Preamble

FINITELY BASED VARIETIES LIE ALGEBRAS V. ZAICEV

Abstract

In this paper the author investigates the question of when the following objects finitely based: 1) the product of two varieties of Lie algebras; 2) the commutator of two varieties; 3) the union of two varieties, one of which is nilpotent.

Bibliography: 4 titles. is known [1] that over any infinite field, the product of two finitely based varieties of Lie algebras is itself finitely based. In the first section of this paper, we continue to try to determine when the product of finitely based varieties of Lie algebras over an arbitrary field is finitely based.

§2 deals with the question of when the commutator of varieties of

Lie algebras over an arbitrary field is finitely based, and

§3 deals with the question of

the union of two varieties is finitely based, given that one of them is nilpotent. be the base field, let be an absolutely free Lie algebra of countable rank over and let . . } be a set of free generators of use some of the notation of [1]. Let be a set of elements of I(S, L), which we refer to as the weak closure of is the smallest vector subspace of which contains and which is invariant under all endomorphisms of as a Lie algebra; and J(S, L), which we refer as the strong closure of is the smallest verbal ideal of containing 5". is any Lie algebra over we let denote the subspace of which is spanned by the homomorphic images of I(S, L) under all possible homomorphisms of G. S(G) denote the smallest ideal in which is spanned by the images of under all possible homomorphisms of an identity follows from an identity /,, then, in our notation, }, L).

If /, is equivalent to / /({/]}, is a subalgebra of equivalent to S(G). say that a polynomial / = ,,..., normal polynomial if every monomial in / depends on all of the variables , . . ., x say that the row of whole numbers , . . . , d is the structure of the normal polynomial / = /(*,, . . . , if 4 is the maximum of the degrees of the monomials of / with respect to the variable x, for / = 1, . . . , , . . . be a row of whole numbers such that 0 < s, < d for / = 1, ...,«. we consider the polynomial mathematics subject classification.

Primary 17B99; Secondary 17B30. American Mathematical Society 1979

sum of all the monomials of this polynomial which contain all of the generators where / = 1, . . ., andy = 1, . . ., be denoted by the symbol usefulness of the system of polynomials which we obtain in this way from / lies in fact that if /({/}, L), then is a linear combination of polynomials of the form (Gil, · · · » where 1 < here the are monomials (with coefficients) of fory = 1,. . ., j / = 1, . . . , convenience of notation, we use the symbol abc . . . d to refer to the product ((ab)c) ...)d. product varieties of Lie algebras THEOREM finitely based varieties of Lie algebras P, and suppose that the verbal ideal V is the weak closure finite system polynomials in L; that is, V = I(S, L), where \S\ < Then the variety finitely based.

PROOF. If the variety U corresponds to the verbal ideal U — U(L) and 93 corresponds the verbal ideal = K(L), then the verbal ideal of U · 93 is the closure of U{V{L)) verbal ideal of Therefore, it is sufficient to show that U(V(L)) is equivalent to a finite set. If U = J(R, L), then, in accordance with our notation, we have U( V(L)) R(V(L)). can be assumed to be a finite set consisting of normal polynomials: , . . . , u where w, is a normal polynomial for 1, . . . , As we have already observed, R(V(L)) is equivalent to R(V(L)), every element of which is a linear combination of the values in which we obtain by substituting elements of in place of generators in the w,; furthermore, I(S, L), which is to say that is the set of linear combinations of the images of elements o £ S under endomorphisms of assume that 5 is a set consisting of a single element:

S = {v(x . . . , * „ ) } . M,(X,, . . ., and suppose that [d{, . . ., d^] is the structure of «,. We consider the system (λ? · 0 l' « ' " " /!,·! where ν(χγ\ . . . , 1, . . . , t\' + . . . +/{; here [/{,... all possible choices of integers such that 1 < 1, . . . , «,·; also, = 1, . . . , Γ, with λ/ £ Ρ, and χ{ , . . . , < > · · · <> are distinct generators. determine and the coefficients λ/\ We set M(JC,, . . . , = iZ; + · · · where ΰ- is a monomial (with coefficient) for 1, . . . , and for any choice , . . . ,\ of numbers from we have

u (λ!*!, . . . , λ^χ^) = Υι · «ι + . . . + Yc · «c.

We now have a transformation ψ: of vector spaces over ψ(λ) = γ, where λ = (Aj, . . . , λ^) and γ = (γ^ . . . , The image of this transformation con-

tains a finite number of linearly independent vectors γ , . . ., < c). We choose , . . ., such that ψ(λ') = γ' for = 1, . . ., Then for any λ = (λ,, . . . , λ^) we Ύ Γ ) .

Consequently,

ΰ (λ**!, . .. , λΑΤ^Λτ) = Υι · «ι + . · . + Yc · «Γ

(ηιΥΪ + . . . + ητΥί) · «ι + . · · + (ηιΥ? + . _ . . + ητΥί) · « _ ( Υ Γ is this and these λ," that we take in the system (). is not hard to see that all of the values of «, in are generated by the system (•) that this system is finite. If we construct such systems for all «,·(/= 1, . . ., m) and take their union, we obtain a finite set which is equivalent to R(V(L)).

This proves Theorem A verbal ideal is called homogeneous implies that all of the homogeneous components of also lie in LEMMA finitely based variety of Lie algebras P, and suppose that its verbal ideal V is homogeneous.

Then V = I(S, L), where < oo. PROOF. Because of its homogeneity, decomposes into the direct sum of subspaces: e ... e e ..., where is the vector space spanned by the homogeneous polynomials of which have degree We also consider the subspace generated by the homogeneous polynomials of degree which depend on only the generators , . . . , x Then one can easily see that is equivalant to and that is a finite-dimensional vector space. Let ...,%} basis consisting of homogeneous polynomials. We show that , L) = J(S , L).

To establish this equality, it be sufficient for us to show that Vj · ζ G I(S where ζ is a generator which does not appear in Note that in this case the equivalence of implies that Vj = Vj(x . . . , Then if . . ., d is the structure of as a normal polynomial, it follows is the degree of all of the monomials of with respect to generator / = 1, . . . , k. consider the polynomial + x\, . . . , x where , . . . , x\, . . . are distinct generators. Then contains the following homogeneous components of tJ: 1) Wj, which has structure 1, 1, . . . , with respect to the generators , x\, which has structure 1, 1, , . . with respect to the generators . . , which has structure , . . . — 1, 1] with respect to the generators . . . , x If φ is an endomorphism of such that φ(χ,) = χ, and φ(χ-) for / = 1, . . . ,

φ ( α θ + φ ( α ; 2 ) + ... + q>(w h ) =v r z£I(S n , L).

Since the variety is finitely based, there exists a number such that is equivalent to Φ · · · θ which, in turn, is equivalent to θ · · · θ Set 5 = · · · U 5 . Then I(S, L) = J{S, L) = V, and | S | < oo. Lemma 1 is proved.

COROLLARY finitely based varieties of Lie algebras P, and suppose that the verbal ideal of homogeneous.

U · 33 finitely based variety. LEMMA Let V be the verbal ideal of a nilpotent variety of Lie algebras.

I(S,L), where \S\ < PROOF. Since 33 is nilpotent, there exists a natural number such that 33 It is clear that L c + 1 = 7({xj - . . . 'X +}> L).

Let us now consider vector subspace such that Then every element of linear combination of monomials whose degree does not exceed But no normal polynomial in contains more than generators. Now consider the vector space spanned by the normal polynomials in which depend on the generators . . . , x is a finite-dimensional vector space, since there are only a finite number of monomials (with coefficient 1) of given bounded degrees which depend on a given finite number of variables. It is clear that is equivalent to Let 5" = . . ., w basis and let We consider Since c + 1 follows w · ;c c + 1 a + b, where c + 1 . We now decompose into a sum of normal polynomials: + · · · Then a, = <p (6,), where <p is an automorphism that is, fy = w, fory = 1, . . ., Hence the whole polynomial is a linear combination of images of that is, E 7(5", L). But 6 Ε /({u}, L), where · . . . consequently, for the set = 5" υ {*! U * we have I(S, L) = J(S, L) = V \S\ < oo. Lemma 2 is proved.

COROLLARY 2. 7/ U W finitely based variety 33 w α nilpotent variety, then U · finitely based variety.

LEMMA 33 6e variety of Lie algebras where Ρ φ 2, and let V be the verbal ideal of Suppose 33 C 9? · 31.

Then V = I(S, L), where \S\ < PROOF. C = 7({(X )}, L), Since for char every subvariety of 9? is finitely based (see [2]), it follows = J(S, L), where a finite set which contains the polynomial / = and which has the property that every other polynomial is both normal and a linear combination of monomials of the form . . . · * · */, · ... · - x r ... · * / ; , ) · Let φ be an endomorphism of the algebra which maps each appearing in where ζ is some other generator. Then <p(u) where sum of monomials whose degree with respect to ζ is 1 (i.e., — w z), u is a sum of monomials whose degree with respect to is 2, and u Similarly, <p(w) — w + w · ζ + where Ε C and the degree with respect to ζ of the monomials in is 2. But then φ(υ) = + υ · ζ + , where the degree of

with respect to ζ is 2 and Ε C. Consequently, 1({ν], + C. But since char 2, we can assume ν • ζ individually lie in /({«}, L) + C.

Hence, /({«}, J({v], L) (mod C). From what we earlier, it follows J(S, L) I(S, L).

Lemma 3 is proved. REMARK. If 33 is a metabelian variety of Lie algebras, then 93 is finitely based field (see, for example [4]). Moreover, J({v], field any polynomial Consequently, the lemma is true for metabelian varieties field.

COROLLARY is a finitely based variety and is a metabelian variety, then XX · 93 a finitely based variety.

COROLLARY Ρ φ 2, and suppose that is a finitely based variety of Lie algebras over Ρ and · 21. the variety XX · 93 is finitely based.

§2. Commutators of

varieties of Lie algebras Let U and 93 be varieties of Lie algebras and let be their verbal ideals. Then we define the commutator [XX, 93] to be the variety corresponding to the verbal ideal [ U, V].

THEOREM Let the conditions of Theorem be satisfied for the varieties [XX, 93] is a finitely based variety of Lie algebras over P.

PROOF. /({«}, J({v}, where u = u(x . . .,

ν ( χ η +ι> · · · ' x n+m)> t n e n t ^' V\ xs generated by the system

where 0, 1, . . . and 0, 1, . . . ; distinct generators which differ *,, . . . , for / = 1, . . ., = 1, . . ., But since ( « · / ) · ν = (u v)-f-w(v ·/), it follows [U, V] = 7(5', L), where 5 ' = {u · v, u · . . . , « · · · · ), . . . }. By hypothesis, Κ = I(S, L), where | 5 | < oo. We can assume {€>}. Therefore, for all 1, 2, . . ., we have

Ό · Z l · . . . · Ζ* = λ{ • V lk + . · . + λί Λ · ϋα^,

where λ,* G Ρ and the % are images of under homomorphisms for ι = 1, . . ., Hence, the whole set is generated by u · v.

Thus, [U, V] = /({«· v], L). Theorem proved. COROLLARY THEOREM be a finitely based variety. In each of the following cases, the variety [XX, 93] is finitely based:

P is an infinite field, and is a finitely based variety. Ρ is finite, is finitely based, and the verbal ideal corresponding to is homogeneous. Ρ is finite and is nilpotent. Ρ is finite and is metabelian. Ρ is finite, Ρ φ 2, and All of these

results

follow from Lemmas 1, 2, and 3 of the preceding section (in the of an infinite field, one should bear in mind that all verbal ideals are homogeneous).

Unions of varieties of Lie algebras the theory of varieties of groups, we that if a variety groups is nilpotent, its union with another variety groups finitely based if and only if the other

variety finitely based (see [3]). this section, prove similar result varieties algebras over arbitrary field the variety nilpotent algebras whose nilpotence class does exceed THEOREM variety finitely based and only finitely based.

PROOF. verbal ideal corresponding to U, verbal ideal

33. Then the verbal ideal corresponding

theorem follows from lemmas which prove. LEMMA verbal ideal suppose finitely generated verbal ideal finitely generated verbal ideal.

PROOF. generated finite system elements consider I " " u = I where / is a linear combina- monomials degree equivalent finite system identities equivalent and the lemma proved. Suppose equivalent to the infinite system . . , and we can assume that , . . . expressed terms generators . . . , finite subset these can linearly independent. Hence, there number such that for all O-N+k- t = l Consequently,

α^ + Λ = ^ λ ! · Μ ί + «>*, where w k = CiN+k — 2 λ * " fli '

But H^ lies therefore dependent Lemma proved. LEMMA verbal ideal suppose finitely generated verbal ideal. finitely generated verbal ideal, also.

PROOF. Suppose generated by the system identities . . . , l . . . , where , . . . « , , . . . , « „ , £ Let us extend system . . . , generating set for verbal ideal. assume finitely generated verbal ideal. Then the extension {«,,..., is necessarily infinite system identities . . . }. for all polynomial depends {/,,..., . . . ,u Let us see which elements depend {/,,..., . . . , normal polynomial in L " " be the structure Instead /, we consider following system, which equivalent /[,„ ...,^Oii' · · · **,,)> + · ' · + · · ·

\ / _ 1 2 > . . · > *[s,, Sq} \ · · · > · · · >

1 ( Υ Υ Υ y\ * [ s 1 Sq] ^ Λ 1 1 > · · · > A q,Sq-i, *qSq ' £)·,

/ [ . , always have that 1 < < / , , . . . , 1 < moreover, we assume that JC,,, . . . , are distinct generators and that x, , . . . , x, assume all values in We denote by 5" the set consisting of those polynomials which have the form 1), 2a), 2b). The set 5" is finite and lies in Κ η L". The set of polynomials having the form be denoted by We set / = /, and construct sets and 5,". We do the same for the polynomials . . . , /,. We set

2 LJ

· · · LJ ^ f i -Ί LJ - · · · We have already observed that for k > m, every polynomial depends on the system {/„ . . . , . . . , where . . . , ), L) /({/,, . . . , / }, L). From the way we constructed however, it follows that where depends on is a linear combination of elements of 5 moreover, we can take to be a normal polynomial which depends on the generators . . . , where k' < n. system «,,..., . . . , . . . is equivalent to the system {«!, . . . , U 5i U .. ., e, ...}.

But there are at most a finite number of different linearly independent Consequently, system (**) (and, hence, the system {«,,..., . . . , . . . }) is equivalent to a finite set. This contradiction proves Lemma 5. theorem follows trivially from Lemmas 4 and 5. Q.E.D.

COROLLARY. IftBisa nilpotent variety of Lie algebras, then the variety U U 23 is finitely based if and only if is a finitely based variety.

Since 33 C 9? , we have (U υ 33) U U υ 9? , and the corollary follows. closing, the author would like to express his gratitude to Ju. A. Bahturin for posing problem and for assisting the author in his work.

BIBLIOGRAPHY 1. Ju. A. Bahturin, Identities in Lie algebras.

I, II, Vestnik Moskov. Univ. Meh. 1973, no. 1, 12-18; no. 30-37, (Russian); English transl.

Moscow Univ. Math. Bull. (1973). R. M. Bryant and M. R.

Vaughn-Lee, Soluble varieties algebras, Quart. Math. Oxford Ser. (2) 23 (1972), Roger M. Bryant, varieties groups, Bull. London Math. Soc. (1969), 60-64.

Ju. A. Bahturin, Identities metabelian algebras, Trudy Sem. Petrovsk.

Vyp. 1 (1975), 45-56. (Russian) Translated by Τ. Β.

GREGORY