Local structure - 4 — Quiver Representations

4 — Quiver Representations

4.2 Local structure

vjm =ij(πj(m)) alm =ij(Va(πi(m)))

for all arrows^joo ^a^l i. A computation verifies that these two operations are inverse to each

other and induce an equivalence of categories.

4.2 Local structure

In this section we give some applications of the slice theorem to the local structure of quotient varieties of representation spaces. We will first handle the case of an affineC-algebraAleading to a local description ofR

nA. Next, we will refine this slightly to prove similar results for an arbitrary affineC-algebraB inalg@n.

WhenAis an affineC-algebra generated bymelements{a1, . . . , am}, its levelnapproximation R

nAis trace generated bymdetermining a trace preserving epimorphismT^mn -- ^R_nA. Thus we have aGLn-equivariant closed embedding of affine schemes

rep_nA=trep_n isomorphism class of a semi-simplen-dimensional representation ofA, say

Mξ=S^⊕e₁ ¹⊕. . .⊕S_k^⊕e^k

where theSi are distinct simpleA-representations, say of dimensiondi and occurring inMξ with multiplicityei. These numbers determine therepresentation type τ(ξ) ofξ (or of the semi-simple representationMξ), that is

τ(ξ) = (e1, d1;e2, d2;. . .;ek, dk)

4.2. Local structure 177

where eachm⁽ⁱ⁾_j ∈Md_j(C). Using this description we can compute the stabilizer subgroupStab(x) ofGLnconsisting of those invertible matricesg∈GLncommuting with everyXi. That is,Stab(x) is the multiplicative group of units of the centralizer of the algebra generated by theXi. It is easy to verify that this group is isomorphic to

Stab(x)'GLe₁×GLe₂×. . .×GLe_k=GL(αξ)

A different choice of point in the orbitO(Mξ) gives a subgroup ofGLnconjugated toStab(x).

We know that the normal space Nx^sm can be identified with the self-extensions Ext¹_A(M, M) and we will give a quiver-description of this space. The idea is to describe first theGL(α)-module structure ofNx^big, the normal space to the orbitO(Mξ) inMn^m(see figure 4.2) and then to identify the direct summandNx^sm. The description ofNx^big follow from a book-keeping operation involving GL(α)-representations. Forx= (X1, . . . , Xm), the tangent spaceTx O(Mxi) inM_n^mto the orbit is equal to the image of the linear map

gl_n=Mn - Mn⊕. . .⊕Mn=Tx M_n^m A 7→ ([A, X1], . . . ,[A, Xm])

Observe that the kernel of this map is the centralizer of the subalgebra generated by theXi, so we have an exact sequence ofStab(x) =GL(α)-modules

0 - gl(α) =Lie GL(α) - gl_n=Mn - Tx O(x) - 0

BecauseGL(α) is a reductive group everyGL(α)-module is completely reducible and so the sequence splits. But then, the normal space inM_n^m=Tx M_n^m to the orbit is isomorphic asGL(α)-module to

Nx^big=Mn⊕. . .⊕Mn

| {z }

m−1

⊕gl(α)

with the action of GL(α) (embedded as above inGLn) is given by simultaneous conjugation. If we consider theGL(α)-action onMn depicted in figure 4.2 we see that it decomposes into a direct sum of subrepresentations

• for each 1 ≤ i ≤ k we have d²i copies of the GL(α)-module Me_i on which GLe_i acts by conjugation and the other factors ofGL(α) act trivially,

•

2222 2222 2222 2222 2222 2222 222

O(Mξ)

N smx Nbig

repn A

N_x^sm= Txrep_nA

TxO(Mξ) / N_x^big= Tx Mn^m

Tx O(Mξ) Figure 4.2: Big and small normal spaces to the orbit.

• for all 1≤i, j≤kwe havedidjcopies of theGL(α)-moduleMe_i×e_j on whichGLe_i×GLe_j

acts viag.m=gimg⁻¹_j and the other factors ofGL(α) act trivially.

TheseGL(α) components are precisely the modules appearing in representation spaces of quivers.

Theorem 4.2 Let ξ be of representation type τ = (e1, d1;. . .;ek, dk) and let α = (e1, . . . , ek).

Then, theGL(α)-module structure of the normal spaceN_x^big inM_n^mto the orbit of the semi-simple n-dimensional representationO(Mξ)is isomorphic to

rep_α Q^big_ξ

where the quiver Q^big_ξ has k vertices (the number of distinct simple summands of Mξ) and the

4.2. Local structure 179

| { z }

Figure 4.3: TheGL(α)-action onMn

subquiver on any two verticesvi, vj for1≤i6=j≤khas the following shape

8?9>:=;<ei 8?9>:=;<^ej ^(m⁻^1)d²j+ 1 (m−1)d2

i+ 1

(m−1)didj

jj77 gg

That is, in each vertexvithere are(m−1)d²_i+ 1-loops and there are(m−1)didjarrows from vertex vito vertex vj for all1≤i6=j≤k.

Example 4.2 Ifm= 2 andn= 3 and the representation type isτ = (1,1; 1,1; 1,1) (that is,Mξ

is the direct sum of three distinct one-dimensional simple representations) then the quiverQξ is

8?9>:=;<1 8?9>:=;<1

8?9>:=;<1

**jj ;;

{{ SS

-- MM qqQQ

qq--

qq--We haveGLn-equivariant embeddingsO(Mξ)^⊂ - trep_n R

nA^⊂ - M_n^m and corresponding embeddings of the tangent spaces inx

Tx O(Mξ)^⊂ - Tx trep_n Z

A^⊂ - Tx Mn^m

BecauseGL(α) is reductive we then obtain that the normal spaces to the orbit is a direct summand ofGL(α)-modules.

Nx^sm= Txtrep_n R

TxO(Mξ) / Nx^big = Tx M_n^m Tx O(Mξ)

As we know the isotypical decomposition ofN_x^big as the GL(α)-modulerep_α Qξ this allows us to controlN_x^sm. We only have to observe that arrows in Qξ correspond to simpleGL(α)-modules, whereas a loop at vertexvi decomposes asGL(α)-module into the simples

Me_i=Me⁰_i⊕C^triv

whereCtriv is the one-dimensional simple with trivialGL(α)-action andM_e⁰_i is the space of trace zero matrices inMe_i. AnyGL(α)-submodule ofNx^big can be represented by amarked quiverusing the dictionary

• a loop at vertexvicorresponds to theGL(α)-moduleMe_ion whichGLe_iacts by conjugation and the other factors act trivially,

• a marked loop at vertexvicorresponds to the simpleGL(α)-moduleM_e⁰_i on whichGLe_i acts by conjugation and the other factors act trivially,

• an arrow from vertex vi to vertex vj corresponds to the simple GL(α)-module Me_i×e_j on whichGLe_i×GLe_j acts viag.m=gimg⁻¹_j and the other factors act trivially,

Combining this with the calculation that the normalspace is the space of self-extensions Ext¹_A(Mξ, Mξ) or the trace preserving self-extensions Ext^tr_B(Mξ, Mxi) (in case B ∈ Ob(alg@n)) we have.

Theorem 4.3 Consider the marked quiver on k vertices such that the full marked subquiver on any two verticesvi6=vj has the form

8?9>:=;<ei 8?9>:=;<^ej

ajj aii

mii mjj

aij

aji

•

where these numbers satisfyaij≤(m−1)didj andaii+mii≤(m−1)d²i+ 1. Then,

4.2. Local structure 181

GLn×^GL(α)Nx^sm ^GLⁿ

×^GL(α)φ

GLn×^GL(α)Sx ψ

- rep_nA

Nx^sm/GL(α)

^φ/GL(α) Sx/GL(α)

?? _ψ/GL(α)

- issnA

?? Figure 4.4: Slice diagram for representation space.

1. LetAbe an affineC-algebra generated bymelements, letMξ be ann-dimensional semisimple A-module of representation-typeτ = (e1, d1;. . .;ek, dk) and let α= (e1, . . . , ek). Then, the normal space N_x^sm in a pointx∈ O(Mξ)to the orbit with respect to the representation space rep_n Ais isomorphic to the GL(α)-module of quiver-representations rep_α Qξ of above type with

• aii=dim_C Ext¹_A(Si, Si) andmii= 0for all1≤i≤k.

• aij=dim_C Ext¹_A(Si, Sj)for all1≤i6=j≤n.

2. Let B be a Cayley-Hamilton algebra of degree n, trace generated by m elements, let Mξ be a trace preserving n-dimensional semisimple B-module of representation type τ = (e1, d1;. . .;ek, dk)and letα= (e1, . . . , ek). Then, the normal spaceNx^tr in a pointx∈ O(Mξ) to the orbit with respect to the trace preserving representation space trep_n B is isomorphic to theGL(α)-module of marked quiver-representationsrep_α Q^•ξ of above type with

• aij=dim_C Ext¹_B(Si, Sj)for all1≤i6=j≤k.

and the (marked) vertex loops further determine the structure ofExt^tr_B(Mξ, Mξ).

By amarked quiver-representation we mean a representation of the underlying quiver (that is, forgetting the marks) subject to the condition that the matrices corresponding to marked loops have trace zero.

Consider the slice diagram of figure 4.4 for the representation spacerep_nA. The left hand side exists whenxis a smooth point ofrep_nA, the right hand side exists always. The horizontal maps are ´etale and the upper onesGLn-equivariant.

Definition 4.3 A point ξ ∈ issn A is said to belong to the n-smooth locus of A iff the repre-sentation space rep_n A is smooth in x ∈ O(Mξ). The n-smooth locus of A will be denoted by Smn(A).

To determine the ´etale local structure of Cayley-Hamilton algebras in theirn-smooth locus, we need to investigate the special case ofquiver orders. We will do this in the next section and, at its

end, draw some consequences about the ´etale local structure. We end this section by explaining the remarkable success of these local quiver settings and suggest that one can extend this using the theory ofA∞-algebras.

The category alg has a topological origin. Consider thetiny interval operad D1, that is, let D1(n) be the collection of all configurations

i1 i2 in

0 1

• ^{. . .} ^{. . .} •

consisting of the unit interval with n closed intervals removed, each gap given a label ij where (i1, i2, . . . , in) is a permutation of (1,2, . . . , n). Clearly,D1(n) is a real 2n-dimensionalC^∞-manifold havingn! connected components, each of which is a contractible space. The operad structure comes from the collection of composition maps

D1(n)×(D1(m1)×. . . D1(mn)) ^m^(n,m^1,...,mn)- D1(m1+. . .+mn)

defined by resizing the configuration in theD1(mi)-component such that it fits precisely in the i-th gap of the configuration of the D1(n)-component, see figure 4.5. We obtain a unit interval havingm1+. . .+mngaps which are labeled in the natural way, that is the firstm1 labels are for the gaps in the D1(m1)-configuration fitted in gap 1, the next m2 labels are for the gaps in the D1(m2)-configuration fitted in gap 2 and so on. The tiny interval operadD1consists of

• a collection of topological spacesD1(n) forn≥0,

• a continuous action ofSn onD1(n) by relabeling, for everyn,

• an identity elementid∈D1(1),

• the continuous composition mapsm(n,m₁,...,m_n) which satisfy associativity and equivariance with respect to the symmetric group actions.

By taking thehomology groupsof these manifoldsD1(n) we obtain alinear operadassoc. Because D1(n) has n! contractible components we can identify assoc(n) with the subspace of the free algebra Chx1, . . . , xni spanned by the multilinear monomials. assoc(n) has dimension n! with basisxσ(1). . . xσ(n)forσ∈Sn. Eachassoc(n) has a natural action ofSnand asSn-representation it is isomorphic to the regular representation. The composition mapsm(n,m₁,...,mn)induce on the homology level linear composition maps

assoc(n)⊗assoc(m1)⊗. . .⊗assoc(mn) ^γ^(n,m^1,...,mn)- assoc(m1+. . .+mn)

obtained by substituting the multilinear monomialsφi∈assoc(mi) in the place of the variablexi

into the multilinear monomialψ∈assoc(n).

4.2. Local structure 183

j1 j2 jmi1

0 1

• ^{. . .} ^{. . .} •

k1 k2 kmi2

0 1

• ^{. . .} ^{. . .} •

l1 l2 lmin

0 1

• ^{. . .} ^{. . .} •

i2 in

0 1

• ^{. . .} ^{. . .} •

Figure 4.5: The tiny interval operad.

In general, aC-linear operadPconsists of a family of vectorspacesP(n) each equipped with an Sn-action,P(1) contains an identity element and there are composition linear morphisms

P(n)⊗P(m1)⊗. . .⊗P(mn) ^c^(n,m^1,...,mn)- P(m1+. . .+mn)

satisfying the same compatibility relations as the maps γ(n,m₁,...,m_n) above. An example is the endomorphism operadendV for a vectorspaceV defined by taking

endV(n) =Hom_C(V^⊗n, V)

with compositions and Sn-action defined in the obvious way and unit elementrr

V ∈ endV(1) = End(V). Amorphismof linear operadsP ^f- P⁰is a collection of linear maps which are equivariant with respect to theSn-action, commute with the composition maps and take the identity element ofPto the identity element ofP⁰.

Definition 4.4 Let Pbe a C-linear operad. AP-algebra is a vectorspaceA equipped with a mor-phism of operadsP ^f- endA.

For example,assoc-algebras are just associativeC-algebras, explaining the topological origin of alg. Instead of considering thehomology operad assoc of the tiny intervals D1 we can consider itschain operad chain. For a topological spaceX, let chains(X) be the complex concentrated in non-positive degrees, whose−k-component consists of the finite formal additive combinations Pci.fiwhere ci∈Cand fi : [0,1]^k - X is a continuous map (asingular cubeinX ) modulo the following relations

• For anyσ∈Skacting on [0,1]^kby permutation, we havef◦σ=sg(σ)f.

• Forpr^k_k−1: [0,1]^k ^k−1-- the projection on the firstk−1 coordinates and any continuous map [0,1]^k−1 ^f

- X we havef⁰◦pr^k_k−1= 0.

Then, chain is the collection of complexes chains(D1(n)) and is an operad in the category of complexes of vectorspaces with cohomology the homology operadassoc. Again, we can consider chain-algebras, this time as complexes of vectorspaces. These are theA∞-algebras.

Definition 4.5 AnA∞-algebra is aZ-graded complex vectorspace B=⊕p∈ZBp

endowed with homogeneousC-linear maps

mn:B^⊗n - B of degree2−nfor alln≥1, satisfying the following relations

• We havem1◦m1= 0, that is(B, m1) is a differential complex . . . ^m-¹ Bi−1

m₁

- Bi m₁

- Bi+1 m₁

- . . .

• We have the equality of mapsB⊗B - B

m1◦m2=m2◦(m1⊗rr₊rr_⊗_m

whererris the identity map on the vectorspaceB. That is,m1 is a derivation with respect to the multiplicationB⊗B ^m-² B.

• We have the equality of mapsB⊗B⊗B - B m2◦(rr_⊗_m

2−m2⊗rr₎

=m1◦m3+m3◦(m1⊗rr_⊗rr₊rr_⊗_m

1⊗1 +rr_⊗rr_⊗_m

where the right second expression is the associator for the multiplicationm2 and the first is a boundary ofm3, implying thatm2 is associative up to homology.

4.2. Local structure 185

P ± = 0

bi+1oooooooo oo

oo OOOOOOOOOOOO

////

Figure 4.6: A∞-identities.

• More generally, forn≥1we have the relations X(−1)^i+j+kml◦(rr^⊗i_⊗_m

j⊗rr^⊗k_{) = 0}

where the sum runs over all decompositions n =i+j+k and wherel =i+ 1 +k. These identities are pictorially represented in figure 4.6.

Observe that anA∞-algebra B is in general not associative for the multiplication m2, but its homology

H^∗B=H^∗(B, m2)

is an associative graded algebra for the multiplication induced bym2. Further, ifmn= 0 for all n ≥ 3, then B is an associative differentially graded algebra and conversely every differentially graded algebra yields anA∞-algebra withmn= 0 for alln≥3.

LetAbe an associativeC-algebra andM a leftA-module. Choose aninjective resolutionofM 0 - M - I⁰ - I¹ - . . .

with theI^k injective leftA-modules and denote byI^•the complex I^• : 0 - I⁰ ^d- I¹ ^d- . . .

Let B =HOM_A^•(I^•, I^•) be the morphism complex. That is, its n-th component are the graded A-linear mapsI^• - I^•of degreen. This space can be equipped with a differential

d(f) =d◦f−(−1)ⁿf◦d forf in then-th part

Then,B is a differentially graded algebra where the multiplication is the natural composition of graded maps. The homology algebra

H^∗B=Ext^∗A(M, M)

is theextension algebra ofM. Generalizing the description of Ext¹A(M, M) given in section 4.3, an element ofExt^k_A(M, M) is an equivalence class of exact sequences ofA-modules

0 - M - P1 - P2 - . . . - Pk - M - 0

and the algebra structure on the extension algebra is induced by concatenation of such sequences.

This extension algebra has acanonical structure of A∞-algebra with m1 = 0 and m2 he usual multiplication.

Now, let M1, . . . , Mk beA-modules (for example, finite dimensional representations) and with f ilt(M1, . . . , Mk) we denote the full subcategory of allA-modules whose objects admit finite filtra-tions with subquotients among theMi. We have the following result, for a proof and more details we refer to the excellent notes by B. Keller [40,§6].

Theorem 4.4 Let M =M1⊕. . .⊕Mk. The canonical A∞-structure on the extension algebra Ext^∗_A(M, M)contains enough information to reconstruct the categoryf ilt(M1, . . . , Mk).

If we specialize to the case when M is a semi-simple n-dimensional representation of A of representation typeτ = (e1, d1;. . .;ek, dk) say with decomposition

Mξ=S^⊕e₁ ¹⊕. . .⊕S_k^⊕e^k Then, the first two terms of the extension algebraExt^∗_A(Mξ, Mξ) are

• Ext⁰_A(Mξ, Mξ) = EndA(Mξ) = Me1(C) ⊕ . . . ⊕Me_k(C) because by Schur’s lemma HomA(Si, Sj) =δijC. Hence, the 0-th part ofExt^∗_A(Mξ, Mξ) determine the dimension vector α= (e1, . . . , ek).

• Ext¹_A(Mξ, Mξ) = ⊕^k_i,j=1Me_j×e_i(Ext¹_A(Si, Sj)) and we have seen that dim_C Ext¹_A(Si, Sj) is the number of arrows from vertexvitovjin the local quiverQξ.

Summarizing the results of the previous section, we have :

Proposition 4.7 Letξ∈Smn(A), then the first two terms of the extension algebraExt^∗_A(Mξ, Mξ) contain enough information to determine the ´etale local structure ofrep_nAandissn AnearMξ. If one wants to extend this result to noncommutative singular pointsξ /∈Smn(A), one will have to consider the canonicalA∞-structure on the full extension algebraExt^∗_A(Mξ, Mξ).

Im Dokument noncommutative geometry@n, volume 1 : the tools (Seite 184-195)