Geometric reconstruction

Proposition 2.10 (separation property) The quotient X ^π- Y separates disjoint closed GLn-stable subvarieties ofX.

Proof. Let Zj be closed GLn-stable subvarieties ofX with defining ideals Zj =VX(Ij). Then,

∩jZj=VX(P

jIj). Applying lemma 2.7(3) we obtain π(∩jZj) =VY((X

j

Ij)∩C[Y]) =VY(X

j

(Ij∩C[Y]))

=∩jV^Y(Ij∩C[Y]) =∩jπ(Zj)

The onto property implies thatπ(Zj) =π(Zj) from which the statement follows.

It follows from the universal property that the quotient varietyY determined by the ring of polynomial invariants C[Y]^GLn is the best algebraic approximation to the orbit space problem.

From the separation property a stronger fact follows.

Proposition 2.11 The algebraic quotientX ^π- Y is the best continuous approximation to the orbit space. That is, points of Y parameterize the closed GLn-orbits in X. In fact, every fiber π⁻¹(y)contains exactly one closed orbitC and we have

π⁻¹(y) ={x∈X |C⊂GLn.x}

Proof. The fiberF =π⁻¹(y) is aGLn-stable closed subvariety ofX. Take any orbitGLn.x⊂F then either it is closed or contains in its closure an orbit of strictly smaller dimension. Induction on the dimension then shows thatG.xcontains a closed orbitC. On the other hand, assume thatF contains two closed orbits, then they have to be disjoint contradicting the separation property.

2.6 Geometric reconstruction

In this section we will give a geometric interpretation of the reconstruction result of theorem 1.17.

LetAbe a Cayley-Hamilton algebra of degreen, with trace maptrA, which is generated by at most m elementsa1, . . . , am. We will give a functorial interpretation to the affine scheme determined by the canonical idealNA/C[Mn^m] in the formulation of theorem 1.17. First, let us identify the reduced affine variety V(NA). A point m= (m1, . . . , mm) ∈V(NA) determines an algebra map

fm:C[M_n^m]/NA - _Cand hence an algebra mapφm

A ...^φm- Mn(C)

Mn(C[Mn^m]/NA)

?

∩

Mⁿ (f^m

)

-which is trace preserving. Conversely, from the universal property it follows that any trace preserv-ing algebra morphismA - Mn(C) is of this form by considering the images of the trace generators a1, . . . , am ofA. Alternatively, the points ofV(NA) classifyn-dimensionaltrace preserving repre-sentations ofA. That is, n-dimensional representations for which the morphismA - Mn(C) describing the action is trace preserving. For this reason we will denote the variety V(NA) by trepnAand call it thetrace preserving reduced representation varietyofA.

Assume thatAis generated as aC-algebra bya1, . . . , am(observe that this is no restriction as trace affine algebras are affine) then clearlyIA(n)⊂NA. That is,

Lemma 2.8 ForA a Cayley-Hamilton algebra of degreengenerated by {a1, . . . , am}, the reduced trace preserving representation variety

trepnA^⊂ - repnA is a closed subvariety of the reduced representation variety.

It is easy to determine the additional defining equations. Write any trace monomial out in the generators

trA(ai₁. . . ai_k) =X

αj₁...j_laj₁. . . aj_l

then for a pointm= (m1, . . . , mm)∈repnAto belong totrepnA, it must satisfy all the relations of the form

tr(mi₁. . . mi_k) =X

αj₁...j_lmj₁. . . mj_l

withtrthe usual trace on Mn(C). These relations define the closed subvarietytrepn(A). Usually, this is a proper subvariety.

Example 2.11 LetAbe a finite dimensional semi-simple algebraA=Md₁(C)⊕. . .⊕Md_k(C), then Ahas preciselykdistinct simple modules{M1, . . . , Mk}of dimensions{d1, . . . , dk}. Here,Mican be viewed as column vectors of sizedion which the componentMd_i(C) acts by left multiplication and the other factors act as zero. BecauseAis semi-simple everyn-dimensionalA-representation M is isomorphic to

M =M₁^⊕e1⊕. . .⊕M_k^⊕ek

2.6. Geometric reconstruction 87

for certain multiplicitiesei satisfying the numerical condition n=e1d1+. . .+ekdk

That is, repn Ais the disjoint union of a finite number of (closed) orbits each determined by an integral vector (e1, . . . , ek) satisfying the condition called thedimension vector ofM.

repnA' G

Via this embedding,A becomes a Cayley-Hamilton algebra of degree n when equipped with the induced tracetrfromMn(C).

LetM be then-dimensionalA-representation with dimension vector (e1, . . . , ek) and choose a basis compatible with this decomposition. LetEi be the idempotent of A corresponding to the identity matrixId_iof thei-th factor. Then, the trace of the matrix defining the action ofEionMis clearlyeidi.In. On the other hand,tr(Ei) =fidi.In, hence the only trace preservingn-dimensional A-representation is that of dimension vector (f1, . . . , fk). Therefore,trepn Aconsists of the single closed orbit determined by the integral vector (f1, . . . , fk).

trepn A'GLn/(GLf₁×. . .×GLf_k)

Consider the scheme structure of the trace preserving representation variety trep_n A. The corresponding functor assigns to a commutative affineC-algebraR

trep_n(R) =Alg_C(C[Mn^m]/NA, R).

Composing with the canonical embedding

A ...^φ- Mn(R)

Mn(C[Mn^m]/NA)

?

∩

Mⁿ (ψ)

-determines the trace preserving algebra morphismφ : A - Mn(R) where the trace map on Mn(R) is the usual trace. By the universal property any trace preserving map A - Mn(R) is also of this form.

Lemma 2.9 LetA be a Cayley-Hamilton algebra of degreen which is generated by {a1, . . . , am}.

The trace preserving representation varietytrep_nA represents the functor trep_n A(R) ={A ^φ- Mn(R) | φis trace preserving} Moreover,trep_nA is a closed subscheme ofrep_nA.

Recall that there is an action of GLn onC[Mn^m] and from the definition of the ideals IA(n) and NA it is clear that they are stable under the GLn-action. That is, there is an action by automorphisms on the quotient algebrasC[Mn^m]/IA(n) andC[Mn^m]/NA. But then, their algebras of invariants are equal to

(

C[rep_nA]^GLn = (C[M_n^m]/IA(n))^GLn=N^mn/(IA(n)∩N^mn) C[trep_nA]^GLn = (C[M_n^m]/NA)^GLn=N^mn/(NA∩N^mn)

That is, these rings of invariants define closed subschemes of the affine (reduced) variety associated to the necklace algebraN^mn. We will call these schemes thequotient schemes for the action ofGLn

and denote them respectively by

issn A=rep_nA/GLn and trissnA=trep_nA/GLn.

We have seen that the geometric points of the reduced varietyissnAof the affine quotient scheme issnAparameterize the isomorphism classes ofn-dimensional semisimpleA-representations. Sim-ilarly, the geometric points of the reduced variety trissn A of the quotient scheme trissn A parameterize isomorphism classes oftrace preservingn-dimensional semisimpleA-representations.

Proposition 2.12 LetAbe a Cayley-Hamilton algebra of degreenwith trace maptrA. Then, we have that

trA(A) =C[trissn A],

the coordinate ring of the quotient schemetrissnA. In particular, maximal ideals oftrA(A) param-eterize the isomorphism classes of trace preservingn-dimensional semi-simpleA-representations.

2.6. Geometric reconstruction 89

By definition, aGLn-equivariant map between the affineGLn-schemes trep_nA ^f- Mn=Mn

means that for any commutative affineC-algebraRthe corresponding map trep_nA(R) ^f(R)- Mn(R)

commutes with the action ofGLn(R). Alternatively, the ring of all morphismstrep_nA - Mn

is the matrixalgebraMn(C[Mn^m]/NA) and those that commute with theGLn action are precisely the invariants. That is, we have the following description ofA.

Theorem 2.6 LetAbe a Cayley-Hamilton algebra of degreen with trace maptrA. Then, we can recoverAas the ring ofGLn-equivariant maps

A={f:trep_n A - Mn GLn-equivariant } Summarizing the results of this and the previous section we have Theorem 2.7 The functor

alg@n ^trep-ⁿ GL(n)-affine

which assigns to a Cayley-Hamilton algebraAof degreentheGLn-affine schemetrep_nAof trace preservingn-dimensional representations has a left inverse. This left inverse functor

GL(n)-affine ^⇑

n

- alg@n

assigns to aGLn-affine schemeX its witness algebra ⇑ⁿ [X] =Mn(C[X])^GLn which is a Cayley-Hamilton algebra of degreen.

Note however that this functor isnot an equivalence of categories. For, there are many affine GLn-schemes having the same witness algebra as we will see in the next section.

We will give an application of the algebraic reconstruction result, theorem 1.17, to finite dimen-sional algebras.

LetA be a Cayley-Hamilton algebra of degreenwit trace map tr, then we can define anorm maponAby

N(a) =σn(a) for alla∈A.

Recall that the elementary symmetric functionσn is a polynomial functionf(t1, t2, . . . , tn) in the Newton functions ti = Pn

j=1xⁱ_j. Then, σ(a) = f(tr(a), tr(a²), . . . , tr(aⁿ)). Because, we have a trace preserving embedding A^⊂ - Mn(C[trep_n A]) and the norm map N coincides with the determinant in this matrix-algebra, we have that

N(1) = 1 and ∀a, b∈A N(ab) =N(a)N(b).

Furthermore, the norm map extends to a polynomial map onA[t] and we have that χ⁽ⁿ⁾a (t) = N(t−a). In particular we can obtain the trace by polarization of the norm map. Consider a finite dimensional semi-simpleC-algebra

A=Md₁(C)⊕. . .⊕Md_k(C),

then all the Cayley-Hamilton structures of degreenonAwith trace values inCare given by the following result.

Lemma 2.10 LetA be a semi-simple algebra as above andtra trace map onA making it into a Cayley-Hamilton algebra of degreen with tr(A) =C. Then, there exist a dimension vector α= (m1, . . . , mk)∈N^k+ such thatn=Pk

i=1midi and for anya= (A1, . . . , Ak)∈AwithAi∈Md_i(C) we have that

tr(a) =m1T r(A1) +. . .+mkT r(Ak) whereT rare the usual trace maps on matrices.

Proof. The norm-map N on A defined by the trace map tr induces a group morphism on the invertible elements ofA

N :A^∗=GLd₁(C)×. . .×GLd_k(C) - _C^∗

that is, a character. Now, any character is of the following form, let Ai ∈ GLd_i(C), then for a= (A1, . . . , Ak) we must have

N(a) =det(A1)^m1det(A2)^m2. . . det(Ak)^mk

for certain integersmi∈Z. SinceN extends to a polynomial map on the whole ofAwe must have that allmi≥0. By polarization it then follows that

tr(a) =m1T r(A1) +. . .+mkT r(Ak)

and it remains to show that nomi = 0. Indeed, if mi = 0 then tr would be the zero map on Md_i(C), but then we would have for anya= (0, . . . ,0, A,0, . . . ,0) withA∈Md_i(C) that

χ⁽ⁿ⁾_a (t) =tⁿ

whenceχ⁽ⁿ⁾a (a)6= 0 wheneverAis not nilpotent. This contradiction finishes the proof.

We can extend this to all finite dimensional C-algebras. Let Abe a finite dimensional algebra with radicalJ and assume there is a trace maptronAmakingAinto a Cayley-Hamilton algebra of degreenand such thattr(A) =C. We claim that the norm mapN :A - _Cis zero onJ.

2.6. Geometric reconstruction 91

Indeed, anyj∈J satisfiesj^l= 0 for somelwhenceN(j)^l= 0. But then, polarization gives that tr(J) = 0 and we have that the semisimple algebra

A^ss=A/J=Md₁(C)⊕. . .⊕Md_k(C)

is a semisimple Cayley-hamilton algebra of degreenon which we can apply the foregoing lemma.

Finally, note thatA'A^ss⊕JasC-vectorspaces. This concludes the proof of

Proposition 2.13 LetAbe a finite dimensionalC-algebra with radicalJ and semisimple part A^ss=A/J=Md₁(C)⊕. . .⊕Md_k(C).

Lettr :A - _C^⊂ - A be a trace map such that A is a Cayley-Hamilton algebra of degree n.

Then, there exists a dimension vectorα= (m1, . . . , mk)∈N^k+ such that for all a= (A1, . . . , Ak, j) withAi∈Md_i(C)andj∈J we have

tr(a) =m1T r(A1) +. . .+mkT r(Ak) withT rthe usual traces onMd_i(C)andP

imidi=n.

Fix a trace maptr onAdetermined by a dimension vectorα= (m1, . . . , mk)∈N^k. Then, the trace preserving varietytrep_nAis the scheme ofA-modules ofdimension vector α, that is, those A-modulesM such that

M^ss=S^⊕m₁ ¹⊕. . .⊕S_k^⊕mk

whereSiis the simpleA-module of dimensiondidetermined by thei-th factor inA^ss. An immediate consequence of the reconstruction theorem 2.6 is

Proposition 2.14 LetAbe a finite dimensional algebra with trace maptr:A - _Cdetermined by a dimension vectorα= (m1, . . . , mk)as before with all mi>0. Then,Acan be recovered from theGLn-structure of the affine schemetrep_nAof allA-modules of dimension vectorα.

Still, there can be other trace maps onAmakingAinto a Cayley-Hamilton algebra of degreen.

For example letCbe a finite dimensional commutativeC-algebra with radicalN, thenA=Mn(C) is finite dimensional with radicalJ=Mn(N) and the usual trace maptr:Mn(C) - CmakesA into a Cayley-Hamilton algebra of degreensuch thattr(J) =N6= 0. Still, ifAis semi-simple, the centerZ(A) =C⊕. . .⊕C(as many terms as there are matrix components inA) and any subring ofZ(A) is of the formC⊕. . .⊕C. In particular,tr(A) has this form and composing the trace map with projection on thej-th component we have a trace maptrjon which we can apply lemma 2.10.

Im Dokument noncommutative geometry@n, volume 1 : the tools (Seite 93-100)