50 NOTES(4) HomC(C, Y ) =?First, students realized that HomC(D, Z) = 0 since the only pathfrom D to Z is the composition g ◦ f which is equal to zero because ofthe dotted line from D to Z.Next, we looked at HomC(C, Z). There are two paths from C to Z.The first is zero since g ◦ f = 0 by the previous calculation.g ◦ f!"# $0◦h =0The other path isg ◦ ! ◦ k = g ◦ f ◦ h =0which is equal to the previous path and is thus zero by the mesh relationf ◦ h = ! ◦ kStudents got the hang of this and they realized that HomC(C, X) = 0since all paths from C to X are equivalent, by repeated use of meshrelations, to the pathm ◦ g ◦ f ◦ h =0Finally, we came to HomC(C, Y ). This is nonzero. There are threepath from C to Y and they are all equal. But none of these goesthrough a zero relation. So, they are not necessarily zero. The ruleabout these diagrams is that all of the information is in the diagram.If a composition of paths is zero, it must be as a consequence of theinformation in the drawing. This not being the case, the three pathsare not zero.Students asked: Where are the clusters?It seems that I need one more lecture to show where the clustersenter in this category. This will tie everything together and we can goon to discuss questions and problems.MATH 47A FALL 2008 INTRO TO MATH RESEARCH 517.5. Clusters in the cluster category . In the cluster category oftype An, the clusters are defined to be sets of n “indecomposable”objects which do not “extend” each other. I will explain the (difficult)definitions and give really simple interpretations, just as I did for theadditive categories, so that you can understand it.I explained the concept of “rigidity” both geometrically and alge-braically. I think the geometry went over better. Rigid means thereare no deformations. For example a triangle is rigid but a quadrilateralis not rigid. If you take four sticks and tie them together at the endsthen the object you get is flexible or “deformable.” A rectangle is avery special case and is called a “specialization” of the parallelogramand the parallelogram is a “deformation” of the rectangle.For groups, there are two groups of order 4, the cyclic group and theKlein 4-group:Z/4, Z/2 × Z/2Z/4 is a “deformation” of Z/2 × Z/2. So, Z/2 × Z/2 is not “rigid.”Similarly, there are two groups of order 6, the symmetric group andthe cyclic group:S3, Z/6∼=Z/2 × Z/3However, the symmetric group (permutation group on 3 letters) is non-abelian. And the two abelian (commutative) groups of order 6 areisomorphic (the same). So, commutative groups of order 6 are rigid.If p is a prime numb er there is only one group of order p, namely thecyclic group /ZZ/p. So Z/p is rigid if p is prime.The set of (commutative) deformations of A × B forms a groupExt(A, B). For example,Ext(Z/2, Z/2) = {0,e}where 0 represents Z/2 × Z/2 and e represents ZZ/4.Ext(Z/3, Z/2) = 0because there are no commutative deformations of Z/3 ×Z/2. There isa difference between Ext(A, B) and Ext(B, A). They classify differentkinds of deformations. An element of Ext(A, B) is a group E whichhas a nonzero homomorphism B → E and another one E → A. In thecluster category this means that A must be to the right of B in orderfor Ext(B, A) to be nonzero.7.5.1. definition of clusters.Definition 7.6. In a cluster category of type An,acluster is a set ofn distinct indecomposable objects C1,C2, · · · ,CnwhereExt(Ci,Cj) = 052 NOTESfor all i, j.To figure out what the clusters are, we need a formula for Ext(A, B).This is called either “Serre duality” or “Auslander-Reiten duality” be-cause it was discovered by the late Brandeis University professor Mau-rice Auslander and his collaborator Idun Reiten.Theorem 7.7. Ext(A, B)∼=D Hom(τ−1B, A)∼=Hom(τ−1B, A) whereD is vector space dual.Here τ−1is the inverse of the Auslander-Reiten translation functorτ which shifts everything to the left. (Thus τ−1shifts everything tothe right.) Note that in the Auslander-Reiten quiver, there is always adotted line going from τX to X.The symbol∼=means “isomorphic” although many mathematicianswrite equality:Ext(A, B)=D Hom(τ−1B, A)7.5.2. example: A5. First we did calculations on A5:•!!!!!!!!!!"""""""•""#########"""""""""W##$$$$$$$$"""""""•!!!!!!!!!!"""""""A!!!!!!!!!"""""""••$$%%%%%%%%!!&&&&&&&&""""""""•%%'''''''''&&((((((((((""""""""X''))))))))##!!!!!!!!"""""""•$$))))))))!!!!!!!!!!"""""""B$$%%%%%%%%!!&&&&&&&&•$$))))))))!!&&&&&&&&"""""""•((*********""+++++++++"""""""""•'',,,,,,,,##---------"""""""Y$$))))))))!!&&&&&&&&"""""""•$$))))))))!!&&&&&&&&"""""""•B$$........!!////////"""""""τ−1B(('''''''''&&(((((((((""""""""•$$))))))))!!!!!!!!!!"""""""Z$$%%%%%%%%!!&&&&&&&&"""""""•$$........!!////////A$$%%%%%%%%"""""""•((000000000"""""""""•''111111111""""""""•$$%%%%%%%%"""""""•$$%%%%%%%%"""""""•(1) Ext(X, B) =?(2) Ext(W, B) =?(3) Ext(Y, B) =?(4) Ext(Z, B) =?First you use Serre duality:Ext(A, B)∼=Hom(τ−1B, A)Then you look at the rectangle starting at τ−1B. Only X, Y are in therectangle. So,(1) Ext(X, B) = Hom(τ−1B, X) %=0(2) Ext(W, B) = Hom(τ−1B, W )=0MATH 47A FALL 2008 INTRO TO MATH RESEARCH 53(3) Ext(Y, B) = Hom(τ−1B, Y ) %=0(4) Ext(Z, B) = Hom(τ−1B, Z) = 07.5.3. example: A2. Next we did A2and found all of the clusters. Hereis the cluster category:X"""""""""""########Y"""""""!!&&&&&&&&AA"""""""$$))))))))τ−1A"""""""((********Z$$%%%%%%%Since this is A2, a cluster consists of two objects C1,C2so thatExt(C1,C2) = 0 = Ext(C2,C1)We took C1= A and asked what are the possible C2.(1) Ext(X, A) = Hom(τ−1A, X) = 0(2) Ext(Y, A) = Hom(τ−1A, Y ) %=0(3) Ext(Z, A) = Hom(τ−1A, Z) = 0In other words, A, X form a cluster and A, Z form a cluster. Theobjects X, Z are the ones that come right after A and right before A.So, two objects form a cluster if and only if they are consecutive. If wecall B = τ−1A then the clusters are:{A, X}{X, B}{B, Y, }{Y, Z, }{Z, A}There are C(3) = 5 clusters.7.5.4. Problems. 1) Do the same for A3.2) Prove that Ext(A, B)∼=Ext(B, A) for any two objects in thecluster category.8. ResearchWhen a new area of mathematics opens up, not all of the easytheorems have been discovered. So, students have a chance to findsomething new. We will try to find something new about clusters andCatalan