If

a1a2. . . an∼ a1a2. . . a_{i −1}a_{i +1}aia_{i +2}a_{i +3}. . . an

then there exist edges (directed or undirected) a_{i −1}a_{i +1}and a_ia_{i +2}inΓI^⊥. Proof. Assume

a1a2. . . an∼ a1a2. . . a_{i −1}a_{i +1}aia_{i +2}a_{i +3}. . . an.

Then we have an allowed transposition (aia_{i +1}) which implies that aia_{i +1}− a_{i +1}ai ∈ I . By Corollary 2.1.14 this gives that ai and a_{i +1} are loops. By Lemma 3.1.20 we have that there exist edges a_{i −1}a_{i +1}and aia_{i +2}in either ΓI^⊥orΓI. Since we assumed that a1a2. . . an∉ I we get that the edges have to lie inΓI^⊥.

It might be good to visualize what this proposition acctually tells us. As- sume that a1a2. . . an∉ I . Then, by Lemma 3.1.19(iii) we have that there exist edges aia_{i +1}in the generator graphΓI^⊥, they can be either directed or undi- rected depending on the generators of I , but for example it might look like this:

a₁ //a₂ a₃ a₄ //. . . //a_n.

Now, what Proposition 3.1.22 says is that whenever we have an undirected edge aia_{i +1}, we also have edges a_{i −1}a_{i +1}and aia_{i +2}. Since the edge a2a₃is undirected in the graph above we get two more edges

a₁

Q V Z _ d h m//a₂^{m h d _ Z V Q}a₃ a₄ //. . . //an.