In this section we define the categoryalg@nof Cayley-Hamilton algebras of degreen.
Definition 1.1 Atrace mapon an (affine)C-algebraA is aC-linear map tr:A - A
satisfying the following three properties for alla, b∈A: 1. tr(a)b=btr(a),
2. tr(ab) =tr(ba) and 3. tr(tr(a)b) =tr(a)tr(b).
Note that it follows from the first property that the imagetr(A) of the trace map is contained in thecenter ofA. Consider two algebrasAandB equipped with a trace map which we will denote bytrA respectivelytrB. A trace morphismφ:A - B will be aC-algebra morphism which is compatible with the trace maps, that is, the following diagram commutes
A φ - B
A
trA
? φ - B
trB
?
This definition turns algebras with a trace map into a category, denoted byalg@. We will say that an algebraAwith trace maptr istrace generated by a subset of elementsI ⊂Aif the C-algebra
1.8. Cayley-Hamilton algebras 53
generated byBandtr(B) is equal toAwhereBis theC-subalgebra generated by the elements of I. Note thatAdoes not have to be generated as aC-algebra by the elements fromI.
Observe that for T∞ the formal trace t : T∞ -- N∞⊂ - T∞ is a trace map. Property (1) follows becauseN∞commutes with all elements ofT∞, property (2) is the cyclic permutation property fortand property (3) is the fact thattis aN∞-linear map. The formal trace algebraT∞ is trace generated by the variables{x1, x2, . . . , xi, . . .}butnotas aC-algebra.
Actually,T∞is thefree algebrain the generators{x1, x2, . . . , xi, . . .}in the category of algebras with a trace map, alg@. That is, if A is an algebra with trace tr which is trace generated by {a1, a2, . . .}, then there is a trace preserving algebra epimorphism
T∞
π
-- A .
For example, defineπ(xi) =ai andπ(t(xi1. . . xil)) =tr(π(xi1). . . π(xil)). Also, the formal trace algebraTm, that is the subalgebra ofT∞ trace generated by{x1, . . . , xm}, is the free algebra in the category of algebras with trace that are trace generated by at mostmelements.
Given a trace maptr onA, we can define for any a∈Aaformal Cayley-Hamilton polynomial of degreen. Indeed, express
f(t) =
n
Y
i=1
(t−λi)
as a polynomial intwith coefficients polynomial functions in the Newton functionsPn
i=1λki. Re-placing the Newton functionP
λki bytr(ak) we obtain the Cayley-Hamilton polynomial of degree n
χ(n)a (t)∈A[t] .
Definition 1.2 An (affine) C-algebra A with trace map tr : A - A is said to be a Cayley-Hamilton algebra of degree nif the following two properties are satisfied :
1. tr(1) =n, and
2. For alla∈A we haveχ(n)a (a) = 0inA.
alg@nis the category of Cayley-Hamilton algebras of degree nwith trace preserving morphisms.
Observe that if R is a commutative C-algebra, then Mn(R) is a Cayley-Hamilton algebra of degreen. The corresponding trace map is the composition of the usual trace with the inclusion of R -- Mn(R) via scalar matrices. As a consequence, the infinite trace algebra T∞n has a trace map induced by the natural inclusion
T∞n ⊂ - Mn(C[Mn∞])
N∞n tr
? ...
...
....
⊂ - C[Mn∞]
tr
?
which has imagetr(T∞n) the infinite necklace algebraN∞n. Clearly, being a trace preserving inclu-sion,T∞n is a Cayley-Hamilton algebra of degreen. With this definition, we have the following categorical description of the trace algebraT∞n.
Theorem 1.16 The trace algebra T∞n is the free algebra in the generic matrix generators {X1, X2, . . . , Xi, . . .}in the category of Cayley-Hamilton algebras of degreen.
For anym, the trace algebraTmn is the free algebra in the generic matrix generators{X1, . . . , Xm} in the categoryalg@nof Cayley-Hamilton algebras of degreenwhich are trace generated by at most melements.
Proof. Let Fn be the free algebra in the generators {y1, y2, . . .}in the category alg@n, then by freeness ofT∞there is a trace preserving algebra epimorphism
T∞ π- Fn with π(xi) =yi.
By the universal property ofFn, the idealKer πis the minimal idealI ofT∞such thatT∞/I is Cayley-Hamilton of degreen.
We claim thatKer πis substitution invariant. Consider a substitution endomorphismφofT∞ and consider the diagram
T∞
φ
- T∞
T∞/Ker χ
? ...
...
...
⊂ - Fn π
?? ...
...
χ
-thenKer χis an ideal closed under traces such thatT∞/Ker χis a Cayley-Hamilton algebra of degree n (being a subalgebra of Fn). But then Ker π ⊂Ker χ (by minimality of Ker π) and thereforeχfactors overFn, that is, the substitution endomorphismφdescends to an endomorphism φ: Fn - Fn meaning that Ker π is left invariant under φ, proving the claim. Further, any formal Cayley-Hamilton polynomial χ(n)x (x) of degree n of x ∈ T∞ maps to zero under π. By substitution invariance it follows that the ideal of trace relationsKer τ ⊂Ker π. We have seen thatT∞/Ker τ =T∞n is the infinite trace algebra which is a Cayley-Hamilton algebra of degree n. Thus, by minimality ofKer π we must haveKer τ =Ker π and henceFn'T∞n. The second
assertion follows immediately.
1.8. Cayley-Hamilton algebras 55
Let A be a Cayley-Hamilton algebra of degree n which is trace generated by the elements {a1, . . . , am}. We have a trace preserving algebra epimorphismpA defined byp(Xi) =ai
Tmn pa
-- A
Tmn tr
? pa
-- A
trA
?
and hence apresentationA'Tmn/TAwhereTA/Tmn is theideal of trace relationsholding among the generatorsai. We recall thatTmn is the ring of GLn-equivariant polynomial mapsMnm
f
- Mn, that is,
Mn(C[Mnm])GLn=Tmn
where the action ofGLnis the diagonal action onMn(C[Mnm]) =Mn⊗C[Mnm].
Observe that ifR is a commutative algebra, then any twosided idealI / Mn(R) is of the form Mn(J) for an idealJ / R. Indeed, the subsetsJij of (i, j) entries of elements ofI is an ideal ofR as can be seen by multiplication with scalar matrices. Moreover, by multiplying on both sides with permutation matricesone verifies thatJij=Jklfor alli, j, k, lproving the claim.
Applying this to the induced idealMn(C[Mnm])TA Mn(C[Mnm])/ Mn(C[Mnm]) we find an ideal NA/C[Mnm] such that
Mn(C[Mnm])TA Mn(C[Mnm]) =Mn(NA)
Observe that both the induced ideal andNAare stable under the respectiveGLn-actions.
Assume thatV andW are two (not necessarily finite dimensional)C-vectorspaces with a locally finiteGLn-action (that is, every finite dimensional subspace is contained in a finite dimensional GLn-stable subspace) and thatV f- W is a linear map commuting with theGLn-action. In section 2.5 we will see that we can decomposeV andW uniquely in direct sums of simple repre-sentations and in their isotypical components (that is, collecting all factors isomorphic to a given simple GLn-representation) and prove that V(0) = VGLn respectivelyW(0) = WGLn where (0) denotes the trivialGLn-representation. We obtain a commutative diagram
V f - W
VGLn
R
??
f0
- WG
R
??
whereRis theReynolds operator, that is, the canonical projection to the isotypical component of the trivial representation. Clearly, the Reynolds operator commutes with theGLn-action. More-over, using complete decomposability we see thatf0 is surjective (resp. injective) iff is surjective (resp. injective).
BecauseNA is aGLn-stable ideal ofC[Mnm] we can apply the above in the situation Mn(C[Mnm]) π -- Mn(C[Mnm]/NA)
Tmn R
?? π
0 -- Mn(C[Mnm]/NA)GLn
R
?? and the bottom map factorizes throughA=Tmn/TA giving a surjection
A -- Mn(C[Mnm]/NA)GLn.
In order to verify that this map is injective (and hence an isomorphism) it suffices to check that Mn(C[Mnm])TAMn(C[Mnm])∩Tmn =TA.
Using the functoriality of the Reynolds operator with respect to multiplication inMn(C[Mn∞]) with an elementx∈Tmn or with respect to the trace map (both commuting with theGLn-action) we deduce the following relations :
• For allx∈Tmn and allz∈Mn(C[Mn∞]) we haveR(xz) =xR(z) andR(zx) =R(z)x.
• For allz∈Mn(C[Mn∞]) we have R(tr(z)) =tr(R(z)).
Assume thatz =P
itixini ∈Mn(C[Mnm]) TA Mn(C[Mnm])∩Tmn with mi, ni ∈Mn(C[Mnm]) and ti∈TA. Now, considerXm+1∈T∞n. Using the cyclic property of traces we have
tr(zXm+1) =X
i
tr(mitiniXm+1) =X
i
tr(niXm+1miti) and if we apply the Reynolds operator to it we obtain the equality
tr(zXm+1) =tr(X
i
R(niXm+1mi)ti)
For anyi, the termR(niXm+1mi) is invariant so belongs toTm+1n and is linear inXm+1. Knowing the generating elements ofTm+1n we can write
R(niXm+1mi) =X
j
sijXm+1tij+X
k
tr(uikXm+1)vik
with all of the elementssij, tij, uik and vik inTmn. Substituting this information and again using the cyclic property of traces we obtain
tr(zXm+1) =tr((X
i,j,k
sijtijti+tr(vikti))Xm+1)
1.8. Cayley-Hamilton algebras 57
and by the nondegeneracy of the trace map we again deduce from this the equality z=X
i,j,k
sijtijti+tr(vikti)
Becauseti∈TA andTA is stable under taking traces we deduce from this thatz∈TAas required.
BecauseA =Mn(C[Mnm]/NA)GLn we can apply functoriality of the Reynolds operator to the setting
Mn(C[Mnm]/NA)
tr
-
⊃C[Mn]/NA
A
R
?? trA
-
⊃(C[Mn]/NA)
GLn R
?? Concluding we also have the equality
trA(A) = (C[Mnm]/JA)GLn.
Summarizing, we have proved the following invariant theoretic reconstruction result for Cayley-Hamilton algebras.
Theorem 1.17 LetAbe a Cayley-Hamilton algebra of degreen, with trace maptrA, which is trace generated by at mostmelements. Then , there is a canonical idealNA/C[Mnm]from which we can reconstruct the algebrasAandtrA(A)as invariant algebras
A=Mn(C[Mnm]/NA)GLn and trA(A) = (C[Mnm]/NA)GLn
A direct consequence of the above proof is the followinguniversal propertyof the embedding A⊂iA- Mn(C[Mnm]/NA).
LetRbe a commutativeC-algebra, thenMn(R) with the usual trace is a Cayley-Hamilton algebra of degree n. If f : A - Mn(R) is a trace preserving morphism, we claim that there exists a natural algebra morphismf:C[Mnm]/NA - Rsuch that the diagram
A f- Mn(R)
Mn(C[Mnm]/NA)
iA
?
∩
...
Mn (f)
-whereMn(f) is the algebra morphism defined entrywise. To see this, consider the composed trace preserving morphism φ:Tmn -- A f- Mn(R). Its image is fully determined by the images of the trace generatorsXk ofTmn which are say mk = (mij(k))i,j. But then we have an algebra morphismC[Mnm] g- Rdefined by sending the variablexij(k) tomij(k). Clearly, TA⊂Ker φ and after inducing toMn(C[Mnm]) it follows that NA ⊂ Ker g proving that g factors through C[Mnm]/JA - R. This morphism has the required universal property.
References
The first fundamental theorem of matrix invariants, theorem 1.6, is due independently to G. B.
Gurevich [31], C. Procesi [67] and K. S. Siberskii [78]. The second fundamental theorem of matrix invariants, theorem 1.10 is due independently to C. Procesi [67] and Y. P. Razmyslov [69]. Our treatment follows the paper [67] of C. Procesi, supplemented with material taken from the lecture notes of H-P. Kraft [52] and E. Formanek [26]. The invariant theoretic reconstruction result, theorem 1.17, is due to C. Procesi [68].