Siberian Mathematical Journal, Vol. 42, No. 3, pp. 610–612, 2001Original Russian Text Copyright c© 2001 Chirkov I. V. and Shevelin M. A.

IDEALS OF FREE METABELIAN LIEALGEBRAS AND PRIMITIVE ELEMENTS

I. V. Chirkov and M. A. Shevelin UDC 517.55

1. Introduction. The following theorem was proven in M. Evans’s article [1]:

Theorem 1. Let q be a primitive element of a free metabelian group G with n free generators. If qbelongs to the normal closure of some element y ∈ G then q is conjugate with y or y−1.

In this article we prove a similar theorem for free metabelian Lie algebras.Let K be an arbitrary field and let Mn be a free metabelian Lie algebra over K with free generators

y1, . . . , yn. Denote the universal enveloping algebra for Mn by U = U(Mn) and denote by ε : U → Kthe homomorphism of the free associative algebras which is defined by the rule yiε = 0. Also, denote by

: Mn → Mn/M ′n the natural homomorphism and put yi = xi. Recall that U(Mn/M ′

n) is isomorphic tothe polynomial ring P = K[x1, . . . , xn]. The homomorphism extends to a homomorphism : U → P ofthe associative algebras. We let W stand for the free right module of rank n over P with basis e1, . . . , en.Recall that an element of Mn is said to be primitive if it can be included in some system of free generators.Henceforth the ideal generated in Mn by a set X is denoted by 〈X〉.

The space M ′n is furnished with the natural structure of a P -module (we denote this action by the

dot sign) according to which c.xi = c.(yi +M ′n) = [c, yi] for c ∈ M ′

n and yi +M ′n ∈ Mn/M ′

n. Moreover, bythe universal property of U the right regular action of Mn onto itself extends to an action of U on Mn:

m ◦ (m1m2 . . .mk) = [. . . [[m,m1],m2], . . . ,mk], m ◦ 1 = m

for m ∈ Mn and m1 . . .mk ∈ U(Mn).

Definition. An automorphism λ of Mn is called inner if λ acts on the generators as follows:

yiλ = yi ◦ (ξ + c).

Here c ∈ M ′n and ξ ∈ K \ {0}.

The mapping in this definition is really an automorphism. Indeed, Mn ◦ c2 = 0 implies the equalities

yi ◦ (ξ + c)(ξ−1 − ξ−2c) = yi

for i = 1, . . . , n. This proves injectivity.To check surjectivity, put zi = yiλ. We have to verify that the subalgebra generated by z1, . . . , zn

contains yi (i = 1, . . . , n). Since

yi = ξ−1zi − ξ−1[yi, c] = ξ−1zi − ξ−2[ξyi + [yi, c], c] = ξ−1zi − ξ−2[zi, c]

and c ∈ M ′n, it suffices to establish that M ′

n lies in the subalgebra generated by z1, . . . , zn. This is reallyso, for

[zi, zj ] = [ξyi + [yi, c], ξyj + [yj , c]] = ξ2[yi, yj ] + ξ[yi, [yj , c]]− ξ[yj , [yi, c]]

= ξ2[yi, yj ] + ξ[[c, yj ], yi]− ξ[[c, yi], yj ] = ξ2[yi, yj ]− ξ[c, [yi, yj ]] = ξ2[yi, yj ].

We also note, although need not this, that the inner automorphisms form an abelian normal subgroupin the automorphism group of the algebra Mn.

Now, we formulate our main result:

The research was supported by the Russian Foundation for Basic Research (Grant 98–01–00932).

Omsk. Translated from Sibirskiı Matematicheskiı Zhurnal, Vol. 42, No. 3, pp. 720–723, May–June, 2001.Original article submitted December 28, 1999.

610 0037-4466/01/4203–0610 $25.00 c© 2001 Plenum Publishing Corporation

Theorem 2. Let z be a primitive element of the algebra Mn and let y ∈ Mn. If z ∈ 〈y〉 thenthe elements z and y are conjugate by means of an inner automorphism of Mn.

We will use V. A. Artamonov’s construction as exposed, for instance, in [2]. We briefly recall it.Denote by P0 the ideal in P that is generated by x1, . . . , xn. Let l : W → P0 be the epimorphism of

the P -modules which is defined by the rule l(ei) = xi. The module W is endowed with the multiplicationab = al(b) − bl(a) for a, b ∈ W which makes W into a metabelian Lie K-algebra. Put L = {a ∈ W |l(a) ∈ Kx1 + · · ·+ Kxn}.

Theorem 3 [2]. L is a free metabelian Lie algebra with the set of free generators e1, . . . , en.

Corollary [2]. The commutant L′ of the algebra L coincides with Ker(l).In particular, the P -module M ′

n is a submodule of the free module and is consequently a torsion-free module, and the action of P on W extends to the natural action of P on M ′

n. We denote by θthe embedding of Mn in W for which yiθ = ei.

2. Proof of Theorem 2. Take y ∈ Mn and let z be a primitive element of Mn; z belongs tothe ideal 〈y〉 generated in Mn by y. We may assume that z = y1. Factoring by the commutant andreplacing y with a proportional element if need be, we obtain y = y1 + c (c ∈ M ′

n). If c = 0 then we havenothing to prove. Therefore, we henceforth assume that c 6= 0. Since y1 ∈ 〈y〉, for some f ∈ U such thatfε = 0 we have y1 = y1 + c + (y1 + c) ◦ f . Hence, (y1 + c) ◦ f = y1 ◦ c1 + y1 ◦ f ′ + c ◦ f ′, where c1 ∈ M ′

n

and f ′ is a linear combination of nonempty products of yj (1 ≤ j ≤ n); in particular, f ′ε = 0. Applythe embedding θ to c + (y1 + c) ◦ f = 0. Let

cθ =∑

1≤i≤n

eiαi, αi ∈ P, c1θ =∑

1≤i≤n

eiβi, βi ∈ P, f ′ =∑

1≤i≤n

yif′i , f ′i ∈ U.

Then ∑1≤i≤n

eiαi −∑

1≤i≤n

eiβix1 + e1f ′ −∑

1≤i≤n

eix1f ′i +∑

1≤i≤n

eiαif ′ = 0

according to the definition of multiplication on the module W . Collecting the coefficients of ei, we obtain

αi − x1βi − x1f ′i + αif ′ = 0 (2 ≤ i ≤ n),

α1 − x1β1 + f ′ − x1f ′1 + α1f ′ = 0.

The first equality implies that (1 + f ′)αi ∈ x1P . Hence, αi = x1α′i (α′i ∈ P, i = 2, . . . , n), f ′ε = 0.

The second equality implies that α1 +f ′+α1f ′ ∈ x1P . Therefore, (1+α1)(1+f ′) ≡ 1(modx1P ). Scalarsare the only invertible elements of the ring P/x1P . Thereby 1 + α1 = ξ + x1α

′1 (ξ ∈ K \ {0}). Since

c ∈ M ′n, by the Corollary to Theorem 3 we have

0 =∑

1≤i≤n

αixi = (ξ − 1 + x1α′1)x1 +

∑2≤i≤n

α′ix1xi.

Hence,∑

1≤i≤n α′ixi = 1 − ξ; whence ξ = 1 and α1 = x1α′1 for some α′1 ∈ P . In other words, c ∈ 〈y1〉.

Therefore, we can write down c = y1 ◦ g (g ∈ U), y1 = y1 + y1 ◦ g +(y1 + y1 ◦ g) ◦ f , which is equivalent tothe equality y1 ◦ (g + f + gf) = 0. This means that the right multiplication by f acts on y1 nilpotently;for instance, y1 ◦ fk = 0. Since y1 ◦ f ∈ M ′

n, it follows that y1 ◦ fk = (y1 ◦ f).fk−1.If y1 ◦ f = 0 then c + c ◦ f = c.(1 + f) = 0, which is impossible, since fε = 0, c 6= 0, and W is

a torsion-free P -module.We again use the absence of torsion and zero divisors to deduce the equalities f

k−1 = 0 and f = 0.Therefore, f ∈ M ′

nU . Since each element of M ′nU acts on Mn in the same way as some element of M ′

n

(by the metabelianity identity), we may assume that f ∈ M ′n. Since y1 = y ◦ (1 + f); therefore, y1 = yλ

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for the inner automorphism λ defined by the rule yiλ = yi ◦ (1+f) (1 ≤ i ≤ n). This completes the proofof the theorem.

3. The conjugation problem. Note that the conjugation problem for the elements of the algebraMn by an inner automorphism is solved in the positive. Indeed, assume that s1, s2 ∈ Mn, s1 = l1 + c1,s2 = l2 + c2, where l1 and l2 are linear combinations of generators and c1 and c2 are elements inthe commutant. Factoring by the commutant, we see that proportionality of the elements l1 and l2 isa necessary condition for conjugation. Since multiplication by a nonzero scalar is an inner automorphism,we assume that l1 = l2. If l1 = 0 then the elements s1 and s2 are conjugate if and only if they coincide.But if l1 6= 0 then we may assume that l1 coincides with the free generator y1. Suppose that s1◦(1+c) = s2,where c ∈ M ′

n. This implies that [y1, c] = c2 − c2. Apply the embedding θ. Let

cθ =∑

1≤i≤n

eiαi, c1θ =∑

1≤i≤n

eiβi, c2θ =∑

1≤i≤n

eiγi.

Then ( ∑1≤i≤n

eiαi

)x1 =

∑1≤i≤n

ei(γi − βi).

Equating the coefficients of ei, we obtain αix1 = γi − βi (i = 1, . . . , n). By the Corollary to Theorem 3,∑1≤i≤n xi(γi − βi) = 0. Therefore, x1

∑1≤i≤n xiαi = 0, and so

∑1≤i≤n xiαi = 0. Thus, if the elements

s1 and s2 are conjugate then(1) γi − βi = x1αi for some αi ∈ P ;(2)

∑1≤i≤n xiαi = 0.

Clearly, these conditions are also sufficient. Summing up the above, we come to the following

Proposition. Let s1 = l1 + c1 and s2 = l2 + c2 be elements of the algebra Mn. These elements areconjugate by means of an inner automorphism if and only if

(1) l1 and l2 are proportional;(2) γi − βi = l1αi for some αi ∈ P ;(3)

∑1≤i≤n xiαi = 0.

The authors are grateful to V. A. Roman′kov for the interest evinced.

References

1. Evans M. J., “Presentations of the free metabelian group of rank 2,” Canad. Math. Bull., 37, No. 4, 468–472 (1994).2. Artamonov V. A., “The categories of free metabelian groups and Lie algebras,” Comment. Math. Univ. Carolin., 18,

No. 1, 143–159 (1977).

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