Unformatted text preview:

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


View Full Document

Berkeley MATH 252 - Lecture Notes

Download Lecture Notes
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture Notes and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture Notes 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?