Siberian Mathematical Journal, Vol. 44, No. 6, pp. 969–980, 2003Original Russian Text Copyright c© 2003 Belov A. Ya.

NO ASSOCIATIVE PI-ALGEBRACOINCIDES WITH ITS COMMUTANT

A. Ya. Belov UDC 512.552.4+512.554.32+512.664.2

Abstract: We prove that each (possibly not finitely generated) associative PI-algebra does not coincidewith its commutant. We thus solve I. V. L′vov’s problem in the Dniester Notebook. The result followsfrom the fact (also established in this article) that, in every T -prime variety, some weak identity holdsand there exists a central polynomial (the existence of a central polynomial was earlier proved byA. R. Kemer). Moreover, we prove stability of T -prime varieties (in the case of characteristic zero, thiswas done by S. V. Okhitin who used A. R. Kemer’s classification of T -prime varieties).

Keywords: PI-algebra, variety of algebras, identity, stable variety, weak identity, identity with trace,forms, Capelli identity, T -prime variety, Hamilton–Cayley equation, central polynomial

§ 1. Introduction

An important example of an associative algebra over a field of characteristic zero which coincideswith its commutant is the algebra of differential operators with polynomial coefficients. However, thisalgebra does not have nontrivial polynomial identities. If the characteristic of the field is p > 0 then somenontrivial identities arise (the algebra of differential operators of n variables generates the same variety asthe algebra of matrices of order pn with infinite center) but the coincidence with the commutant is lost.

However, in the case of positive characteristic p > 0, it is possible to construct an associative algebrathat coincides with its commutant. Such algebra is easy to obtain inductively as follows: The commutantof a matrix algebra is formed by the matrices with zero trace, and the trace of a matrix consisting of pidentical blocks on the principal diagonal is equal to zero. Therefore, every finite-dimensional algebra Aadmits, for every x ∈ A, a finite-dimensional extension by elements z, t such that [z, t] = x. However,the algebra obtained in this way has no nontrivial identities.

In this connection, I. V. L′vov posed the following question in the Dniester Notebook [1]:Question. Does there exist a PI-algebra A coinciding with its commutant [A,A]?The main theorem of this article answers this question.

Theorem 1. No associative PI-algebra coincides with its commutant.

Recall that an algebra A is called a PI-algebra if some nontrivial identity is fulfilled in it. Thisimplies that the polynomial

f(x1, . . . , xm) =∑

σ∈Sm

ασxσ(1) . . . xσ(m)

(where not all ασ ∈ F are zeros) vanishes on substituting arbitrary elements of A for xi’s.The set of all polynomials that vanish identically on an algebra is closed under the substitutions of

polynomials for variables and, moreover, is an ideal in the ring of (noncommutative) polynomials. Suchideal is called a T -ideal. It is known that the set of all T -ideals is the set of ideals stable under allendomorphisms. A class of algebras that satisfy a certain set of identities is called a variety, and a freeobject in a variety is called a relatively free algebra. The T -ideals of a relatively free algebra are alsoideals stable under all endomorphisms.

Dedicated to the memory of Igor′ Vladimirovich L′vov.

Bremen; Moscow. Translated from Sibirskiı Matematicheskiı Zhurnal, Vol. 44, No. 6, pp. 1239–1254,November–December, 2003. Original article submitted May 12, 2003.

0037-4466/03/4406–0969 $25.00 c© 2003 Plenum Publishing Corporation 969

A. R. Kemer introduced the notion of T -prime ideal [2, 3]. This is a T -ideal such that the factor by Ihas no nonzero T -ideals with zero product (an equivalent definition: the inclusion I1I2 ⊂ I, where I1 andI2 are T -ideals, implies I1 ⊂ I or I2 ⊂ I). A T -prime variety is a variety of algebras corresponding toa T -prime ideal of identities. This notion, first introduced by A. R. Kemer, is an analog of the classicalnotion of primeness.

Let Var(A) be the variety generated by A, i.e., the variety corresponding to the set of identitiesfulfilled in A. The first step in the proof of the main theorem (Theorem 1) consists in reducing thesituation to the case when Var(A) is a T -prime variety. To this end, we need to extend some classical“ideal” constructions to T -ideals.

Proposition 2. Every T -ideal defining a nonnilpotent variety of associative algebras is included ina maximal T -ideal having this property.

Proof. The condition that a variety is nonnilpotent is equivalent to the fact that the correspondingT -ideal contains a multilinear word. Clearly, the union of an increasing chain of T -ideals not containingmultilinear words also possesses this property. Therefore, by Zorn’s lemma, every T -ideal Γ with a non-nilpotent factor is included in a maximal T -ideal J with a nonnilpotent factor. T -primeness of J followsfrom the fact that nilpotency of A/J1 and A/J2 implies that of A/J1J2 and if J1 and J2 properly includeJ then their product cannot be included in J .

Remark. Let A be a “quasiprime” algebra in the following invariant sense. Suppose that X =xi∞i=1 is a collection of elements in A such that every permutation X → X extends to an endomorphismof A. Hence, if X does not generate a nilpotent ideal then there exists a factor A = A/I generatinga T -prime variety; moreover, the projection of X generates a nonnilpotent ideal. We can take as Ian arbitrary maximal ideal among the ideals invariant under the permutations of elements in X and notcontaining multilinear words of the xi’s.

Let A = [A,A] be a PI-algebra coinciding with its commutant. Among all such algebras, there isa universal algebra B generating Var(A) and meeting the conditions of the preceding remark. Thus, weobtain the following assertion.

Proposition 3. If there exists a PI-algebra coinciding with its commutant then there exists aPI-algebra coinciding with its commutant and generating a T -prime variety.

Remark. Developing noncommutative algebraic geometry by directly carrying over the classicalconstructions, we must confine ourselves only to “sensible” ideals. So are, first of all, the ideals stableunder the action of a group or subgroup of endomorphisms. A most invariant ideal is a T -ideal; the Spechtproblem gives an analog of Hilbert’s basis theorem and Amitsur’s theorem of the nilpotency of the radicalyields an analog to Hilbert’s zero theorem (as another analog to the latter theorem, we can suggest the“Razmyslov–Kemer–Brown T -ideal theorem of the nilpotency of the radical” proved by A. R. Kemer forthe case of characteristic zero). The above reduction to the T -prime case lies in the framework of thisnoncommutative algebro-geometric approach. Kemer’s “supertrick” and the appearance of analogs of theGrassmann algebra in the case of characteristic 2 in the study of the problem of finite basability are ofthe same spirit.

Definition 4. A multilinear polynomial f is called a weak identity if it is not an identity butbecomes an identity after substituting a commutator [z, t] for some variable.

If we show that a T -prime variety contains weak identities (Theorem 5) then the algebra generatingsuch a variety cannot coincide with its commutant and Theorem 1 will be proved. So, our aim is to provethe following

Theorem 5. A T -prime variety of associative algebras contains weak identities.

By the way, we also obtain the following results.

Theorem 6. A T -prime variety has a central polynomial.

970

A polynomial is called central if it is not an identity and all its values are in the center of the algebra.

Theorem 7. A T -prime variety M of associative algebras is unitarily closed (on the multilinearlevel in the case of positive characteristic).

This theorem was earlier proved by A. R. Kemer [4].

Definition 8. A variety M is called stable if, for every identity f =∑uixvi linear in x, the

fulfillment of f is equivalent to the fulfillment of the identity f∗ =∑vixui.

The notion of stability was suggested by V. N. Latyshev [5–7].

Theorem 9. A T -prime variety M is stable.

Here we consider the varieties other than the variety of all associative algebras (However, the lastvariety is also unitarily closed and stable; hence, this remark is in fact not necessary for Theorems 7 and 9.)

In the case of characteristic zero, there is a classification of T -prime varieties. These are the varietiesgenerated by the infinitely generated Grassmann algebra G, the algebra Mn ⊗G, the algebra of generalmatrices Mn, and the algebra M0

n,k ⊗ G0 + M1n,k ⊗ G1 which is the Grassmann envelope of the simple

superalgebra Mn,k. All these algebras are unital, and so the above theorems are well known to hold forthem [8]. (Theorem 6 was proved by A. R. Kemer [4] and earlier, in the matrix case, independentlyby Yu. P. Razmyslov [9] and E. Formanek [10]; moreover, Yu. P. Razmyslov proved this theorem in thecase of characteristic zero [8]. Theorem 7 in the case of characteristic zero is a direct corollary of theavailable classification of T -prime varieties. Theorem 9 was proved by S. V. Okhitin [11] in the case ofcharacteristic zero.

Remark. The notion of stability was originally defined in the case of characteristic zero; therefore, [8]has a reference that “stability of T -prime varieties is proved in [11].” In fact, the mentioned paper dealsonly with the case of characteristic zero.

All above theorems are well known in the case of characteristic zero. For this reason, we alwaysassume that the characteristic p of the field F is positive. In the case of positive characteristic, thesystem of Capelli identities of some order is always fulfilled in a PI-algebra (see [12]). Therefore, wealways have this in mind. In particular, instead of Theorem 7, we prove the following assertion.

Proposition 10. A T -prime variety M of associative algebras in which the system of Capelli iden-tities of some order is fulfilled is unitarily closed.

A polynomial Cn of order n is a polynomial of the form

Cn =∑σ∈Sn

(−1)σxσ(1)y1xσ(2) · . . . · yn−1xσ(n).

Here the yi’s are called layers. In the nonassociative case (including algebras of arbitrary signature Ω), bythe system Cn of Capelli polynomials of order n we mean a collection of polynomials which are multilinearand skew-symmetric with respect to some collection of n variables xi. If, in an algebra B, each Capellipolynomial of order n vanishes then we say that the system of Capelli identities is fulfilled in B. Thesystem Cn is fulfilled in all algebras of dimension less than n. For example, in the algebra of matrices oforder n, the identity Cn2+1 (but not Cn2) is fulfilled.

We remark that many results on this topic can be found in [4, 13–21].Acknowledgment. The author thanks L. A. Bokut′, who attracted the author’s attention to

I. V. L′vov’s problem.

§ 2. Traces, Forms, and Capelli Polynomials

This section addresses the technique of dealing with polynomials which is due to Yu. P. Razmyslov,C. Procesi, A. R. Kemer, and K. A. Zubrilin (a disciple of Yu. P. Razmyslov); see the correspondingarticles in the list of references.

971

Denote byD(A) the operator algebra for an algebra A. It is generated by left and right multiplicationsby elements of A, and so D(A) ' A ⊗ Aop, where Aop is the algebra anti-isomorphic to A. If a ∈ D(A)then the element a(x) ∈ A is well defined for every x ∈ A.

Yu. P. Razmyslov established the following equalities for the algebra of matrices:

Φk(a)Cn2(x1, . . . , xn2 , y0, . . . , xn2) =∑

i1<···<ik

F (~y, x1, . . . , xn)|xi1=axi1

,...,xik=axik

. (1)

Here Φk(a) is the sum of principal (i.e., symmetric with respect to the principal diagonal) kth-order minorsof a matrix a considered as a linear operator in the n2-dimensional space, Φ1 = Tr, and Φn2 = det.

In particular,

nTr(Z)C(x1, . . . , xn2 ; y1, . . . , yn2) =n2∑i=1

C(x1, . . . , xn2 ; y1, . . . , yn2)|xi=Zxi , (2)

det(Z)C(x1, . . . , xn2 ; y1, . . . , yn2) = C(Zx1, . . . , Zxn2 ; y1, . . . , yn2)

= C(x1, . . . , xn2 ; y1, . . . , yn2)|xi=Zxi∀i, (3)

Tr(Z1) Tr(Z2)C(x1, . . . , xn2 ; y1, . . . , yn2) =n2∑i=1

C(x1, . . . , xn2 ; y1, . . . , yn2)|xi=Z1xiZ2 . (4)

The Hamilton–Cayley identity ξn is defined as

ξn(X) = Xn − Φ1(X)Xn−1 + · · ·+ (−1)nΦn(X),

where Φk(X) are the elementary forms; Φk(X) is the sum of principal (i.e., symmetric with respect tothe principal diagonal) minors of order k of a matrix X.

It is well known that all invariants of general matrices are polynomials of Φk(ai1 . . . aik) (the firstfundamental theorem) and the relations between them mean the vanishing of Φs, s > n, and follow fromthe Hamilton–Cayley identity (the second fundamental theorem). In the case of characteristic zero, thiswas proved by Yu. P. Razmyslov [18] and C. Procesi [22] and, in the case of positive characteristic, byS. Donkin and A. N. Zubkov [23–25].

The theory developed for the matrix algebra makes it possible to deal with Capelli’s identities ina rather general situation.

Suppose that a polynomial F (~y, x1, . . . , xn) is multilinear and skew-symmetric in xi’s; a ∈ A. Definethe operators of interior forms δk

a by the formulas

δka(F ) =

∑i1<···<ik

F (~y, x1, . . . , xn)|xi1=axi1

,...,xik=axik

; δ0a(F ) = F. (5)

The polynomial δka(F ) is the component of the result of the substitution F |(a(xi)+xi)→xi;i=1,...,n which

is homogeneous of degree k in a.It is easy to check that

δka(Cn) =

∑i1<···<ik

∑σ∈Sn

(−1)σxσ(1)y1 . . . xσ(i1)ayi1 . . . xσ(ik)ayik . . . yn−1xσ(n).

Therefore, δka(F ) is skew-symmetric in xin

i=1.Put Tr(a) = δ1a. Clearly, Tr(a+ b) = Tr(a) + Tr(b).The operators δk

a are defined only on representations of elements, and hence, generally speaking, theresult of their application can depend on the representation of A as a polynomial F and the choice ofxi. If F is multilinear and skew-symmetric with respect to several collections then, while speakingof δk, we will indicate the collection with respect to which it is calculated.

972

Let n+1 be the minimal order of Capelli identities that hold in A. Suppose that k ≤ n. Given a pairof k-dimensional vectors (~a,~b), define the operator of “mixed volume” δk(~a,~b):

δk(~a,~b)[Cn(~x,~y)] =∑

i1<···<ik

Cn(~x,~y)|xiα=aαxiαbα . (6)

Put Tr(a, b) = δ1(a, b).Lemma 13 implies that, in a T -prime variety, the operators of interior forms commute and, moreover,

their definitions do not depend on the representations.The operators δk(~a,~b) are the linearizations of the operators of interior forms serving as the coefficients

of the Hamilton–Cayley polynomial. More exactly, δk(~a,~b) is the full linearization of δk(a) (a ∈ D(A)).We now formulate the main lemma of [26], which is an analog of the Hamilton–Cayley theorem for

operators defined internally.

Lemma 11. Suppose that a polynomial F (y,~z, x1, . . . , xn) is multilinear and skew-symmetric in xi

and a ∈ D(A) is an element of the operator algebra (for example, multiplication by a). The followingequality holds modulo Cn+1 (the Hamilton–Cayley theorem)

F (an(y),~z, x1, . . . , xn) =n∑

k=1

(−1)kδka(F (an−k(y),~z, x1, . . . , xn)). (7)

We also need the following two assertions of [26].

Proposition 12 (on transfer). (a) Suppose that the system of Capelli identities of order n + 1 isfulfilled in A and that a polynomial F is multilinear and skew-symmetric in xin

i=1 and in zini=1. Then

δka(F ) does not depend on the group of variables xi or zi; moreover, δk

a and δsb commute;

(b) Furthermore, in this case Tr(ab) = Tr(ba).

Lemma 13. Let f(x1, . . . , x2n) be a polynomial multilinear and skew-symmetric with respect tocollections x1, . . . , xn and xn+1, . . . , x2n and possibly dependent on other variables. Then

f(x1, . . . , xn, xn+1, . . . , x2n)− f(xn+1, . . . , x2n, x1, . . . , xn) ≡ 0 mod I,

where I is the sum of the T -ideals generated by the polynomials corresponding to the diagrams D withone column of length n− k and the other column of length n+ k.

Corollary 14. Let M be a T -prime variety in which the Capelli identity of order n+ 1 is fulfilled.Suppose that a polynomial F (~z, x1, . . . , xn) is multilinear and skew-symmetric in xi’s and a, b ∈ D(A)are elements in the operator algebra. Then δk

a and δsb commute.

The operators δk(x) are internally defined forms.We need the following technical assertion.

Lemma 15 (on absorption of a variable). Suppose that the system of Capelli identities of order n+1is fulfilled and a polynomial F is multilinear and skew-symmetric in x1, . . . , xn and linear in z. Then

F (z, x1, . . . , xn,~y) =n∑

i=1

F (z, x1, . . . , xn,~y)|z=xi;xi=z. (8)

Proof. The difference between the right-hand and left-hand sides of the equality is a polyno-mial T(Cn+1) which is multilinear and skew-symmetric with respect to the collection of n + 1 variablesz, x1, . . . , xn.

We need one more auxiliary assertion. The properties of Martindale’s centroid and the centralclosure (see [21]) imply the following

973

Proposition 16. Let A ∈ M be a prime algebra in which the system of Capelli identities of ordern+ 1 is fulfilled but the Capelli system of order n does not hold. Then A embeds in a finite-dimensionalalgebra B over an associative commutative ring K so that, for every a ∈ D(A), there exists λ(a) ∈ Ksuch that the equalities

n∑i=1

F (~y, x1, . . . , xn)|a(xi)→xi= λ(a) · F (~y, x1, . . . , xn)

hold for every polynomial F (~y, x1, . . . , xn) multilinear and skew-symmetric in x1, . . . , xn. Moreover, K isgenerated by these λ(a) and is Noetherian.

Proof. By the properties of Martindale’s central closure, it suffices to demonstrate that if a poly-nomial F~y(~x,~z) is multilinear and skew-symmetric in x1, . . . , xn and in z1, . . . , zn then the operation

F~y(~x,~z) →n∑

i=1

F~y(~z, x1, . . . , xn)|a(xi)→xi

defines a morphism of D(A)-modules (D(A) is the operator algebra) generated by the values of F on theT -ideal A generated by Cn. This follows from Lemma 13.

§ 3. Connection Between Traces and Forms

In the case of positive characteristic, the operators of forms are generally not expressible throughtraces. Therefore, the theory of identities with forms is necessary for proving local finite basability in thecase of positive characteristic (see [27, 28]). (This is due to the fact that all symmetric polynomials areexpressible in terms of polynomials of the form

∑xk

i only in the case of characteristic zero.) However,we have such a representation on the multilinear level (see Proposition 17).

In the study of T -ideals, a representation of the symmetric group σ → ϕ(σ) is used which relatesto invariants. It was introduced by Yu. P. Razmyslov [8] and C. Procesi [22]. By this correspondence,a decomposition of a permutation σ into cycles

σ = (i11, . . . , i1k1), . . . , (is1, . . . , isks)

corresponds to the monomial with trace

ϕ(σ) = Tr(xi11 . . . xi1k1) . . .Tr(xis1 . . . xisks

).

The correspondence ϕ is convenient, because the multiplication by the elements of Sn agrees with thesubstitutions and the identity corresponding to a table D is a corollary to some identity correspondingto D′ whenever D′ ⊂ D. (The bad thing about ϕ is that the traces are sometimes undefined. If this isthe case, they should be imitated by finding “trace killers” or polynomials whose multiplication by thetrace does not lead out of the initial algebra (without traces).)

The following proposition is the main result of this section.

Proposition 17. Let δk(~a) be the full linearization of the form Φk expressing the kth coefficient ofthe Hamilton–Cayley polynomial. Then

δk(~a) = ψ

(∑τ

(−1)ττ

)|ai→xi . (9)

A similar assertion holds for the linearizations of the forms σk defined internally.

Here ψ is the operator from the group algebra of Sk into the space generated by the multilinearwords in x1, . . . , xk which is defined in this section.

974

Proof. To an expression h = δk(u1, . . . , uk), assign the product f(h) = Tr(u1) . . .Tr(uk) of traces.If h is multilinear in x1, . . . , xn and does not depend on other variables, then we assign the elementϕ(h) ∈ Sn to h by putting ϕ(h) = ψ(f(h)).

Suppose that σ, τ ∈ Sn, f = ψ(−1)(σ), and h = ϕ(−1)(τ). Consider the actions of the correspondinginterior forms. Then the terms corresponding to h are among the terms corresponding to f with coeffi-cients 0 or 1. Moreover, all these coefficients are equal to 1 if σ is obtained from τ as follows: each cycle τis cut into several pieces and each piece is closed into its own cycle. Otherwise, all coefficients are zero.

A cycle of length n can be cut into k pieces in(n− 1k − 1

)ways, and a collection of cycles of lengths

n1, . . . , ns can be simultaneously cut into k1, . . . , ks pieces in∏s

i=1

(ni − 1ki − 1

)ways.

Also, observe that the parity of a permutation in Sn is equal to the number of cycles with signregarded. Hence, the following hold for the decomposition corresponding to the right-hand side of (9):

(α) The terms corresponding to the left-hand side of (9) are encountered with coefficient +1.(β) Let τ ∈ Sn be a permutation with cycles n1, . . . , ns. Then the terms corresponding to the

decomposition of ϕ−1(τ) are encountered with the coefficient λ equal to

∑k1,...,ks

(−1)n+∑

ki

s∏i=1

(ni − 1ki − 1

).

It remains to note that

∑k1,...,ks

(−1)n+∑

ki

s∏i=1

(ni − 1ki − 1

)= (−1)n

s∏i=1

ni∑ki=1

(ni − 1ki − 1

)= (−1)n+

∑ni

s∏i=1

(1− 1)ni−1. (10)

This expression vanishes if at least one of ni’s is greater than 1 (we put (1− 1)0 = 1).

§ 4. Proofs of the Main Theorems

It is well known that the trace function in a T -prime variety can be zero (an example is given bythe variety Mn(G) generated by the matrices over the infinite Grassmann algebra). However, we willdemonstrate the existence of a nonzero form.

Lemma 18. Suppose that a polynomial f(~y, x1, . . . , xn) is multilinear and skew-symmetric in xi.Let ui w zi be the word with exactly one occurrence of a variable zi and not containing the occurrencesof zj for j 6= i, ui = ui1ziui2.

Then the polynomial

f =∑σ∈Sn

(−1)σf(~y, u1, . . . , un)|zi→σ(zi)

is a linear combination of polynomials of the form δi(~a,~b)(g).

Proof. It is easy to see that the sign depends only on the disposition of variables of zi inside themonomial corresponding to a monomial of f and, moreover, for each such disposition of the zi’s, eachkind of entourages uτ(i)1 ∗ uτ(i)2 is encountered once for every permutation τ .

Thus, if a polynomial f meeting the conditions of Lemma 18 is nonzero then there is a nonzerooperator of an interior form.

Recall equality (8) of Lemma 15:

F (z, x1, . . . , xn,~y) =n∑

i=1

F (z, x1, . . . , xn,~y)|z=xi;xi=z.

975

Construct a nonzero polynomial of the kind of f . Let f be an arbitrary polynomial multilinear andskew-symmetric in x1, . . . , xn. Fix xj and represent f as

f =∑

i

bixjdi.

Suppose that the polynomials R1 and R2 are multilinear and skew-symmetric respectively in z1, . . . , znand y1, . . . , yn. Using (8), we infer

fR1R2 =∑

i

bixjdiR1R2 =∑i,k

bixjykR1R2|di→yk=

∑k

∑i

bixjykR1ekdigk =∑

k

f |xjykR1ek→xj · gk.

Thus, we have achieved that an expression skew-symmetric in a group of variables is substituted fora chosen variable xj . We have proved the following

Proposition 19. Suppose that M is a T -prime variety in which the system of Capelli identities oforder n + 1 (but not of order n) is fulfilled. Put gi = Cn(~yi, ~xi), f = Cn(~t,~z), and h = f |gi→zi . Thenh 6≡ 0 in M.

Remark. In the proof of Proposition 19, we substantially use associativity. The rest of the argumentsof this section can be easily carried over to the nonassociative situation. Furthermore, if the precedingproposition holds for every T -prime subvariety of M and the adjunction of unity does not lead outof M then this implies the unitary closure of T -prime subvarieties of M and the existence of centralpolynomials.

Example. Suppose that M is generated by a simple Lie algebra sln with adjoined unity (the obtainedalgebra is not a Lie algebra). Proposition 19 does not hold for M but it holds for sln. We conclude thatM does not have central polynomials.

Proposition 19 and Lemma 15 imply the following

Corollary 20. Suppose that M is a T -prime variety in which the Capelli identity of order n + 1(but not of order n) is fulfilled, g is a multilinear polynomial not vanishing on M, and h is obtained fromg by inserting Cn(~xi,~yi) into g instead of the ith variable; moreover, the collections of variables of thesepolynomials do not intersect. Then h does not vanish on M either.

If we adjoin a unity to a free countably generated algebra in a T -prime variety then we also obtain a T -prime variety. Suppose that A′ is obtained from A by the adjunction of unity. Then A′[A′, A′]A′ ⊆ A. Inparticular, if n > 1 then all values of Cn in A′ lie in A. Therefore, by Proposition 19 the system of Capelliidentities of the same order is fulfilled in M′. Now, we apply Corollary 20. Thus, we have establishedthe unitary closure of a T -prime variety in which a Capelli system of some order is fulfilled (i.e., we haveproved Proposition 10) and hence the unitary closure of an arbitrary t-prime variety (Theorem 7).

Suppose that F is multilinear and skew-symmetric with respect to two collections of variables Λ1 =xin

i=1 and Λ2 = zini=1.

Lemma 15 makes it possible to absorb not only a variable but also a position. Namely, fix a positionfor variables of Λ1. Represent F as the sum F =

∑ni=1 Fi, where Fi corresponds to the terms having

xi at this position. Assume that H = Ψ(F,G,~y), where G = Cn(~t,~z) and Ψ is linear in the first twoarguments, H =

∑i Ψ(Fi, G,~y). Apply equality (8) of Lemma 15 to each Fi and make a regrouping so

that the terms with zj , j = 1, . . . , n, stand in each group at the fixed position.Now, let H be the polynomial of Proposition 19. Distributing n− 1 positions for variables in which

g1 is skew-symmetric over g2, . . . , gn (with one position left in g1), we obtain a linear combination ofpolynomials of the kind of

f =∑σ∈Sn

(−1)σf(~y, u1, . . . , un)|zi→σ(zi)

meeting the conditions of Lemma 18. By Proposition 19, these terms cannot all be zero. We have provedthe following

976

Theorem 21. Let M be a T -prime variety. If the basic field is of positive characteristic or M isregular then not all operators of (two-sided) interior forms are zero in M.

(In the case of characteristic zero, Theorem 21 follows from the classification of T -prime varieties).We now turn to the proof of existence of a central polynomial (i.e., of Theorem 6). Extend the free

algebra of the T -prime variety M by the operators of forms and a unity. The ideal generated by theCapelli polynomial Cn =

∑(−1)σxσ(1)t1 . . . tn−1xσ(n) is the same for the initial and extended algebras.

(Such ideal is called a “form killer.”) Hence, by T -primeness, Cn+1 holds in the algebra with the extendedsignature (some alternated variables can be under the operators of forms).

Let Ψ be a nonzero form (i.e., a polynomial in elementary forms). Instead of the variables occurringin Ψ, insert the values of the Capelli polynomials Cn(yij, xij) (i is the index of a collection and j is theindex of a variable in the collection) of pairwise disjoint collections of variables. By Proposition 19, we havea nonzero value of the form, which we denote by Ψ′. By the unitary closure, we have Ψ′ · 1 =

∑j Ψj ·x1j .

The polynomial with the forms∑

j Ψj · x1j is in the center of the extended algebra. We are left withinserting, instead of x1j , the values of Cn at disjoint collections of variables (i.e., pairwise disjoint andnonintersecting with the rest of the variables in Ψ). By Proposition 19, we obtain a nonzero polynomial.Moreover, it is in the initial algebra and is central.

We have thus demonstrated the existence of a central polynomial in a T -prime variety.We now formulate another theorem which follows from the considerations of this section.

Theorem 22. Suppose that a variety M of (in general, nonassociative) algebras is T -prime andunitarily closed and the operator algebra PI and the operators of interior forms are not identically zero.Then M has a central polynomial.

This theorem and Proposition 16 on the possibility of defining interior traces for operator algebrasyield the following assertion.

Corollary 23. Suppose that a variety M of (in general, nonassociative) algebras is generated bya prime finite-dimensional algebra with nontrivial center. Then M has a central polynomial.

Remarks. 1. Note that the associative algebra generated by the operators ad in a simple finite-dimensional Lie algebra is simple and has a nontrivial center. Considering the triple algebra (L, ad(L),U(L)) and dealing with the interior traces of the operators in ad(L) applied to L, in the same way wecan establish the existence of a central polynomial in the variety of pairs (L,U(L)), where L is a freeLie algebra in the variety generated by a finite-dimensional simple Lie algebra and U(L) is its universalenvelope.

2. If a polynomial F is multilinear and skew-symmetric in variables of two groups X = x1, . . . , xnand Y = y1, . . . , yn then F |axib→xi∀i = F |ayi→yi; xib→xi ∀i modulo T(Cn+1). Therefore, we can “shift”the “left” and “right” substitutions from one group of n anticommuting variables to the other. (Thecorresponding operator in the matrix algebra is det(a)n det(b)n.) However, the approach connectedwith the linearization of the “interior determinant,” i.e., of the substitution axib → xi ∀i, encountersobstacles, since the left and right substitutions appearing in such linearizations (xi → xibj , xi → ajxi)behave independently and the expressions are symmetric.

3. Thus, every T -prime multilinear variety can be extended by operators of forms with preserva-tion of the collection of identities. If a polynomial F has the shape gz, where g =

∑αkΨk ∈ T(Cn)

is a polynomial in the operators of forms, and the extended algebra satisfies the system of identitiesCn+1 then (8) means that it is possible to leave out of the operators of forms only variables of a fixedfinite collection (occurring in g). In a sense, this can be considered as “weak representability.” Underlocalization with respect to g, it becomes the usual representability and all identities coincide with thoseof the matrix algebra.

Unitary closure implies that if a T -prime variety is defined by multilinear identities then the equalityxn = 0 holds for no n in it. Since every nonnilpotent variety has a T -prime factor, the considerations inthe beginning of this section imply the following assertion.

977

Proposition 24. Let M be a nonnilpotent variety defined by its multilinear identities. Then theidentity xn = 0 holds for no n in M.

Hence nonnilpotent locally nilpotent varieties cannot be generated by multilinear identities alone,and we have another proof of the Nagata–Higman theorem for infinitely generated algebras over a field ofcharacteristic zero. In this case, every variety is defined by multilinear identities, and the identity xn ≡ 0cannot hold in a nonnilpotent variety.

Combining Proposition 24 and Lemma 11, we infer that if M is a T -prime variety in which a Capelliidentity of order n is fulfilled and which is defined by its multilinear identities then the operator δk(a)is nonzero for some k ≤ n. If M is nonregular then δ1(x) ≡ 0. In any case (as shown below) the fulllinearization of δk is expressed in terms of the traces, i.e., through δ1. Therefore, in the nonregular casethe full linearization of δk’s is zero and hence δk(x) ≡ 0 for k < p = char(F).

Now, we continue the study of T -prime varieties.Since not all forms δk are zero, from Proposition 17 it follows that the operator δ(a, b) = δ1(a, b) is

nonzero.

Proposition 25. The equality

δ([a1, a2], b) = δ(a, [b1, b2]) = 0 (11)

holds.

Proof. The definition of the interior trace immediately implies the equality

[δ(a, b), δ(c, d)] = δ([ac], bd) + δ(ca, [bd]);

moreover, [δ(a, b), δ(c, d)] = 0.Suppose that w = w1w2 = u1u2. Then the preceding identity implies that δ(ca, [w1, w2]) =

δ(ca, [u1, u2]) and, by the unitary closure of T -prime varieties, we have

δ(a, [u1, u2]− [w1, w2]) ≡ 0.

It remains to observe that [x, y · 1] = [x, y]− [xy, 1].The equality δ([a1, a2], b) = 0 is proved likewise.Thus, we have defined a nonzero operator Trb(a) = δ(a, b) enjoying some useful properties of Tr; in

particular, Trb(a1a2) = Trb(a2a1). Furthermore, Lemma 15, Proposition 19, and Corollary 20 imply thefollowing assertion.

Proposition 26. Let M be a T -prime variety over a field of positive characteristic and let A bea relatively free infinitely generated algebra in M. Then h 6= 0 in A if and only if Try(ax) 6= 0 for somex and y.

Since Trb([c, d]) ≡ 0 and Trb(c) 6≡ 0, the polynomial

g = Trb(a)Cn =∑

i

Cn(~x,~y)|xi→axib

is a weak identity in a T -prime variety of rank n (this means that it satisfies Cn+1 but not Cn).Thus, we have established the existence of weak identities and so proved Theorem 5.Remark. Inspection of the arguments of this section shows that, to prove the existence of weak

identities, it suffices that the system Cn+1 be fulfilled and the nth power of the ideal generated by Cn

be nonzero. For the existence of a central polynomial, it is sufficient that this ideal be nonzero in thepower n2.

Therefore, the duality technique works for T -prime varieties in the case of positive characteristic.We now prove the stability (i.e., Theorem 9).

978

Proof. Recall that we only need to deal with the case of positive characteristic and the case inwhich some system of Capelli identities is fulfilled.

Suppose that g =∑

i cixdi = 0 in M with g linear in x. By Proposition 26, this is equivalent tothe fact that Trz(

∑i cixdiy) = 0 for all z. It suffices to prove the equality

∑i dizci = 0 for all z. Then,

specializing z in x and involving Proposition 26, we conclude that∑

i dixci = 0 and so the stabilitycondition is checked.

On the other hand, Trz(∑

i cix · diy) = Trz(∑

i diy · cix), whence the theorem follows.Theorem 9 and the nonvanishing of δ1 for T -prime varieties over a field of positive characteristic

imply the following assertion.

Corollary 27. If M is a T -prime variety then M⊗M is regular.

Proof. In the case of positive characteristic, the assertion follows from stability and existence ofa nonzero trace in the multiplication algebra L[A] ⊗ R[A] for a relatively free algebra A. Furthermore,L[A] ' A, R[A] ' Aop and, by stability, Aop ' A.

If the characteristic is zero then the assertion follows from the classification of T -prime varietiesbecause G⊗G = M1,1, Mp,q⊗Mr,s = Mpr+qs,ps+qr, Mpq⊗G = Mp+q⊗G, and Mn⊗G⊗Mm⊗G = Mnm,nm.

Remark. To define the trace independently of b, it suffices to prove the identity δ(a, b)δ(c, d) =δ(c, b)δ(a, d). However, in the case of the Grassmann algebra, a permutation of generators leads to mul-tiplying by −1 and this assertion does not hold. For the same reason, δ(a, b) 6≡ δ(b, a) in the Grassmannalgebra.

On the other hand, since Tr(L(a) ⊗ R(b)) = Tr(δ1(L(a) ⊗ 1 · 1 ⊗ R(b)) = Tr(δ1(L(a) ⊗ 1) Tr(1 ⊗R(b))− δ2(1⊗R(b)), L(a)⊗ 1), it follows that a and b are separated in the weak sense. They cannot beseparated in the strong sense, since otherwise the operators of interior trace would be nonzero, which isimpossible in the nonregular case.

In the nonregular case, we have Trb(Z) = 0 for every central element Z. Otherwise, one-sided tracescan be defined.

Examples. 1. Let char(F) > 2 and let G = G0 + G1 be an infinitely generated Grassmann algebrawith unity over F. Then the Capelli identity of order p+1 (but not of order p) is fulfilled in G. If we insertspecializations of variables in G0 and G1 into Cp then the result is nonzero if one of the variables has theeven specialization and p−1, uneven. It is easy to see that if a ∈ G0 or b ∈ G0 then Trb(a) = 0. If a, b ∈ G1

then Trb(a) is the operator of multiplication by 2ab. (If a, b ∈ G1 then in p − 1 cases axib = −abxi andin one case axib = −abxi for a variable xi with the even specialization ab ∈ G0 ⊂ Z(G).) It is immediatethat Trb(a) the operator of multiplication by the commutator [a, b].

2. Suppose that char(F) = 2 and G is the Grassmann algebra in characteristic 2, i.e., an infinitelygenerated algebra with unity, generators xi, and relations [xi, xj ] = εiεjxixj Here ε2i = 0, εi are centralvariables. It is readily checked that then G satisfies C5 but not C4. Furthermore, Trxi(xj) = εiεjR(xixj),where R(u) is the right multiplication by u. Thus, in this case we also see that Trb(a) is the operator ofmultiplication by [a, b].

3. Suppose that char(F) > 2 and M = Var(Mnk). Then M meets the Capelli identity of ordern2 + k2 + 2nk(p − 1) + 1 but not of order n2 + k2 + 2nk(p − 1) ≡ (n − k)2(mod p). In this case,Tr(1) = δ1(1) = (n − k)2 if it is defined internally in terms of permutations. If we normalize it bydividing it by the maximal order of Capelli polynomials not fulfilled in the algebra (see (2)) then we inferTr(1) = n− k, which we expected.

4. Suppose that char(F) > 2 and M = Var(Mn ⊗G). Then M satisfies the Capelli identity of ordern2p+ 1 but not of order n2p. In this case,

Trx(y) =∑

i

[xii, yii]

and Try(1) = Tr1(y) = 0.

979

It is known that, in a stable variety, the existence of central polynomials is equivalent to the existenceof weak identities [8]. This enables us to find the central polynomials explicitly.

Let Z =∑

i cixdi 6= 0 be a central polynomial, i.e., [Z, t] ≡ 0 or

0 ≡ Trb

(∑i

cixdity −∑

i

tcixdiy)

= Trb

(∑i

di[y, t]cix),

which is equivalent to the condition∑

i di[y, t]ci ≡ 0. In view of Proposition 26,

Trb(Zy) 6≡ 0, 0 6= Trb(Z) = Trb

(∑i

diycix)6≡ 0.

Again by Proposition 26, this inequality is equivalent to the condition∑

i diyci 6= 0. We have constructeda weak identity h =

∑i diyci.

It is possible to act in the opposite direction. If h =∑

i diyci 6= 0 but∑

i di[y, t]ci ≡ 0 then h =∑i cixdi is a central polynomial. Put h = Trb(y) ·Cn(~t,~z). Then the central polynomial h corresponding

to h can be written down explicitly:

h =∑σ∈Sn

∑σ(i)=k

(−1)σ(−1)kbtkzσ(k+1) . . . zσ(n)tnxt0zσ(1) . . . zσ(k−1)tk−1.

References

1. Dnestrovskaya Tetrad ′ [in Russian], Inst. Mat., Novosibirsk (1993).2. Kemer A. R., Nonmatrix Varieties, Varieties with Power Growth and Finitely Generated PI-Matrices, Dis. Kand.

Fiz.-Mat. Nauk, Novosibirsk (1981).3. Kemer A. R., Ideals of Identities of Associative Algebras [in Russian], Dis. Dokt. Fiz.-Mat. Nauk, Barnaul (1988).

English transl.: Ideals of Identities of Associative Algebras, Amer. Math. Soc., Providence RI (1991). (Transl. Math.Monogr.; 87.)

4. Kemer A. R., “On the multilinear components of the regular prime varieties,” in: Methods in Ring Theory, M. Dekker,New York, 1998, pp. 171–183. (Lecture Notes Pure Appl. Math.; 138).

5. Latyshev V. N., Nonmatrix Varieties of Associative Algebras [in Russian], Dis. Dokt. Fiz.-Mat. Nauk, Moscow (1977).6. Latyshev V. N., “On some varieties of associative algebras,” Izv. Akad. Nauk SSSR Ser. Mat., 37, No. 5, 1010–1037

(1973).7. Latyshev V. N., “Stable ideals of identities,” Algebra i Logika, 20, No. 5, 563–570, 600 (1981).8. Razmyslov Yu. P., Identities of Algebras and Their Representations [in Russian], Nauka, Moscow (1989).9. Razmyslov Yu. P., “On one Kaplansky’s problem,” Izv. Akad. Nauk SSSR Ser. Mat., 37, No. 3, 483–501 (1973).

10. Formanek E., “Central polynomials for matrix rings,” J. Algebra, 23, No. , 129–133 (1972).11. Okhitin S. V., “On stable T -ideals and central polynomials,” Vestnik MGU Ser. Mat. Mekh., No. 3, 85–89 (1986).12. Kemer A. R., “Multilinear identities of the algebras over a field of characteristic p,” Internat. J. Algebra Comput., 5,

No. 2, 189–197 (1995).13. Belov A. Ya., “On non-Specht varieties,” Fund. i Prikl. Mat., 5, No. 1, 47–66 (1999).14. Zubkov A. N., “A generalization of the Razmyslov–Procesi theorem,” Algebra i Logika, 35, No. 4, 433–457 (1996).15. Zubrilin K. A., “Algebras satisfying Capelli identities,” Mat. Sb., 186, No. 3, 53–64 (1995).16. Razmyslov Yu. P., “Algebras satisfying identity relations of Capelli type,” Izv. Akad. Nauk SSSR Ser. Mat., 45, No. 1,

143–166 (1981).17. Razmyslov Yu. P., “On the Jacobson radical in PI-algebras,” Algebra i Logika, 13, No. 3, 337–360 (1974).18. Razmyslov Yu. P., “Trace identities of full matrix algebras over a field of characteristic zero,” Izv. Akad. Nauk SSSR

Ser. Mat., 38, No. 4, 723–756 (1974).19. Braun A., “The radical in a finitely generated PI-algebra,” Bull. Amer. Math. Soc., 7, No. 2, 385–386 (1982).20. Procesi C., Rings with Polynomial Identities, Acad. Press, New York (1973).21. Rowen L. H., Polynomial Identities in Ring Theory, Acad. Press, New York (1980).22. Procesi C., “The invariant theory of n× n matrices,” Adv. Math., 19, No. 3, 306–381 (1976).23. Donkin S., “On tilting modules for algebraic groups,” Math. Z., 212, No. 1, 39–60 (1993).24. Donkin S., “Invariant functions on matrices,” Math. Proc. Cambridge Philos. Soc., 113, No. 1, 23–43 (1993).25. Donkin S., “Invariants of several matrices,” Invent. Math., 110, No. 2, 389–401 (1992).26. Zubrilin K. A, “On class of nilpotency of obstruction to embeddability of algebras satisfying Capelli identities,” Fund.

Prikl. Mat., 1, No. 2, 409–430 (1995).27. Kemer A. R., “Identities of finitely generated algebras over the infinite field,” Izv. Akad. Nauk SSSR Ser. Mat., 54,

No. 4, 726–753 (1990).28. Belov A. Ya., Algebras with Polynomial Identities: Representations and Combinatorial Methods [in Russian], Dis. Dokt.

Fiz.-Mat. Nauk, Moscow (2002).

980