REPRESENTATION THEORY.WEEK 141. Applications of quiversTwo rings A and B are Morita equivalent if the categories of A− modules andB-modules are equivalent. A projective finitely generated A-module P is a projectivegenerator i f any other projective finitel y gener ated A-module is isomorphic to a dire ctsummand of P⊕nfor some n.Theorem 1.1. A and B are Morita equivalent iff there exists a projective generatorP in A− mod such that B∼=EndA(P ). The functor X 7→ HomA(P, X) establishesthe equivalence between A − mod and B − mod .For the proof see, for example, Bass “Algebraic K-theory”.Assume now that C is a finite-dimensional algebra over algebraically closed field k.Let P1, . . . , Pnbe a set of representatives of isomorphism classes of indecomposableprojective C-module s. Then P = P1⊕ · · · ⊕ Pnis a projective generator, and A =EndC(P ) is M orita equivalent to C.Example 1.2. Let C be semisimple, then C∼=Matm1(k) × · · · × Matmn(k), andA∼=kn. LetC =XY0Z∈ Matp+q(k) | X ∈ Matp(k) , Y ∈ Matp,q(k) , Z ∈ Matq(k).ThenA = {(xy0z) | x, y, z ∈ k} .Let R be the radical of C. Then each i ndecomposable projective Pihas the filtrationPi⊃ RPi⊃ R2Pi⊃ · · · ⊃ 0 such that RjPi/Rj+1Piis semisimple for all j. Recall thatPi/RPiis simple (lecture notes 9), hence HomC(Pi, Pj/RPj) = 0 if i 6= j. Define thequiver Q in the following way. Vertices are e numerated by indecomposable proje ctivemodules P1, . . . , Pn, the number of arrows i → j equals dim HomC(Pi, RPj/R2Pj).We construct a surjective homomorphism φ: k (Q) → A. (This construction is notcanonical). Fist set φ (ei) = IdPi. Let γ1, . . . , γsbe the set of arrows from i toj, choose a basis η1, . . . , ηs∈ HomC(Pi, RPj/R2Pj), each ηlcan be lifted to ξl∈HomC(Pi, RPj) as Piis projective. Define φ (γl) = ξl. Now φ ex t ends i n the uniqueway to the whole k (Q) since k (Q) is generated by idempotents eiand arrows.Since φ is surjective, then A∼=k (Q) /I for some two-sided ideal I ⊂ k (Q). Thepair Q and an ideal I in k (Q) is called a quiver with relations. The problem of clas-sification of indecomposable C-modules is equival ent to the problem of classificationDate: December 6, 2005.12 REPRESENTATION THEORY. WEEK 14of indecomposable representations of Q satisfying relations I. In some cases suchquiver approach is very useful.Example 1.3. Let k b e the algebraic closure of F3and C = k [S3]. In lec t urenotes 9 we showed that C has two indecomposable proj ecti ves P+= IndS3S2triv andP−= IndS3S2sgn. The quiver Q is• ⇐⇒αβ•with relations αβα = 0, βαβ = 0. The quiver itself isˆA2, indecomposable representa-tions have dimensi ons (m, m), (m + 1, m) and (m, m + 1). Since we have the precisedescription, it is not difficult to see that only six indecomposable representationssatisfy the relations. They arek ⇐⇒ 0; 0 ⇐⇒ k; k ⇐⇒ k, α = 1, β = 0 or α = 0, β = 1,k2⇐⇒ k, α = (10) , β =01; k ⇐⇒ k2, α =01, β = (10) .The first two representations correspond to irreducible representations triv and sgn,the last two are proje ctives. Two r epresentations of dimension (1,1) correspond tothe quotients of P+and P−by the minimal submo dules.In fact one can apply the quiver approach to any category C which satisfies thefollowing conditions(1) All objects have finite length;(2) Any object has a projective resolution;(3) For any two objects X, Y , Hom (X, Y ) is a vector space over an algebraicallyclosed field k.We do not ne ed the assumption that the number of simple or projective objects isfinite. We illustrate this in t he following example.Example 1.4. Let Λ be the Grassmann algebra with two generators, i.e. Λ = k <x, y > / (x2, y2, xy + yx). Consider the Z-grading Λ = Λ0⊕ Λ1⊕ Λ2, wher e Λ0= k,Λ1is the span of x and y, Λ2= kxy. Let C denote the category of graded Λ-modules.In other words, objects are Λ-modules M =Li∈ZMi, such that ΛiMj⊂ Mi+jandmorphisms preserve the grading. All projective modules are free. An indecomposableprojective module Piis isomorphic to Λ with shifted grading deg (1) = i. Thus, thequiver Q has infinitely many vertices enume r ated by Z:· · · ⇐αiβi• ⇐αi+1βi+1• ⇐αi+2βi+2• ⇐αi+3βi+3. . .Here αi+1, βi+1∈ Hom (Pi+1, Pi), αi+1(1) = x, βi+1(1) = y. Relations are αiαi+1=βiβi+1= 0, αiβi+1+ βiαi+1= 0.Let us classify the indecomposable representations of above quiver. Assume firstthat, that there exists v ∈ Xi+1such that αiβi+16= 0, Then the subrepresentationV spanned by v, αi+1v, βi+1v, αiβi+1v splits as a direct summand in X. If X isindecomposable, then X = V . The corresponding object in C is Pi+1.REPRESENTATION THEORY. WEEK 14 3Now assume that αiβi+1Xi+1= 0 for any i ∈ Z. That is equivale nt to putting thenew relations for Q: every path of length 2 is zero. Consider the subspacesWi= Im αi+1+ Im βi+1⊂ Xi, Zi+1= Ker αi+1∩ Ker βi+1⊂ Xi+1.One can find Ui⊂ Xiand Yi+1⊂ Xi+1such that Xi= Ui⊕ Wi, Xi+1= Zi+1⊕ Yi+1.Check that Wi⊕ Yi+1is a subrepresentation, which splits as a direct summand inX. If X is indecomposable and Wi6= 0, then X = Wi⊕ Yi+1. Thus, we reducedour problem to Kronecker quiver • ⇐ •! There is the obvious bijection betweenindecomposable non-projective objec ts from C and the pairs (Y, i ) , where Y is anindecomposable representation of Kronecker quiver, i ∈ Z (de fines the grading).Remark 1.5. The last example is related to the algebraic geometry as the derivedcategory of C is equivalent to the derived category of coherent sheaves on P1.Remark 1.6. If in the last example we increase the number of generators in Λ, thenthe problem becomes wild (definition below).Let C be a finite-dimensional algebra. We say that C is finitely represented if C hasfinitel y many indecomposable representations. We call C tame if for each d ⊂ Z>0,there exist a finite set M1, . . . , Mrof C − k [ x] bimodules (free of rank d over k [x])such that every indecomposable representation of C of dime nsion d is isomorphic toMi⊗k[x]k [x] / (x − λ) for some i ≤ r, λ ∈ k. Finally, C is wild if there exists aC − k < x, y > bimodule M such that the functor X 7→ M ⊗k<x,y>X preservesindecomposability and is faithful. We formulate here without proof the followingresults.Theorem 1.7. Every finite -dimensional algebra over algebraically closed field k iseither finitely represented or tame or wildTheorem 1.8. Let Q be a c
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