Noncommutative geometry (web draft, version 3)

355 Pages·2000·2.79 MB·English

by  Le Bruyn L.

noncommutative geometry @
n
Lieven Le Bruyn
Universiteit Antwerpen (UIA), B-2610 Antwerp (Belgium)
E-mail address: [email protected]
URL: http://win-www.uia.ac.be/u/lebruyn/
2000 Mathematics Subject Classification. Primary 14A22,16A30 ; Secondary
13A50, 14D20, 14L24, 16G20, 53D30
Research Director of the FWO (Belgium).
Abstract. Thistextwillneverbepublished!
Contents
alg @ 1
n
Chapter 1. Generic Matrices. 3
1.1. Conjugacy classes of matrices 3
1.2. Simultaneous conjugacy classes. 11
1.3. Matrix invariants and necklaces 14
1.4. The trace algebra. 19
1.5. The symmetric group. 23
1.6. Necklace relations. 25
1.7. Trace relations. 31
1.8. Cayley-Hamilton algebras. 35
References. 39
Chapter 2. Reconstructing Algebras. 41
2.1. Representation spaces. 41
2.2. Some algebraic geometry. 43
2.3. The Hilbert criterium. 47
2.4. Semisimple modules 50
2.5. Some invariant theory. 54
2.6. Geometric reconstruction 60
2.7. The Gerstenhaber-Hesselink theorem. 64
2.8. The real moment map. 72
References. 77
top @ 79
n
Chapter 3. Etale Cohomology. 81
3.1. Etale topology. 81
3.2. Central simple algebras 87
3.3. Quiver orders. 90
3.4. Simple roots. 101
3.5. Spectral sequences 106
3.6. Tsen and Tate fields 109
3.7. Coniveau spectral sequence 112
3.8. The Artin-Mumford exact sequence 115
References. 120
Chapter 4. Etale Slices. 121
4.1. C∞ slices. 121
4.2. Normal spaces. 122
4.3. Knop-Luna slices. 128
4.4. Smoothness. 132
4.5. Local structure. 137
4.6. A -algebras. 143
∞
4.7. Indecomposable roots. 146
4.8. Canonical decomposition. 153
vii
viii CONTENTS
References. 162
loc @ 165
n
Chapter 5. Classification. 167
5.1. Type stratification. 167
5.2. Cayley-smooth locus. 172
5.3. Local classification. 179
5.4. Low dimensional orders. 181
5.5. Noncommutative smooth surfaces. 188
5.6. Complex moment map. 193
5.7. Preprojective algebras. 197
5.8. Quantum groups. 199
References. 205
Chapter 6. Nullcones. 207
6.1. Cornering matrices. 207
6.2. General subrepresentations. 211
6.3. Semistable representations. 215
6.4. Optimal corners. 221
6.5. Hesselink stratification. 224
6.6. Cornering quiver representations. 230
6.7. Simultaneous conjugacy classes. 235
6.8. Representation fibers. 241
References. 247
geo @ 249
n
Chapter 7. Noncommutative Geometry 251
7.1. Formal structure. 251
7.2. Semi invariants. 256
7.3. Universal localization. 264
7.4. Compact manifolds. 270
7.5. Differential forms. 277
7.6. deRham cohomology. 288
7.7. Symplectic structure. 294
7.8. Necklace Lie algebras. 299
References. 302
Chapter 8. Moduli Spaces 303
8.1. Moment maps. 303
8.2. Dynamical systems. 306
8.3. Deformed preprojective algebras. 312
8.4. Hilbert schemes. 315
8.5. Hyper K¨ahler structure. 325
8.6. Calogero particles. 329
8.7. Coadjoint orbits. 333
8.8. Adelic Grassmannian. 336
References. 340
Bibliography 341
Index 345
alg @
n
noncommutativegeometry@ isthestudyoffamiliesof(commutative)al-
n
gebraicvarieties(withspecifiedconnectingmorphisms)whicharelocallycontrolled
by noncommutative algebras. In the next two chapters we will clarify the latter
part of this dogmatic statement.
Definethecategoryalgtr of(notnecessarilycommutative)C-algebrasequipped
with a C-linear map tr : A - A which is a formal trace, that is, satisfies for all
a,b∈A
tr(a)b=btr(a) tr(ab)=tr(ba) and tr(tr(a)b)=tr(a)tr(b).
Morphisms in algtr are C-algebra morphisms compatible with the trace structure.
The forgetful functor to the category of all C-algebras has a left adjoint
alg tr- algtr A7→Atr
where Atr is obtained by formally adjoining traces to cyclic words in a generating
setofA. Fora∈Atr andafixednaturalnumbern,wecandefinetheformalCayley-
Hamiltonpolynomial χ(n)(t)∈Atr[t]ofdegree n byexpressing f(t)=Qn (t−λ )
a i=1 i
(inthecommutingvariablesλ )asapolynomialintwithallcoefficientspolynomials
i
in the Newton functions Pn λk and replacing each occurrence of Pn λk by
i=1 i i=1 i
tr(ak). The level n approximation of the algebra A is defined to be the quotient
algebra
Atr
A @ =
n (tr(1)−n, χ(n)(a) ∀a∈Atr)
a
The main results, theorems 1.28 and 2.38, assert that we can recover the alge-
bra A @ and its central subalgebra tr A @ from geometric data. If A is an
n n
affine C-algebra with generating set {a ,...,a }, then the representation space
1 m
rep A of all n-dimensional representations of A is a closed subscheme of m-tuples
n
of n×n matrices Mm = M ⊕...⊕M and is stable under the GL -action on
n n n n
Mm by simultaneous conjugation. The GL -orbits in rep A are the isomorphism
n n n
classes of representations and an orbit is closed if and only if the corresponding
representation is semisimple. Geometric invariant theory asserts the existence of
an algebraic quotient iss A=rep A/GL classifying the isomorphism classes of
n n n
n-dimensional semisimple representations of A. We recover A @ as the algebra of
n
GL -equivariant polynomial maps from rep A to M and the central subalgebra
n n n
tr A @ as the coordinate ring of the quotient variety iss A.
n n
A family (X ) of (commutative) algebraic varieties is said to be locally con-
i i∈I
trolledbyasetofnoncommutativealgebrasAifeveryX islocallyisomorphictoan
i
irreduciblecomponentofiss Aforsomen∈NandA∈A. WhenAisacollection
n
of Quillen-smooth algebras (which implies that all rep A are smooth varieties) we
n
will determine the ´etale local structure of the X in the next part.
i
1

CHAPTER 1
Generic Matrices.
1.1. Conjugacy classes of matrices
WedenotebyM thespaceofalln×nmatricesM (C)andbyGL thegeneral
n n n
linear group GL (C). A matrix A∈M determines by left multiplication a linear
n n
operator on the n-dimensional vectorspace V =Cn of column vectors . If g ∈GL
n n
is the matrix describing the base change from the canonical basis of V to a new
n
basis, then the linear operator expressed in this new basis is represented by the
matrix gAg−1. For a given matrix A we want to find an suitable basis such that
the conjugated matrix gAg−1 has a simple form.
Consider the linear action of GL on the n2-dimensional vectorspace M
n n
GL ×M - M (g,A)7→g.A=gAg−1.
n n n
The orbit O(A) = {gAg−1 | g ∈ GL } of A under this action is called the
n
conjugacy class of A. We look for a particularly nice representative in a given
conjugacy class. The answer to this problem is, of course, given by the Jordan
normal form of the matrix.
With e we denote the matrix whose unique non-zero entry is 1 at entry (i,j).
ij
Recall that the group GL is generated by the following three classes of matrices :
n
rr
• the permutation matrices p = +e +e −e −e for all i6=j,
ij rrn ij ji ii jj
• the addition matrices a (λ)= +λe for all i6=j and 06=λ, and
ij n rr ij
• the multiplication matrices m (λ)= +(λ−1)e for all i and 06=λ.
i n ii
Conjugation by these matrices determine the three types of Jordan moves on n×n
matrices, where the altered rows and columns are dashed.
λ−1.
−λ.
(cid:4)(cid:4) (cid:26)(cid:26) (cid:26)(cid:26) (cid:127)(cid:127)
dd dd __
i i i λ.
+λ.
zz
j j
i j i j i
type p type a type m
Therefore, it suffices to consider sequences of these moves on a given n×n matrix
A ∈ M . The characteristic polynomial of A is defined to be the polynomial of
n
degree n in the variable t
rr
χ (t)=det(t −A)∈C[t].
A n
As C is algebraically closed, χ (t) decomposes as a product of linear terms
A
e
Y
(t−λ )di
i
i=1
3
4 1. GENERIC MATRICES.
where the {λ ,...,λ } are called the eigenvalues of the matrix A. Observe that λ
1 e i
is an eigenvalue of A if and only if there is a non-zero eigenvector v ∈ V = Cn
n
with eigenvalue λ , that is, A.v = λ v. In particular, the rank r of the matrix
rr i i i
A = λ −A satisfies n−d ≤ r < n. A nice inductive procedure using Jordan
i i n i i
moves given in [23] gives a proof of the following Jordan-Weierstrass theorem .
Theorem1.1(Jordan-Weierstrass). LetA∈M withcharacteristicpolynomial
n
χA(t)=Qei=1(t−λi)di. Then, A determines unique partitions
p =(a ,a ,...,a ) of d
i i1 i2 imi i
associated to the eigenvalues λ of A such that A is conjugated to a unique (up to
i
permutation of the blocks) block-diagonal matrix
 
B
1
 
 
 B2 
J = 
(p1,...,pe)  ... 
 
 
B
m
withm=m +...+m andexactlyoneblockB oftheformJ (λ )forall1≤i≤e
1 e l aij i
and 1≤j ≤m where
i
 
λ 1
i
Jaij(λi)= λi ...... 1∈Maij(C)
λ
i
Let us prove uniqueness of the partitions p of d corresponding to the eigen-
i i
value λ of A. Assume A is conjugated to another Jordan block matrix J ,
i (q1,...,qe)
necessarily with partitions q = (b ,...,b ) of d . To begin, observe that for a
i i1 im0 i
i
Jordan block of size k we have that
rk J (0)l =k−l for all l≤k and if µ6=0 then rk J (µ)l =k
k k
for all l. As J is conjugated to J we have for all λ∈C and all l
(p1,...,pe) (q1,...,qe)
rr rr
rk (λ −J )l =rk (λ −J )l
n (p1,...,pe) n (q1,...,qe)
Now, take λ=λ then only the Jordan blocks with eigenvalue λ are important in
i i
the calculation and one obtains for the ranks
l l
X X
n− #{j |a ≥h} respectively n− #{j |b ≥h}.
ij ij
h=1 h=1
Now, for any partition p = (c ,...,c ) and any natural number h we see that the
1 u
number z =#{j |c ≥h}
j
c1
c2
cz
cz+1
cu h
1.1. CONJUGACY CLASSES OF MATRICES 5
Figure 1. Orbit closure for 2×2 matrices
is the number of blocks in the h-th row of the dual partition p∗ which is defined to
bethepartitionobtainedbyinterchangingrowsandcolumnsintheYoung diagram
of p. Therefore, the above rank equality implies that p∗ = q∗ and hence that
i i
p = q . As we can repeat this argument for the other eigenvalues we have the
i i
required uniqueness. Hence, the Jordan normal form shows that the classification
of GL -orbits in M consists of two parts : a discrete part choosing
n n
• a partition p=(d ,d ,...,d ) of n, and for each d ,
1 2 e i
• a partition p =(a ,a ,...,a ) of d ,
i i1 i2 imi i
determining the sizes of the Jordan blocks and a continuous part choosing
• an e-tuple of distinct complex numbers (λ ,λ ,...,λ ).
1 2 e
fixing the eigenvalues. Moreover, this e-tuple (λ ,...,λ ) is determined only up to
1 e
permutationsofthesubgroupofallpermutationsπ inthesymmetricgroupS such
e
that p = p for all 1 ≤ i ≤ e. Whereas this gives a satisfactory set-theoretical
i π(i)
descriptionoftheorbitswecannotputanHausdorfftopologyonthissetduetothe
existence of non-closed orbits in M . For example, if n=2, consider the matrices
n
(cid:20) (cid:21) (cid:20) (cid:21)
λ 1 λ 0
A= and B =
0 λ 0 λ
which are in different normal form so correspond to distinct orbits. For any (cid:15) 6= 0
we have that
(cid:20)(cid:15) 0(cid:21) (cid:20)λ 1(cid:21) (cid:20)(cid:15)−1 0(cid:21) (cid:20)λ (cid:15)(cid:21)
. . =
0 1 0 λ 0 1 0 λ
-
belongstotheorbitofA. Henceif(cid:15) 0,weseethatBliesintheclosureofO(A).
As any matrix in O(A) has trace 2λ, the orbit is contained in the 3-dimensional
subspace
(cid:20) (cid:21)
λ+x y -
z λ−x ⊂ M2
In this space, the orbit-closure O(A) is the set of points satisfying x2+yz =0 (the
determinant has to be λ2), which is a cone having the origin as its top : The orbit
O(B) is the top of the cone and the orbit O(A) is the complement, see figure 1.
Still, for general n we can try to find the best separated topological quotient
space forthe actionofGL onM . Wewillprovethatthisspacecoincide withthe
n n
quotient variety determined by the invariant polynomial functions.
If two matrices are conjugated A∼B, then A and B have the same unordered
n-tuple of eigenvalues {λ ,...,λ } (occurring with multiplicities). Hence any sym-
1 n
metric function in the λ will have the same values in A as in B. In particular this
i
is the case for the elementary symmetric functions σ
l
X
σ (λ ,...,λ )= λ λ ...λ .
l 1 l i1 i2 il
i1<i2<...<il
6 1. GENERIC MATRICES.
Observe that for every A∈M with eigenvalues {λ ,...,λ } we have
n 1 n
n n
Y rr X
(t−λ )=χ (t)=det(t −A)=tn+ (−1)iσ (A)tn−i
j A n i
j=1 i=1
rr
Developingthedeterminantdet(t −A)weseethateachofthecoefficientsσ (A)is
n i
infactapolynomialfunctionintheentriesofA. Afortiori,σ (A)isacomplexvalued
i
continuous function on M . The above equality also implies that the functions
n
σ :M - C are constant along orbits. We now construct the continuous map
i n
M π- Cn
n
sending a matrix A∈M to the point (σ (A),...,σ (A)) in Cn. Clearly, if A∼B
n 1 n
then they map to the same point in Cn. We claim that π is surjective. Take any
point (a ,...,a )∈Cn and consider the matrix A∈M
1 n n
 
0 a
n
−1 0 an−1
(1.1) A= ... ... ... 
 
 −1 0 a2 
−1 a
1
then we will show that π(A)=(a ,...,a ), that is,
1 n
rr
det(t −A)=tn−a tn−1+a tn−2−...+(−1)na .
n 1 2 n
rr
Indeed, developing the determinant of t −A along the first column we obtain
n
t 0 0 ... 0 −a t 0 0 ... 0 −a
n n
1 t 0 0 −a 1 t 0 0 −a
07162534 n-1 n-1
0 1 t 0 −a 0 1 t 0 −a
n-2 07162534 n-2
−
... ... ... ... ... ... ... ... ... ...
0 0 1 t −a 0 0 1 t −a
2 2
0 0 1 t−a 0 0 1 t−a
1 1
Here, theseconddeterminantisequalto(−1)n−1a andbyinductiononnthefirst
n
determinant is equal to t.(tn−1−a tn−2+...+(−1)n−1a ), proving the claim.
1 n−1
Next, we will determine which n×n matrices can be conjugated to a matrix in
thecanonicalformAasabove. WecallamatrixB ∈M cyclicifthereisa(column)
n
vector v ∈ Cn such that Cn is spanned by the vectors {v,B.v,B2.v,...,Bn−1.v}.
Let g ∈ GL be the basechange transforming the standard basis to the ordered
n
basis
(v,−B.v,B2.v,−B3.v,...,(−1)n−1Bn−1.v).
In this new basis, the linear map determined by B (or equivalently, g.B.g−1) is
equal to the matrix in canonical form
 
0 b
n
−1 0 bn−1
 ... ... ... 
 
 −1 0 b2 
−1 b
1
whereBn.vhascoordinates(b ,...,b ,b )inthenewbasis. Conversely,anymatrix
n 2 1
in this form is a cyclic matrix.