On Herstein's Lie map conjectures, I

AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 10, Pages 4235–4260 S 0002-9947(01)02731-3

Article electronically published on June 6, 2001

ON HERSTEIN’S LIE MAP CONJECTURES, I

K. I. BEIDAR, M. BREˇ SAR, M. A. CHEBOTAR, AND W. S. MARTINDALE III

Abstract. We describe surjective Lie homomorphisms from Lie ideals of skew elements of algebras with involution onto noncentral Lie ideals (factored by their centers) of skew elements of prime algebras D with involution, provided that char( D) 6= 2 and D is not PI of low degree. This solves the last remaining open problem of Herstein on Lie isomorphisms module cases of PI rings of low degree. A more general problem on maps preserving any polynomial is also discussed.

1. Introduction

Let F be an associative commutative ring with 1 and B an associative F-algebra.

It is well–known that defining new products in B, the Lie product and the Jordan product, by

[x, y] = xy − yx and x ◦ y = xy + yx,

B becomes a Lie and a Jordan algebra, respectively. A Lie (Jordan) subalgebra of B is any F-submodule closed under the Lie (Jordan) product. A submod- ule T of B satisfying a stronger condition [T , B] ⊆ T is called a Lie ideal of B. Given any subset R of B, we define the center Z(R) of R to be the set {r ∈ R | [x, r] = 0 for all x ∈ R}. Next, we denote by [R, R] the F-submodule of B generated by {[x, y] | x, y ∈ R}. Note that Z = Z(B) and [B, B] are Lie ideals of B.

Now assume that B is an algebra with involution ∗. Let

S(B) = {x ∈ B | x ^∗ = x} and K = K(B) = {x ∈ B | x ^∗ = −x}

be the set of symmetric and skew elements in B, respectively. Clearly, K is a Lie subalgebra and S(B) is a Jordan subalgebra of B. A Lie ideal of K is, of course, an F-submodule T of K satisfying [T , K] ⊆ T . For instance, [K, K] is a Lie ideal of K.

The study of the relationship between the associative and the Lie and Jordan structure of an associative ring B was initiated in the 1950’s by Herstein (see, e.g., [38, 39]). Introducing elementary and clever new methods (which, incidentally, are still reflected even in the present work) he obtained, in particular, rather definitive results concerning the Lie and Jordan ideal structure in the case where B was a simple ring (with or without involution), thereby extending classical results of Car- tan and Killing on simple finite–dimensional complex Lie algebras. Some related natural questions, however, remained unsolved, and among them, notably, all the basic questions on Lie isomorphisms. Given Lie subalgebras T and U of associative

Received by the editors October 6, 1999 and, in revised form, June 1, 2000.

1991 Mathematics Subject Classification. Primary 16W10, 16W20, 16R50.

The second author was partially supported by a grant from the Ministry of Science of Slovenia.

2001 American Mathematical Societyc

4235

algebras, one defines a Lie homomorphism of T into U to be an F-module map α : T → U satisfying [x, y] ^α = [x ^α , y ^α ] for all x, y ∈ T (we shall always write Lie maps as exponents). For example, an isomorphism or a negative of an anti- isomorphism of one algebra onto another is also a Lie isomorphism. One can ask whether the converse is true in some special cases. That is, does every Lie isomor- phism of certain Lie subalgebras of associative algebras arise (modulo maps whose range is central) from (anti-)isomorphisms? So, in particular, this question asks whether the algebras should be (anti-)isomorphic provided that some of their Lie subalgebras are isomorphic as Lie algebras. At his 1961 AMS Hour Talk [38, pp.

528–529], Herstein conjectured that this should hold true for Lie isomorphisms of B, [B, B], [B, B]/[B, B] ∩ Z and, in the case where B was a ring with involution, Lie isomorphisms of K, [K, K], and [K, K]/[K, K] ∩ Z, for the case when B was, neglect- ing some low dimensional counterexamples, an arbitrary simple ring. He also posed similar problems for another Lie type map, namely, for Lie derivations. The cases of [ B, B]/[B, B] ∩Z and [K, K]/[K, K] ∩Z seem to be of special interest since these two Lie rings are, except in some very special situations, simple [38, Theorems 4 and 10]

(we remark here that K can differ from [K, K] even when every element of Z is left fixed by the involution [48]). Note that the presence of anti-isomorphisms can be avoided in results on Lie isomorphisms of skew elements—namely, if β were a neg- ative of an anti-isomorphism of a ring B with involution, then the map a 7→ −(a ^∗ ) ^β would be a homomorphism which coincides on K with β.

The resolution of Herstein’s Lie map problems in the classical case when B = M n (F ), F a field, has been well–known for a long time (see, e.g., [43, Chapter 10]). In 1951 Hua [42] described Lie automorphisms of a simple Artinian ring B = M n (D), D a division ring, n ≥ 3. Later on Martindale, a student of Herstein, together with some of his students considered Herstein’s problems in a series of papers [41, 50, 51, 53, 54, 55, 56, 57, 67]. Basically, the problems have been solved provided that the rings contain certain nontrivial idempotents. Also, the treatment of the problems has been extended from simple to prime rings (we mention, as a curiosity, that the utility of the concept of the extended centroid of a prime ring was realized for the first time when treating Lie maps of prime rings). Lie map problems have also been considered in operator algebras [2, 3, 4, 5, 36, 62, 63, 64]

and the techniques there also rest heavily on the presence of idempotents. The question whether the results on Lie maps can be obtained in rings containing no nontrivial idempotents has been open for a long time.

It seems appropriate at this point to say a few words about the analogous Jor-

dan map project. Generally speaking, the Jordan case is usually easier than the

Lie case; in Herstein’s words [38, p. 518]: “... in general, in considering such ques-

tions as we shall, be they about the appropriate ideal structure, homomorphisms,

derivatives, the Jordan situation is much easier to study than the corresponding

Lie one. One reason for this is that in the Lie case the center of R constantly gets

in our way, so much so, that many questions, completely answered for the Jordan

case are virtually untouched in their Lie analogs.” The definition of a Jordan homo-

morphism is analogous to that of a Lie homomorphism. Jordan homomorphisms of

an (associative) ring onto a prime (associative) ring were characterized already by

Herstein [37]. In 1967, Martindale described Jordan homomorphisms of a Jordan

ring of symmetric elements in a ring with involution, containing some nontrivial

orthogonal idempotents [52] (see also an extension of Jacobson [44]). Though the

Jordan case is supposed to be easier, the proofs of the results replacing the condi- tion on idempotents by more intrinsic conditions [58, 60] are quite involved and are based on Zelmanov’s path–breaking work on Jordan algebras [71].

The first idempotent free result on Lie maps was obtained in 1993 by Breˇsar [27]. Under some mild technical assumptions (which were removed somewhat later [22, 33]), he described the form of a Lie isomorphism between arbitrary prime rings.

This was also the first paper based on applications of the theory of functional iden- tities. The main idea of the proof can be easily described. Every Lie isomorphism α of an associative ring B clearly satisfies [x ^α , (x ² ) ^α ] = 0 for every x ∈ B. This identity can be viewed, at least from the present point of view, as a rather simple functional identity (see Section 2). Under suitable assumptions, one can obtain all possible solutions of functional identities. Hence one obtains the form of (x ² ) ^α . Knowing how α acts on squares, and hence on the Jordan product, and at the same time knowing, by the very definition of the Lie map, how α acts on the Lie product, it is then easy to describe the action of α on the initial associative product. The same idea also works in the semiprime case [6]. In 1994, Beidar, Martindale and Mikhalev described Lie isomorphisms of K = K(B), where B was a prime ring with char(B) 6= 2, 3 and with involution of the first kind [17] (see also [18, Chapter 9]).

This proof is considerably more difficult, but its main idea is essentially the same:

every Lie isomorphism α of K gives rise to a functional identity [x ^α , (x ³ ) ^α ] = 0 for all x ∈ K (here, the fact that the cube of a skew element is skew again was used).

The case of char(B) = 3 was investigated by Chebotar [35] who also obtained a considerably shorter proof of the main result of [17]. A consideration of Lie isomor- phisms of Lie ideals of B and K (in particular, [B, B] and [K, K]) is more entangled since there is no such obvious way of how to arrive at appropriate functional identi- ties (in particular, these Lie rings are not closed under any powers). Nevertheless, functional identities can be produced in these cases as well.

Though in the Jordan case surjective homomorphisms were described some time

ago, in the case of Lie maps only the isomorphisms have been investigated. The

first results on surjective Lie maps were obtained recently by Beidar and Chebotar

[12] who described Lie homomorphisms of Lie ideals of algebras onto noncentral

Lie ideals of prime algebras (factored by their centers). As a crucial tool in this

investigation, in [10, 11] the useful concept of a d-free subset of a ring was introduced

which, in particular, allows for a unified approach to a variety of mapping problems

involving different subsets of rings. Very roughly speaking, a subset of a ring is

d-free if every appropriate functional identity in less than d variables has only

the “standard” (i.e., “obvious”) solution. In [10, 11, 12] many basic results were

proved for d-free subsets of rings. These results can be applied to rings, Lie ideals

of rings, skew elements of rings with involution, etc. as soon as the d-freeness of

these subsets has been established. Since the d-freeness of noncentral Lie ideals of

the skew elements of a prime ring with involution was proved in [9], we shall apply

a number of these basic results in the present paper, enabling us in particular

to prove our key result, Theorem 3.5. In a subsequent paper and part II of the

present paper, we shall characterize Lie derivations by reducing the problem to Lie

isomorphisms onto a d-free set, which further illustrates the usefulness of the d-free

concept. We remark that one can usually establish the d-freeness of appropriate

subsets of the ring in question unless the ring is PI of low degree. Therefore it will

turn out that in order to obtain complete solutions of Herstein’s Lie isomorphism

problems classical structure theory (for the PI case) seem to be required along with the d-free methods (for the non–PI case).

As indicated earlier there has been over the years a long series of papers settling Herstein’s Lie map conjectures in a variety of special situations. The main goal of the present paper is to give solutions to all Lie isomorphism problems in rings with involution modulo cases when rings satisfy PI of low degree. In particular, we settle the most difficult of his conjectures, the one involving Lie isomorphisms between Lie rings of the form [K, K]/[K, K] ∩ Z, K the skew elements of a prime ring with involution. In Part III of the present paper, using structure theory of PI rings, we remove these restrictions on degree of PI in both Parts I and II of the present paper and in [12], and thereby solve Herstein’s problem in full generality.

We mention here several recent papers [19, 20, 21, 23, 24, 46, 61, 65, 69, 70]

which are connected in some way with our present work.

The paper is organized as follows. In Section 2 we briefly survey basic concepts and some results on functional identities, thereby providing a necessary tool for at- tacking the Lie map problem. The theory of functional identities has been developed recently and much of this development has been motivated by the Lie map problems;

we refer to [26, 28, 29, 31, 49] for some initial results and to [7, 8, 9, 10, 11, 16, 30, 34]

for the more advanced results. In Section 2 we shall basically present only some ex- tractions from the recent papers [9, 10, 11]. The main reason for including Section 2 in the paper is to make the paper readable and as self-contained as possible.

The body of the paper is Section 3 which treats Lie maps with d-free range.

As a matter of fact, for reasons that become clear in Section 4 and for most of Section 3 we consider not only Lie maps but maps α satisfying a somewhat more general condition [x, y] ^α = λ[x ^α , y ^α ], where λ is a nonzero central element. In our most general result on Lie maps, Theorem 3.5, there is no restriction on the nature of the rings involved, the only essential restriction being that the range of the Lie homomorphism is d-free, specifically d = 9. The d-freeness condition seems to be the “proper condition” when considering such problems, and the class of rings satisfying this condition certainly also includes various nonprime rings.

Nevertheless, because of historic reasons we are primarily interested in prime (and simple) rings (with involution). As an application of the d-free approach we shall then obtain several results which settle Herstein’s Lie isomorphism conjectures for these rings. We are presently going to state these results, the proofs of which are given in Section 3. But first we have to set some notation in place.

In what follows, F is a commutative ring with 1. Given a nonempty subset T of an F-algebra A, we denote by hT i the subalgebra of A generated by T . Next, by B we denote a prime F-algebra with maximal right (left) ring of quotients Q mr = Q mr (B) (respectively Q ml = Q ml (B)) and Martindale extended centroid C = C(B) (see [18, Chapter 2]). We let Q be either Q mr or Q ml . It is well-known that both Q and C are F-algebras and B is a subalgebra of Q. We recall that an involution ∗ on B is said to be of the first kind if ∗ acts as the identity on C;

otherwise, ∗ is of the second kind.

Let x ∈ Q. By deg(x) we shall mean the degree of x over C (if x is algebraic over C) or ∞ (if x is not algebraic over C). Given a nonempty subset R ⊆ Q, we set

deg(R) = sup{deg(x) | x ∈ R}.

If B is prime and deg(B) = n < ∞, then it follows from results of the theory of rings

with polynomial identities [66, 68] that the ring B is isomorphic to a subring D of

the ring of n ×n matrices over C, the algebraic closure of C, such that DC = M n (C).

This is also equivalent to the condition that BC is of dimension n ² over C, as well as to the condition that B satisfies the standard polynomial identity of degree 2n.

The next theorem must be regarded as the principal result of this paper since it constitutes the breakthrough by which all of Herstein’s Lie isomorphism problems for simple and prime rings with involution can be solved (see [38, p. 529, problem 5]).

Theorem 1.1. Let A be an F-algebra with involution, let L be the skew elements of A, and let S be a Lie ideal of L. Let B be a prime F-algebra with involution, let K be the skew elements of B, let R be a noncentral Lie ideal of K, and set R = R/R ∩ C. Further, let α : S → R be a surjective Lie homomorphism. Suppose that deg( B) > 20 and char(B) 6= 2. Then there exists an algebra homomorphism ψ : hSi → hRiC + C such that x ^ψ = x ^α for all x ∈ S. Moreover, if the involution of B is of the first kind, then hSi ^ψ = hRi.

In essence Theorem 1.1 reduces Lie isomorphism problems for prime rings with involution to the case where the ring B is PI of “low” degree. This latter situation is handled separately in [25, 56]. Combining Theorem 1.1 with results from [25, 56]

in Part III of the present paper we shall obtain the the ultimate result that all of Herstein’s Lie isomorphism conjectures for prime rings with involution are finally solved in full generality. Special cases of these conjectures, where, among other assumptions, the existence of nontrivial idempotents was required, were considered by Martindale [56] and Rosen [67], as well as, quite recently, by Ayupov and Azamov [3, 4] who treated Lie isomorphisms of [K(B), K(B)], B being a real factor (and hence a prime real von Neumann algebra). As a corollary to Theorem 1.1 and [17, 25, 35]

we have the following result which solves Herstein’s Lie isomorphism conjectures for simple rings.

Theorem 1.2. Let A be a simple F-algebra with involution and with extended centroid T , let L be the skew elements of A, let S = [L, L] and let S = S/(S∩Z(A)).

Next, let B be a simple F-algebra with involution and with extended centroid C, let K be the skew elements of B, let R = [K, K], and let R = R/(R ∩ Z(B)). Further, let α : S → R be an isomorphism of Lie algebras. Suppose that char(F) 6= 2 and one of the following conditions is fulfilled:

(a) Both involutions are of the first kind and dim _C ( B) 6= 1, 4, 9, 16, 25, 64.

(b) Both involutions are of the second kind and dim _C (B) > 4 (if char(F) 6= 3) or dim _C (B) > 9 (if char(F) = 3).

(c) dim _C (B) > 400.

Then there exists an isomorphism φ : A → B of F-algebras such that x ^α = x ^φ for all x ∈ S.

Theorem 1.3. Let B be a simple F-algebra with involution ∗, with centroid C, and with skew elements K. Further, let R = [K, K], let R = R/(R ∩ Z(B)), let Aur _F (R) and Aut _F (R) be the groups of automorphisms of the Lie algebras R and R respectively, and let

H = {σ ∈ Aut F ( B) | (x ^∗ ) ^σ = (x ^σ ) ^∗ for all x ∈ B}.

Suppose that char(F) 6= 2 and one of the following conditions is fulfilled:

(a) The involution ∗ is of the first kind and dim C (B) 6= 1, 4, 9, 16, 25, 64.

(b) The involution ∗ is of the second kind and dim _C (B) > 4 (if char(F) 6= 3) or dim _C (B) > 9 (if char(F) = 3).

Then H ∼ = Aut _F (R) ∼ = Aut _F (R).

The purpose of our next theorem is to show that in certain situations (that is, those cases not covered in Theorem 1.1) Lie isomorphisms cannot exist in the first place (except for several low-dimensional examples).

Theorem 1.4. Let B be a prime ring with involution ∗ with skew elements K and with Martindale centroid C. Further, let R be a noncentral Lie ideal of R. Suppose that char(B) 6= 2 and deg(B) > 20. Then the Lie ring R = R/(R ∩ C) is not a homomorphic image of a Lie ideal of any ring.

The case R = K of the above theorem was treated in [12, Theorem 1.6]. Further, one can describe a Lie (Jordan) homomorphism as a map preserving the polynomial xy − yx (xy + yx, respectively). In Section 5 we consider a more general problem of characterizing maps that preserve arbitrary polynomials. Some special cases were considered already by Kaplansky [47] (the polynomial xyx), Jacobson and Rickart [45] (the polynomial [[x, y], z]) and Herstein [38] (the polynomial x ⁿ ). The approach with functional identities has already been proved to be efficient in this problem [13, 14, 27, 32]. In this paper we prove

Theorem 1.5. Let B be prime ring with char(B) 6= 2, with involution ∗ and with extended centroid C. Let R be a noncentral Lie ideal of the Lie ring K(B), let F be a subring of S(C) such that ¹ ₂ ∈ F, FR = R and FB = B. Let A be an F- algebra with involution, let S be a Lie ideal of the Lie F-algebra K(A), let 0 6=

f (x 1 , x 2 , . . . , x m ) ∈ FhX i be a multilinear polynomial in x 1 , x 2 , . . . , x m , m ≥ 2, such that f (s 1 , s 2 , . . . , s m ) ∈ S for all s i ∈ S and let α : S → R be an epimorphism of F-modules such that

f (s 1 , s 2 , . . . , s m ) ^α = f (s ^α ₁ , s ^α ₂ , . . . , s ^α _m ) for all s 1 , s 2 , . . . , s m ∈ S.

Suppose that deg( B) > max{4m + 2, 20} and one of the following conditions is fulfilled:

(a) f (x 1 , x 2 , . . . , x m ) is a Lie polynomial;

(b) α is an isomorphism of F-modules and A is a prime ring with deg(A) >

max {2m + 2, 8};

(c) [ S, S] = S.

Then there exist ζ ∈ C, an F-linear map µ : hSi → C and a homomorphism of F-algebras β : hSi → hRiC + C such that ζ ^{m −1} = 1,

x ^α = ζx ^β + µ(x) for all x ∈ S

and µ(f (s 1 , s 2 , . . . , s m )) = 0 for all s i ∈ S. Moreover, if f x

, the partial derivative of f at x i , is nonzero for some 1 ≤ i ≤ m, then µ = 0 and hSi ^φ = hRi.

As a corollary to Theorem 1.5 we will obtain a solution to Herstein’s problem on additive automorphisms of K preserving the polynomial x ²ⁿ⁺¹ [38, Problem 1, p. 528]:

Theorem 1.6. Let B be a prime F-algebra with involution and with skew elements

K, let C be the Martindale centroid of B and let m ≥ 3 be an odd integer. Further,

let A be an F-algebra with involution and with skew elements L and let α : L → K

be a surjective F-module map such that (x ^m ) ^α = (x ^α ) ^m for all x ∈ L. Suppose that

char(B) 6= 2 and deg(B) > max{4m + 2, 20}. Then there exist a homomorphism of F-algebras β : hLi → hKiC + C, an F-linear map µ : hLi → C and an element ζ ∈ C such that ζ ^{m −1} = 1 and x ^α = ζx ^β + µ(x) for all x ∈ L. Furthermore, if char(B) = 0 or char(B) = p > 0 and m is not a power of p, then µ = 0.

2. Functional identities and d-free sets

First we fix the notation. Throughout the section, F will be a commutative ring with 1, Q will be an F-algebra with 1 and C will be its center. Later on we shall be primarily interested in the case when Q is the maximal (right or left) ring of quotients of a prime algebra B, and hence C is the Martindale centroid of B.

However, until further notice, Q may be an arbitrary algebra. By S we denote an arbitrary set, and by R we denote a subset of Q. Let N ^∗ be the set of all positive integers and for n ∈ N ^∗ we let S ⁿ denote the nth Cartesian power of S.

Let m ∈ N ^∗ and E : R ^{m −1} → Q, p : R ^{m −2} → Q be arbitrary maps. In the case when m = 1 it should be understood that E is an element in Q and p = 0. Let 1 ≤ i < j ≤ m, and define E ⁱ , p ^ij , p ^ji : R ^m → Q by

E ⁱ (x m ) = E(x 1 , . . . , x i −1 , x i+1 , . . . , x m ) and

p ^ij (x m ) = p ^ji (x m ) = p(x 1 , . . . , x i −1 , x i+1 , . . . , x j −1 , x j+1 , . . . , x m ).

Here, x m stands for (x 1 , . . . , x m ).

Now let I, J ⊆ {1, 2, . . . , m}, and for each i ∈ I, j ∈ J , let E i , F j : R ^{m −1} → Q be arbitrary maps. The basic functional identities are

X

i ∈I

E ⁱ _i (x m )x i + X

j ∈J

x j F _j ^j (x m ) = 0 for all x m ∈ R ^m , (1)

and a slightly more general one, X

i ∈I

E _i ⁱ (x m )x i + X

j ∈J

x j F _j ^j (x m ) ∈ C for all x m ∈ R ^m . (2)

The goal in the theory of functional identities is to describe the form of the maps appearing in the identity. A natural possibility when (1) (and hence also (2)) is fulfilled is when there exist maps p ij : R ^{m −2} → Q, i ∈ I, j ∈ J , i 6= j, λ k : R ^{m −1} → C, k ∈ I ∪ J , such that

E _i ⁱ (x m ) = X

j∈J , j6=i

x j p ^ij _ij (x m ) + λ ⁱ _i (x m ),

F _j ^j (x m ) = − X

i∈I,i6=j

p ^ij _ij (x m )x i − λ ^j j (x m ) and

λ k = 0 if k 6∈ I ∩ J , (3)

for all x m ∈ R ^m , i ∈ I, j ∈ J . Indeed, one can readily check that (3) implies (1). We shall refer to (3) as a standard solution of (1) (and of (2)). It should be pointed out that the case when one of the sets I or J is empty is not excluded. The sum over the empty set of indices should be simply read as zero. This means that, for example, a standard solution of the functional identity P

i ∈I E _i ⁱ (x m )x i = 0 is E i = 0, i ∈ I.

It turns out that often a standard solution is also the only possible solution.

This is the reason for introducing the following fundamental concept [10].

Definition 2.1. A nonempty subset R ⊆ Q is said to be d-free, where d ∈ N ^∗ , if for all m ∈ N ^∗ and I, J ⊆ {1, 2, . . . , m} the following two conditions are satisfied:

(a) If max{|I|, |J |} ≤ d, (1) implies (3).

(b) If max{|I|, |J |}| ≤ d − 1, (2) implies (3).

Thus, in a loose manner one can say that d-free sets are those subsets R of Q that any functional identity on R in “not too many” variables has only a standard solution. Incidentally, it is easy to see that this standard solution is unique [10, Remark 2.5].

The basic result justifying the introduction of the concept of d-free sets states that a prime ring B with deg(B) ≥ d is a d-free subset of Q, where Q is any of the rings Q mr (B) or Q ml (B) (this is explicitly stated in [10], but essentially already proved in [7] and [16]). For this paper, however, the following deeper result is needed.

Theorem 2.2 ([9, Theorem 1.1]). Let B be a prime ring with involution and let K be the set of skew elements in B. Let Q be either Q mr (B) or Q ml (B). If char(B) 6= 2 and deg(B) ≥ 2d + 3, then any noncentral Lie ideal R of K is a d-free subset of Q.

This theorem makes it possible for us to apply to noncentral Lie ideals of K all the results on d-free subsets obtained in [10, 11]. We will now review some of those results that will be needed in the following sections. The first one will be used frequently.

Theorem 2.3 ([10, Theorem 2.8]). Let D ⊆ R ⊆ Q be nonempty subsets and d ∈ N ^∗ . Suppose that D is d-free. Then R is d-free also.

The concept of a functional identity can be viewed as a generalization of the concept of a polynomial identity. Indeed, assuming that all the maps E i and F j

are “monomials” λx i

. . . x i

, the functional identity (1) reduces to a polyno- mial identity. However, the theory of functional identities can be considered, at this point, more as a complement to that of polynomial identities, rather than its extension. Namely, one can obtain definite results concerning functional iden- tities on d-free sets, which are, in a way, in striking contrast to the sets sat- isfying polynomial identities. Namely, assume that a nonzero multilinear poly- nomial f (X 1 , . . . , X m ) ∈ ChX i, the free algebra over C on the set of noncom- muting variables X = {X 1 , X 2 , . . . }, is a polynomial identity on a set R ⊆ Q.

That is, f (x 1 , . . . , x m ) = 0 for all x 1 , . . . , x m ∈ R. But this can be written as P

i ∈I E _i ⁱ (x m )x i = 0, where each E i ∈ ChX i is a multilinear polynomial in m − 1 variables. Thus we have arrived at a functional identity with a nonstandard solu- tion, unless R satisfies a polynomial identity of a smaller degree. But in the latter case we can repeat the argument just given and so, after a finite number of steps, we certainly get a functional identity in ≤ m variables with a nonstandard solution.

For future reference we record this observation as

Remark 2.4. Let f (X 1 , . . . , X m ) ∈ ChX i be a nonzero multilinear polynomial. If R is a d-free subset of Q with d ≥ m, then f(X 1 , . . . , X m ) is not a polynomial identity on R.

Moreover, from [11, Corollary 2.4] we deduce at once the following somewhat

less obvious fact.

Remark 2.5. Let f (X 1 , . . . , X m ) ∈ ChX i be a nonzero multilinear polynomial at least one of whose coefficients is invertible and let a ∈ Q. If R is a d-free subset of Q with d ≥ m − 1, then f(x 1 , . . . , x m −1 , a) = 0 for all x 1 , . . . , x m −1 ∈ R implies a ∈ C.

As a matter of fact, functional identities that will appear in subsequent sections are not exactly of the same type as (1) and (2). They will involve expressions of the form

π(x m ) = X

i ∈I

E _i ⁱ (x m )x ^α _i + X

j ∈J

x ^α _j F _j ^j (x m ),

where α is a map from an arbitrary set S into Q and E i , F j : S ^{m −1} → Q. Such functional identities may look more complicated than (1) and (2), but fortunately this is only apparently so. One defines standard solutions in this setting in a self–

explanatory way (just replace x i by x ^α _i on appropriate places in (3)). Now assume that the range of α, R = S ^α , is a d-free subset of Q. Then [10, Theorem 2.6]

tells us that if max{|I|, |J |} ≤ d, then the functional identity π(x m ) = 0 for all x m ∈ S ^m has only the standard solution. The same is true for the functional identity π(x m ) ∈ C for all x m ∈ S ^m , provided that max {|I|, |J |}| ≤ d − 1.

When dealing with functional identities, the so–called quasi-polynomials natu- rally get in our way. Let us give a definition in a somewhat loose manner (see [11] for details). A map E : S ² → Q is said to be a multilinear quasi-polynomial (with respect to a map α : S → Q) of degree 2 if there exist λ 1 , λ 2 ∈ C and maps µ 1 , µ 2 : S → C, ν : S ² → C such that

E(x, y) = λ 1 x ^α y ^α + λ 2 y ^α x ^α + µ 1 (x)y ^α + µ 2 (y)x ^α + ν(x, y) for all x, y ∈ S, and at least one of what we shall call the coeffients of E, that is, λ 1 , λ 2 , µ 1 , µ 2 , ν, is nonzero (map or element). A multilinear quasi-polynomial of degree m would, of course, involve summands such as λx ^α 1 . . . x ^α m , µ(x 1 )x ^α 2 . . . x ^α m , ν(x 1 , x 2 )x ^α 3 . . . x ^α m

etc. Adapting the argument establishing Remark 2.4 one easily proves the following statement (which in turn is a special case of [11, Theorem 1.1]).

Lemma 2.6. Let E : S ^m → Q be a multilinear quasi-polynomial (with respect to α) of degree ≤ m. If E(x m ) = 0 for all x m ∈ S ^m and R = S ^α is an (m + 1)-free subset of Q, then all the coefficients of E are zero.

Occasionally we shall have to use a variant of Lemma 2.6 for quasi-polynomials which are not multilinear. Instead of giving a rigorous statement [11, Corollary 2.12] we rather explain what we have in mind by a simple example. Suppose that λ(x ^α ) ² y ^α x ^α +µ(x, x)y ^α x ^α +ν(x, x, x, y) = 0 for all x, y ∈ S, where the coefficients µ and ν are multi-additive maps into C and λ ∈ C. A complete linearization then gives a “multilinear” version of this identity (consisting of 18 terms such as λx ^α 1 x ^α 2 yx ^α 3 , µ(x 1 , x 2 )y ^α x ^α 3 , ν(x 1 , x 2 , x 3 , y) etc.). Now, Lemma 2.6 tells us that if R = S ^α is 5-free, then λ = 0, µ(x 1 , x 2 ) + µ(x 2 , x 1 ) = 0 and P

π ∈S

ν(x π(1) , x π(2) , x π(3) , y) = 0.

Consequently, λ = 2µ(x, x) = 6ν(x, x, x, y) = 0 for all x ∈ S.

Functional identities involving only one map deserve special attention. We shall

need the following result on such identities which is a special case of [11, Theorem

1.2].

Theorem 2.7. Let S be an F-module, let α : S → Q be an F-module map and let E : S ^{m −1} → Q be an F-multilinear map such that

X m i=1

λ i E ⁱ (x m )x ^α _i + µ i x ^α _i E ⁱ (x m ) ∈ C for all x m ∈ S,

where λ i , µ i ∈ C and at least one of them is invertible. If R = S ^α is an (m + 1)-free subset of Q, then E is a multilinear quasi-polynomial of degree ≤ m and each of its coefficient is F-multilinear.

3. Lie homomorphisms and d-freeness

Throughout this section, F will be a commutative ring with 1, Q will be an F-algebra with 1 and with center C, and R will be a subset of Q. Further, we set Q = Q/C and R = R/(R ∩ C). We identify R with (R + C)/C ⊆ Q and remark that Q is the factor Lie algebra of Q by the Lie ideal C.

Our goal is to prove the main result of this section, Theorem 3.5. This will be done in a series of steps. We begin by stating a result from [11].

Theorem 3.1 ([11, Theorem 2.11]). Let U be a Lie subalgebra of an F-algebra A, let α : U → Q be an F-module map and let B : U ^m → Q, m ≥ 2, be any F- multilinear map such that

(a) P n

i=1 B(x i , . . . , x m , x 1 , . . . , x i −1 ) ∈ C for all x 1 , . . . , x m ∈ U,

(b) B(x m −1 , [u, v]) − λ[B(x m −1 , u), v ^α ] − λ[u ^α , B(x m −1 , v)] ∈ C for all x m −1 ∈ U ^{m −1} and u, v ∈ U,

(c) [u, v] ^α − λ[u ^α , v ^α ] ∈ C for all u, v ∈ U,

where λ ∈ C is an invertible element. If R = U ^α is a (2m + 1)-free subset of Q, then there exist a multilinear quasi-polynomial q : U ^{m −1} → Q (with respect to α) of degree ≤ m − 1 and a map µ : U ^m → C such that B(x m ) = [q(x m −1 ), x ^α _m ] + µ(x m ) for all x m ∈ U ^m . Moreover, µ and all the coefficients of q are F-multilinear.

The conditions of this theorem might appear somewhat strange, but actually we shall arrive at them in the proof of the next lemma. This lemma reduces the general case of the Lie homomorphism problem to the case when the domain of the map is closed under the Jordan triple product xyz + zyx (using terms of the classical situation, the case of [K, K] shall be reduced to the case of K).

Lemma 3.2. Let A be an F-algebra, D a Lie subalgebra of A such that xyz+zyx ∈ D for all x, y, z ∈ D and S a Lie ideal of D. Further, let R be a submodule of the F-algebra Q and α : S → Q an F-module map such that S ^α = R and

[x, y] ^α = λ[x ^α , y ^α ] for all x, y ∈ S,

where λ ∈ C is some fixed invertible element. Suppose that R is a 9-free subset of Q and C is a direct summand of the C-module Q. Then there exists a pair (U; γ) such that

(a) U is a Lie ideal of the Lie F-algebra D containing S, (b) γ : U → Q is an F-module map,

(c) [x, y] ^γ = λ[x ^γ , y ^γ ] for all x, y ∈ U, (d) x ^γ = x ^α for all x ∈ S,

(e) xyz + zyx ∈ U for all x, y, z ∈ U.

Proof. We begin with a simple observation. Let V be a submodule of Q containing R. Then V is 9-free (and hence 2-free) by Theorem 2.3. Let c ∈ Z(V). Then cx − xc = 0, x ∈ V, and so c ∈ C by Remark 2.5. Therefore Z(V) = C ∩ V.

Now consider the class Ω of all pairs (U ⁰ ; γ ⁰ ) satisfying (a)–(d). Given (U ⁰ ; γ ⁰ ), (U ⁰⁰ ; γ ⁰⁰ ) ∈ Ω, we shall write (U ⁰ ; γ ⁰ ) ≥ (U ⁰⁰ ; γ ⁰⁰ ) provided U ⁰ ⊇ U ⁰⁰ and x ^γ

= x ^γ

for all x ∈ U ⁰⁰ . By Zorn’s lemma the class Ω has a maximal element, say (U; γ). It remains to show that condition (e) is satisfied. To this end, we set U to be equal to the F-submodule of A generated by the subset {x, xyz + zyx | x, y, z ∈ U}. Since U ⊆ D, U ⊆ D also. Given t ∈ D and x, y, z ∈ U, we have

[xyz + zyx, t] = [x, t]yz + zy[x, t] + x[y, t]z + z[y, t]x + xy[z, t] + [z, t]yx ∈ U, showing that U is a Lie ideal of D containing U (and so containing S). Now our goal is to extend γ to an F-module map U → Q satisfying (c). Then the maximality of ( U; γ) will imply that U = U and the proof will be completed.

Choose a C-submodule W of Q such that Q = W ⊕ C. Since F · 1 ⊆ C, both C and W are F-submodules of Q. Clearly F-modules W and Q are isomorphic and so we may assume, without loss of generality, that γ : U → W. Let π be the canonical projection of the module Q onto W. Then

[x, y] ^γ = λ[x ^γ , y ^γ ] ^π for all x, y ∈ U.

Therefore

[x, y] ^γ = λ[x ^γ , y ^γ ] + (x, y) for all x, y ∈ U where (x, y) ∈ C.

Since we do not know whether U ^γ is 9-free, we do the following. Set A ⁰ = A⊕ C, D ⁰ = D ⊕ C ⊆ A ⁰ and U ⁰ = U ⊕ C ⊆ D ⁰ . Clearly A ⁰ is an F-algebra, D ⁰ is a Lie subalgebra of A ⁰ such that xyz + zyx ∈ D ⁰ for all x, y, z ∈ D ⁰ and U ⁰ is a Lie ideal of D ⁰ . We identify A and C with corresponding subalgebras of A ⁰ . Further, set V = U ^γ ⊕ C ⊆ Q. It follows from (d) that U ^γ ⊇ R = R ^π and so V ⊇ R. By the above observation Z(V) = C ∩ V = C.

Finally, we define maps γ ⁰ : U ⁰ → V and  ⁰ : U ⁰ × U ⁰ → C by the rule (x + a) ^γ

= x ^γ + a for all x ∈ U, a ∈ C and

 ⁰ (x + a, y + b) = (x, y) for all x, y ∈ U, a, b ∈ C.

Clearly x ^γ

= x ^γ for all x ∈ U, (U ⁰ ) ^γ

= V and so (U ⁰ ) ^γ

is a 9-free subset of Q.

We have

[x + a, y + b] ^γ

= [x, y] ^γ = λ[x ^γ , y ^γ ] ^π = λ[x ^γ , y ^γ ] + (x, y)

= λ[(x + a) ^γ

, (y + b) ^γ

] +  ⁰ (x + a, y + b) for all x, y ∈ U, a, b ∈ C and so

[u, v] ^γ

= λ[u ^γ

, v ^γ

] ^π = λ[u ^γ

, v ^γ

] +  ⁰ (u, v) for all u, v ∈ U ⁰ . (4)

Next, setting U ⁰⁰ = U + C ⊆ D ⁰ , we remark that F-submodule U ⁰⁰ is generated by the subset {x, xyz + zyx | x, y, z ∈ U ⁰ }. In particular, U ⁰⁰ ⊇ U ⁰ . To complete the proof, it is enough to construct an F-module map γ ⁰⁰ : U ⁰⁰ → W such that x ^γ

= x ^γ for all x ∈ U and

[x, y] ^γ

= λ[x ^γ

, y ^γ

] ^π for all x, y ∈ U ⁰⁰ .

(5)

To this end, we define a map B : (U ⁰ ) ⁴ → Q by the rule

B(x, y, z, t) = [xyz + zyx, t] ^γ

for all x, y, z, t ∈ U ⁰ .

Since xyz + zyx ∈ D, [xyz + zyx, t] ∈ U ⁰ and so B is a well-defined F-multilinear map. A straightforward computation shows that

[xyz + zyx, t] + [yzt + tzy, x] + [ztx + xtz, y] + [txy + yxt, z] = 0 (6)

and whence

B(x, y, z, t) + B(y, z, t, x) + B(z, t, x, y) + B(t, x, y, z) = 0 (7)

for all x, y, z, t ∈ U ⁰ . It follows from (4) that B(x, y, z, [u, v]) = [xyz + zyx, [u, v]] ^γ

= {[[xyz + zyx, u], v] + [u, [xyz + zyx, v]]} ^γ

= λ[[xyz + zyx, u] ^γ

, v ^γ

] ^π + λ[u ^γ

, [xyz + zyx, v] ^γ

] ^π

= λ[B(x, y, z, u), v ^γ

] ^π + λ[u ^γ

, B(x, y, z, v)] ^π and so

B(x, y, z, [u, v]) − λ[B(x, y, z, u), v ^γ

] − λ[u ^γ

, B(x, y, z, v)] ∈ C (8)

for all x, y, z, u, v ∈ U ⁰ . Since V is a 9-free subset of Q and (U ⁰ ) ^γ

= V, it follows from (4), (7) and (8) that all the conditions of Theorem 3.1 are met (with m = 4).

By Theorem 3.1 there exist a multilinear quasi-polynomial q(x, y, z) (with respect to γ ⁰ ) of degree ≤ 3, all of whose coefficients are F-multilinear maps, and an F- multilinear map µ : (U ⁰ ) ⁴ → C such that

B(x, y, z, t) = [q(x, y, z), t ^γ

] + µ(x, y, z, t) for all x, y, z, t ∈ U ⁰ . (9)

We now define a map γ ⁰⁰ : U ⁰⁰ → W by the rule

x + X n i=1

(x i y i z i + z i y i x i )

! γ

= x ^γ

+ λ ⁻¹ X n i=1

q(x i , y i , z i )

! π

for all x, x i , y i , z i ∈ U ⁰ , i = 1, 2, . . . , n. We claim that γ ⁰⁰ is a well-defined F-module map. Indeed, suppose that x + P n

i=1 (x i y i z i + z i y i x i ) = 0. Let t ∈ U ⁰ . Then

0 =

"

x + X n i=1

(x i y i z i + z i y i x i ), t

# γ

= [x, t] ^γ

+ X n i=1

B(x i , y i , z i , t)

=

"

λx ^γ

+ X n i=1

q(x i , y i , z i ), t ^γ

#

+  ⁰ (x, t) + X n i=1

µ(x i , y i , z i , t) and so [a, t ^γ

] ∈ C for all t ∈ U ⁰ , where a = x ^γ

+ λ ⁻¹ P n

i=1 q(x i , y i , z i ). Since (U ⁰ ) ^γ

= V, [a, V] ⊆ C (forcing [[a, V], V] = 0) and whence a ∈ C (Remark 2.5).

Therefore a ^π = 0 and so γ ⁰⁰ is well-defined. As γ ⁰ is an F-module map and all the coefficients of q are F-multilinear maps, we infer that γ ⁰⁰ is an F-module map which proves our claim.

Finally, we claim that (5) is satisfied. Indeed, let u, v, w, x, y, z, t ∈ U ⁰ . Let us first show that [x, t] ^γ

= λ[x ^γ

, t ^γ

] ^π . Clearly [x, t] ∈ U and so [x, t] ^γ

= [x, t] ^γ ∈ W.

Therefore

[x, t] ^γ

= ([x, t] ^γ

) ^π = [x, t] ^γ

= λ[x ^γ

, t ^γ

] ^π

by (4). Since [a ^π , b ^π ] = [a, b] for all a, b ∈ Q and γ ⁰⁰ = γ ⁰ π, we conclude that [x, t] ^γ

= λ[x ^γ

, t ^γ

] ^π = λ[x ^γ

, t ^γ

] ^π .

Next we show that [xyz +zyx, t] ^γ

= [(xyz +zyx) ^γ

, t ^γ

] ^π . We have [xyz +zyx, t] ∈ U ⁰ because U ⁰ is a Lie ideal of D ⁰ and so

[xyz + zyx, t] ^γ

= ([xyz + zyx, t] ^γ

) ^π = ([q(x, y, z), t ^γ

] + µ(x, y, z, t)) ^π

= λ[λ ⁻¹ q(x, y, z), t ^γ

] ^π = λ[(xyz + zyx) ^γ

, t ^γ

] ^π .

Since the F-module U ⁰⁰ is generated by the subset {x, xyz + zyx | x, y, z ∈ U ⁰ }, we conclude that [s, t] ^γ

= λ[s ^γ

, t ^γ

] ^π for all s ∈ U ⁰⁰ . It remains to show that

[xyz + zyx, uvw + wvu] ^γ

= λ[(xyz + zyx) ^γ

, (uvw + wvu) ^γ

] ^π . Clearly

[xyz + zyx, uvw + wvu] ∈ U ⁰⁰ , [xyz + zyx, t], [uvw + wvu, t] ∈ U ⁰ and so, by what we just proved, we have

λ[[xyz + zyx, uvw + wvu] ^γ

, t ^γ

] ^π = [[xyz + zyx, uvw + wvu], t] ^γ

= ([[xyz + zyx, t], uvw + wvu] + [xyz + zyx, [uvw + wvu, t]]) ^γ

= λ[[xyz + zyx, t] ^γ

, (uvw + wvu) ^γ

] ^π +λ[(xyz + zyx) ^γ

, [uvw + wvu, t] ^γ

] ^π

= λ ² [[(xyz + zyx) ^γ

, t ^γ

] ^π , (uvw + wvu) ^γ

] ^π +λ ² [(xyz + zyx) ^γ

, [(uvw + wvu) ^γ

, t ^γ

] ^π ] ^π

= λ ² [[(xyz + zyx) ^γ

, t ^γ

], (uvw + wvu) ^γ

] ^π +λ ² [(xyz + zyx) ^γ

, [(uvw + wvu) ^γ

, t ^γ

]] ^π

= λ ² [[(xyz + zyx) ^γ

, (uvw + wvu) ^γ

], t ^γ

] ^π . Setting

a = [xyz + zyx, uvw + wvu] ^γ

− λ[(xyz + zyx) ^γ

, (uvw + wvu) ^γ

],

we see that [a, t ^γ

] ^π = 0 for all t ∈ U ⁰ . By Remark 2.5, a ∈ C and so a ^π = 0.

Therefore

[xyz + zyx, uvw + wvu] ^γ

= λ[(xyz + zyx) ^γ

, (uvw + wvu) ^γ

] ^π because γ ⁰⁰ π = γ ⁰⁰ . Thus (5) is satisfied and the proof is thereby complete.

We continue with an elementary lemma whose proof is extracted from that of [39, Theorem 2.3].

Lemma 3.3. Let A be an F-algebra and let U be a Lie subalgebra of A such that xyz + zyx ∈ U for all x, y, z ∈ U. Suppose that ¹ ₂ ∈ F. Then hUi = U + U ◦ U.

Proof. We proceed by induction on n ≥ 2 to show that x 1 x 2 . . . x n ∈ U + U ◦ U for all x 1 , x 2 , . . . , x n ∈ U. Let x 1 , x 2 ∈ U. Then

x 1 x 2 = 1

2 ([x 1 , x 2 ] + x 1 ◦ x 2 ) ∈ U + U ◦ U.

In the inductive case n > 2 we have that

[x 1 , x 2 ]x 3 . . . x n , x 2 [x 1 , x 3 ]x 4 . . . x n ∈ U + U ◦ U

by the induction assumption and whence

(x 1 x 2 x 3 − x 2 x 3 x 1 )x 4 . . . x n = [x 1 , x 2 ]x 3 . . . x n + x 2 [x 1 , x 3 ]x 4 . . . x n ∈ U + U ◦ U.

By assumption x 1 x 3 x 2 + x 2 x 3 x 1 ∈ U. Therefore the induction assumption yields that (x 1 x 3 x 2 + x 2 x 3 x 1 )x 4 . . . x n ∈ U + U ◦ U and so

x 1 (x 2 ◦ x 3 )x 4 . . . x n = (x 1 x 2 x 3 − x 2 x 3 x 1 )x 4 . . . x n

+ (x 1 x 3 x 2 + x 2 x 3 x 1 )x 4 . . . x n ∈ U + U ◦ U.

Thus

x 1 x 2 . . . x n = 1

2 {x 1 (x 2 ◦ x 3 )x 4 . . . x n + x 1 [x 2 , x 3 ]x 4 . . . x n } ∈ U + U ◦ U.

The proof is now complete.

Proposition 3.4. Let A be an F-algebra and let S be a Lie subalgebra of A such that S ∩ (S ◦ S) = 0 and xyz + zyx ∈ S for all x, y ∈ S. Further, let R be a submodule of the F-algebra Q. Let ρ : S → Q be an F-module map with S ^ρ = R and let λ ∈ C be an invertible element such that

[u, v] ^ρ = λ[u ^ρ , v ^ρ ] for all u, v ∈ S.

Suppose that ¹ ₂ ∈ F, C is a direct summand of the C-module Q and R is a 7-free subset of Q. Then there exists a homomorphism of algebras φ : hSi → hRiC + C such that x ^ρ = λ ⁻¹ x ^φ for all x ∈ S.

Proof. Let W ⊆ Q be a C-submodule such that Q = W ⊕ C and let π : Q → W be a canonical projection of modules. Since F · 1 ⊆ C, we conclude that both W and C are F-submodules of Q and π is an F-module map. Clearly F-modules W and Q are isomorphic and so we may assume, without loss of generality, that ρ : S → W is an F-module map such that

[u, v] ^ρ = λ[u ^ρ , v ^ρ ] ^π for all u, v ∈ S and S ^ρ = R ^π .

Set A ⁰ = A ⊕ C, S ⁰ = S ⊕ C and R ⁰ = R ^π + C = R ^π ⊕ C. Clearly S ⁰ is a Lie subalgebra of the algebra A ⁰ such that xyz + zyx ∈ S ⁰ for all x, y, z ∈ S ⁰ and R ⁰ is an F-submodule of Q containing R. We identify A and C with corresponding ideals of the algebra A ⁰ . Since R is 7-free, Theorem 2.3 yields that R ⁰ is 7-free as well. We now define a map α : S ⁰ → R ⁰ by the rule (u + c) ^α = u ^ρ + c for all u ∈ S, c ∈ C. Clearly α is an epimorphism of F-modules and

[u + a, v + b] ^α = [u, v] ^ρ = λ[u ^ρ , v ^ρ ] ^π = λ[(u + a) ^α , (v + b) ^α ] ^π

for all u, v ∈ S, a, b ∈ C. Therefore [x, y] ^α − λ[x ^α , y ^α ] ∈ C for all x, y ∈ S ⁰ . Setting (x, y) = [x, y] ^α − λ[x ^α , y ^α ], x, y ∈ S ⁰ , we conclude that  : (S ⁰ ) ² → C is an F-bilinear map such that

[x, y] ^α = λ[x ^α , y ^α ] + (x, y) for all x, y ∈ S ⁰ . (10)

Define a map B : ( S ⁰ ) ³ → Q by the rule

B(x, y, z) = λ ⁻² (xyz + zyx) ^α − (x ^α y ^α z ^α + z ^α y ^α x ^α ) for all x, y, z ∈ S ⁰ . We first remark that B(x, y, z) is a symmetric map “modulo C”. Indeed, obviously B(x, y, z) = B(z, y, x) for all x, y, z ∈ S ⁰ . Further,

B(x, y, z) − B(y, x, z) = λ ⁻² [[x, y], z] ^α − [[x ^α , y ^α ], z ^α ] ∈ C

which proves our remark. It now follows from both (6) and (10) that [B(x, y, z), t ^α ] + [B(y, z, t), x ^α ] + [B(z, t, x), y ^α ] + [B(t, x, y), z ^α ] ∈ C for all x, y, z ∈ S ⁰ . By Theorem 2.7, B(x, y, z) is a multilinear quasi-polynomial, so that

B(x, y, z) = λ 1 x ^α y ^α z ^α + λ 2 x ^α z ^α y ^α + λ 3 y ^α x ^α z ^α + λ 4 y ^α z ^α x ^α +λ 5 z ^α x ^α y ^α + λ 6 z ^α y ^α x ^α + ν 1 (x)y ^α z ^α + ν 2 (x)z ^α y ^α + ν 3 (y)x ^α z ^α +ν 4 (y)z ^α x ^α + ν 5 (z)x ^α y ^α + ν 6 (z)y ^α x ^α + µ 1 (x, y)z ^α + µ 2 (x, z)y ^α +µ 3 (y, z)x ^α + ω(x, y, z) for all x, y, z ∈ S ⁰ ,

where λ i ∈ C, ν i : S ⁰ → C are F-linear maps, µ j : (S ⁰ ) ² → C are F-bilinear maps and ω : (S ⁰ ) ³ → C is an F-trilinear map. Since B(x, y, z) = B(z, y, x), Lemma 2.6 implies that λ 1 = λ 6 , λ 2 = λ 5 , λ 3 = λ 4 , ν 1 = ν 6 , ν 2 = ν 5 , ν 3 = ν 4 , µ 1 (x, y) = µ 3 (y, x), µ 2 (x, z) = µ 2 (z, x) and ω(x, y, z) = ω(z, y, x). As B(x, y, z) − B(y, x, z) ∈ C, we obtain that

λ 1 = λ 3 , λ 2 = λ 4 , λ 5 = λ 6 , ν 1 = ν 3 , ν 2 = ν 4 , ν 5 = ν 6 , µ 1 (x, y) = µ 1 (y, x) and µ 2 (x, z) = µ 3 (x, z). Set a = λ 1 , ν = ν 1 , µ = µ 1 . We now have

B(x, y, z)

= a {x ^α y ^α z ^α + x ^α z ^α y ^α + y ^α x ^α z ^α + y ^α z ^α x ^α + z ^α x ^α y ^α + z ^α y ^α x ^α } + ν(x)(y ^α z ^α + z ^α y ^α ) + ν(y)(x ^α z ^α + z ^α x ^α ) + ν(z)(x ^α y ^α + y ^α x ^α ) + µ(x, y)z ^α + µ(x, z)y ^α + µ(y, z)x ^α + ω(x, y, z)

(11)

for all x, y, z ∈ S ⁰ . Recalling the definition of B and making use of (11), we get (xyx) ^α = λ ² (x ^α y ^α x ^α + 1

2 B(x, y, x))

= λ ² {(1 + a)x ^α y ^α x ^α + a(x ^α x ^α y ^α + y ^α x ^α x ^α ) + ν(x)(x ^α y ^α + y ^α x ^α ) + ν(y)x ^α x ^α + µ(x, y)x ^α + 1

2 µ(x, x)y ^α + 1

2 ω(x, y, x)}

(12)

for all x, y ∈ S ⁰ . Analogously

(xyz + zyx) ^α = λ ² {(1 + a)(x ^α y ^α z ^α + z ^α y ^α x ^α )

+a(x ^α z ^α y ^α + z ^α x ^α y ^α + y ^α x ^α z ^α + y ^α z ^α x ^α ) + ν(x)(z ^α y ^α + y ^α z ^α ) +ν(z)(x ^α y ^α + y ^α x ^α ) + ν(y)(x ^α z ^α + z ^α x ^α ) + µ(x, y)z ^α + µ(z, y)x ^α +µ(x, z)y ^α + ω(x, y, z)} for all x, y, z ∈ S ⁰ .

(13)

It follows from both (12) and (13) that 2(xyxyx) ^α = {(xyx)yx + xy(xyx)} ^α

= λ ² n

(1 + a) {x ^α y ^α (xyx) ^α + (xyx) ^α y ^α x ^α }

+a{x ^α (xyx) ^α y ^α + (xyx) ^α x ^α y ^α + y ^α x ^α (xyx) ^α + y ^α (xyx) ^α x ^α } +ν(x){(xyx) ^α y ^α + y ^α (xyx) ^α } + ν(xyx)(x ^α y ^α + y ^α x ^α ) +ν(y){x ^α (xyx) ^α + (xyx) ^α x ^α } + µ(x, y)(xyx) ^α + µ(xyx, y)x ^α +µ(x, xyx)y ^α + ω(x, y, xyx)

o

= λ ² a{(xyx) ^α x ^α y ^α + y ^α x ^α (xyx) ^α } + q 1

= 2λ ⁴ a ² y ^α x ^α x ^α x ^α y ^α + q 1 ⁰

(14)