Edward Formanek. My aim in this ... from the Cayley-Hamilton Theorem and the invariant theory of n x n ... M. Dehn [D] in 1922 and his student. W. Wagner [W] in ...
Polynomial Identities and the Cayley-Hamilton Theorem Edward Formanek
My aim in this article is to publicize a theorem of C. Procesi [P] and Y. P. Razmyslov [R] that describes all the polynomial identities (or identical relations) satisfied by the algebra of n x n matrices over a field K. I hope to make clear h o w their theorem was derived from the Cayley-Hamilton Theorem and the invariant theory of n x n matrices. I. Kaplansky defined algebras satisfying a polynomial i d e n t i t y in his seminal article [K]. An area of mathematics does not usually have its origin unequivocally in a single source, but this is true of polynomial identity algebras. M. D e h n [D] in 1922 and his student
W. Wagner [W] in 1937 did work that belongs to the area; however, their contributions were largely overlooked. Definition. Let A be an algebra over a field K. We say that A satisfies a polynomial identity if there is a nonzero polynomial (with coefficients in K) f ( x 1. . . . . xn) in n o n c o m m u t i n g variables xl . . . . . xn such t h a t f(al . . . . . a,) = 0 for all a 1. . . . . a n in A. Here are s o m e examples of algebras satisfying a polynomial identity, or PI-algebras. (For convenience, we sometimes use variables x , y , z instead of x l , x 2. . . . . ) 1. A n y c o m m u t a t i v e algebra satisfies Ix,y] = 0, where [x,y] = x y - y x . 2. The algebra of upper triangular r x r matrices over K satisfies [x,y] r = O. 3. Let V be a K-vector space with basis { v l , v 2 . . . . }, and let A(V) be the exterior algebra on V. (A(V) m a y be characterized as the free algebra on {vi} modulo the relations Vi 2 = 0 , ViVj = - - V j V i ) . A ( V ) satisfies the polynomial [[x,y],z] = ( x y - y x ) z - z ( x y - y x ) = O. 4. M2(K), the algebra of 2 x 2 matrices over K, satisfies [[xy - yx]2,z] = ( x y - y x ) 2 z - z ( x y - y x ) 2 = O. 5. A n y subalgebra or h o m o m o r p h i c image of a PIalgebra is a H-algebra. The remaining examples require a definition. The standard polynomial of degree r is ~r(Xl .....
Xr) = ~
sign(~)x,~(~)x~(2) 99. X.rr(r),
~'r r
where S r is the symmetric group of permutations of {1. . . . . r}. 6. If A is finite dimensional over K of dimension r, t h e n A satisfies ~r+l(xl . . . . . Xr+l) = 0. 7. If A is algebraic over K of b o u n d e d degree r (each THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. 1 9 1989 Springer-Verlag New York
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element of A satisfies a polynomial of degree r over K), then A satisfies 5Dr+x(X, xy, x y 2. . . . . x y r - l , x y r) = O. 8. (Amitsur-Levitzki Theorem [A-L]). M,(K) is annihilated by the standard polynomial of degree 2n and no polynomial of lower degree. In all the examples, except the final one, it takes at most a few lines to verify that the given identity is satisfied. The Amitsur-Levitzki Theorem now has four essentially different proofs, all requiring some work and cleverness. Algebras with polynomial identity are analogous to groups satisfying an identical relation. This area of group theory, called varieties of groups, has developed quite independently of PI-theory, with very little interaction between the two areas. Even basic terminology is sometimes different. For example, the subgroup (of a free group on countably many generators) of identical relations satisfied by a given group is called a f u l l y invariant subgroup, while the ideal (of the free algebra on countably many generators) of identities satisfied by a given algebra is called a T-ideal. Aside from the preferences of researchers in the two areas, an intrinsic reason w h y the two areas have developed differently is that the concept of a prime ideal is fundamental in ring theory, but has no analogue in group theory. It turns out that if K is infinite and A is a prime K-algebra (i.e., the zero ideal is a prime ideal of A), then for some positive integer n, A satisfies exactly the same identities as M,(K), the algebra of n x n matrices over K. Moreover, the T-ideal of identities of M , ( K ) is a prime ideal dR,, the ideals {At,} form a strictly descending chain ~1 D d~2 D ~t 3 D . . . . and they are the only nonzero prime T-ideals. The ideals {~,} have remained a basic object of study. For example, the Amitsur-Levitzki Theorem says that the standard polynomial of degree 2n is the polynomial of least degree in d~,. Although m a n y p r o b l e m s concerning ~ , r e m a i n open, there is a theorem, discovered independently by Procesi [P] and Razmyslov [R], which completely describes ~t,. The theorem says, in a sense that will not be made completely precise here, that all polynomial identities satisfied by n x n matrices are consequences of the Cayley-Hamilton Theorem. A good way to understand their theorem is to look at the case of 2 x 2 matrices. (From now on we will assume that K has characteristic zero. In fact there is no explicit description of ~ , in characteristic p > 0.) Let U be a 2 x 2 matrix over K. Then the Cayley-Hamilton Theorem says that U2 - T(U) + det(U) = 0,
functions of oq and o~2 in terms of the power symmetric functions of a s and o~2 gives det(U)
= c q e 2 = 1/2[(~1 + ~2) 2 -
= 1/2IT(U)2 -
U2 -
T(LOU + V2[T(U)2 - T(U2)] = 0.
(3)
Now we multilinearize (polarize, in the language of the last century) by substituting U = U 1 + U2 in (2) and deleting all but the multilinear terms. The result is equal to (b(U1 + U2) - qb(U1) - (b(U2), where qb(U)
The concept of a prime ideal is fundamental in ring theory, but has no analogue in group theory. denotes the characteristic polynomial of U. UIU2 + U2U1 - T(U1)U2 - T(U2)U~ + T(U1)T(U2) - T(U~U2) = O.
(4)
This formula (4) is the multilinear form of the CayleyHamilton Theorem for 2 x 2 matrices. Note that the denominator in (3) has v a n i s h e d - - a l l of the coefficients in (4) are integers. In fact the coefficients are all _ 1, and this also is true for the multilinear form of the Cayley-Hamilton Theorem for n • n matrices. Multiply (4) on the right by another 2 x 2 matrix U3 and take traces. T(UIU2U3) + T(U2U1U3) - T(U1)T(U2U3) - T(U2)T(U1U3) + T(U1)T(U2)T(U3) - T(UIU2)T(U3) = O.
(5)
Equation (5) is valid for any 2 x 2 matrices U1,U2,U3, and thus gives an identical relation among traces of 2 x 2 matrices. We will now rewrite (5) in a way which makes its form transparent. (123)
(213)
T(U1U2U3) +
T(U2UIU3)
(2)(13)
(1)(23) -
T(U1)T(U2U3)
(1)(2)(3)
-
T(U2)T(UIU 3) + T(UI)T(U2)T(U3)
-
T(U~U2)T(U3)
(6)
(12)(3)
(1)
where T(U) denotes the trace of a ~natrix U. If c,1 and ot2 are the characteristic values of U, then applying N e w t o n ' s formulas for the elementary symmetric
THE MATHEMATICAL INTELLIGENCER VOL. 11, NO. I, 1989
(2)
Then (1) becomes
=
38
(~12 + ~22)]
T(U2)].
~,
sign(~r)T~,(Ul,Uz, U3) = O.
~rES3
Definition: Let "rr e St, where S, is the symmetric group of p e r m u t a t i o n s of 1. . . . . r. Write ~r as a
product of disjoint cycles (including 1-cycles, so that each of the digits 1. . . . . r occurs exactly once): "rr = (a 1 . . . ak, ) (bl . . . bk2) ( q . . . Ck3) . . . .
The associated trace f u n c t i o n of ~r is T,,(U~ . . . . .
U,) = T(U~ . . . LI~,)T(Ub, . . . U G )
T(G,
U~ ....
Note that T~, is a function Mn(K) ~ ~ K, where Mn(K) r is the set of r-tuples of elements of M , ( K ) . Let ~(n,r) denote the K-vector space of all such functions. Then
All polynomial identities satisfied by n x n matrices are consequences of the CayleyHamilton Theorem.
for all invertible n x n matrices P over K. Here the first fundamental theorem is that t h e r i n g of invariants is generated by the traces of monomials in U 1. . . . . U r. A second f u n d a m e n t a l theorem t h e n gives the relations a m o n g the generators. The stated theorem apparently only gives the multilinear relations a m o n g the traces, but it is a general principle that in characteristic zero the multilinear relations generate all relations. For matrix invariants over a field of characteristic p > 0, t h e r e is n o t y e t a first f u n d a m e n t a l theorem, so of course there is also not a second fundamental theorem. Finally, we can give the promised description of ~t n, the T-ideal of polynomial identities satisfied by M,(K), where K has characteristic zero. This requires only a simple but very important observation of B. Kostant [Ko], w h o was clearly aware of the second fundamental theorem although he did not explicitly state it in the above form. Suppose that fix1 . . . . . x~) is a multilinear polynomial with coefficients in K. Then
(*) we can define a K-linear function = qr(n,r) : K[Sr] ~
T~(n,r)
by ~(Xa,,'rr) = EG, T~. Multilinearizing the Cayley-Hamilton Theorem for 2 x 2 matrices showed that G{sign(w)'a-: ~r 9 $3} lies in the kernel of ~(2,3)--i.e., it defines a function which vanishes on triples of 2 x 2 matrices. More generally, multilinearizing the Cayley-Hamilton Theorem for n x n matrices shows that X{sign(~)-~: ~ 9 S,+I} lies in the kernel of ~ ( n , n + 1). We n o w have e n o u g h notation to state the main result of Procesi [P] and Razmyslov [R]. T H E O R E M (Second F u n d a m e n t a l Theorem of Matrix Invariants. [P], Theorem 4.3; [R], Proposition 1). Let K be a field of characteristic zero. Then the kernel of ~(n,r) : K[Sr] ~ @(n,r) is zero if r _-< n, and is the twosided ideal of K[Sr] generated by Z{sign(-~)'m ~ 9 Sn+l} ifr~n
+ 1.
The reason it is called a second f u n d a m e n t a l theorem is the following. In invariant theory, a first f u n d a m e n t a l theorem gives a g e n e r a t i n g set for some ring of invariants. In our set-up the ring of invariants is the set of p o l y n o m i a l functions in @(n,r) that are invariant u n d e r s i m u l t a n e o u s c o n j u g a t i o n - - i . e . , polynomial functions f : M , ( K ) r ~ K that satisfy f(pu1p-1 .....
PUr P - l )
Xr) is a polynomial identity for M , ( K ) if and only if T[f(U1 . . . . . U~) 9 Ur+l] = 0 for all U1, .... UF+ 1 ~- M n ( K ).
f ( x 1. . . . .
= f(U1, . . . , U~)
This is n o t h i n g more than the n o n d e g e n e r a c y of the trace as a bilinear form on M , ( K ) : If U 9 M , ( K ) , then U = 0 if and only if T ( U V ) = 0 for all V e M,(K). Combining (*) with the second fundamental theorem gives a description of all the multilinear polynomial identities satisfied by Mn(K).
References [A-L] S. Amitsur and J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449-463. [D] M. Dehn, Ober die Grundlagen der projectiven Geometrie und allgemeine Zahlsysteme, Math. Ann. 85 (1922), 184-193. [K] I. Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948), 575-580. [Ko] B. Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory, J. of Math. and Mech. 7 (1958), 237-264. [P] C. Procesi, The invariant theory of n x n matrices, Adv. in Math. 19 (1976), 306-381. [R] Y.P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723-756 (Russian). English Translation: Math. USSR Izv. 8 (1974), 727-760. [W] W. Wagner, Uber die Grundlagen der projectiven Geometrie und allgemeine Zahlsysteme, Math. Z. 113 (1937), 528-567. Department of Mathematics The Pennsylvania State University University Park, P A 16802 USA
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