Uniformly distributed systems of elements on metabelian Lie rings
Abstract.
In this paper, the notion of a uniformly distributed systems of elements on the variety of metabelian Lie algebras is introduced. This notion is analogous to one of a measure preserving systems of elements on group varieties. As the main result of the paper it was shown that on the variety of metabelian Lie algebras a system of elements is primitive iff it is uniformly distributed.
1. Introduction
In this paper, we describe systems of elements that uniformly distributed on the variety of Lie rings (over the ring of integers . In [5, 6], elements uniformly distributed on groups were studied. Such elements are called measure preserving. In the papers [13, 14] of the second author, the notion of a measure preserving element on a group variety was introduced and the measure preserving systems of elements on the varieties of nilpotent and metabelian groups were described. It turns out that such systems of elements are primitive i.e. they can be completed to the bases of a free group of the corresponding variety.
Not only the uniform distribution on groups and rings but also other distributions can be considered. Some interesting problems arise in this case. An example of such problem is to describe the sets of elements having the same distribution on a given variety. For this reason, we think that it is appropriate to use the terms “distribution of elements” and correspondingly “distribution of systems of elements” and although the terms “measure preserving elements” and “uniformly distributed elements” have the same meaning we will use the second one.
The main result of the paper is Theorem 3.7. It claims that a system of elements of a free metabelian Lie ring is uniformly distributed on the variety of metabelian Lie rings iff it is primitive. To prove this theorem we had to find a primitivity criterion for this ring variety (Theorem 3.3). The proof of this criterion is similar to one for the varieties of metabelian groups and metabelian Lie algebras, (see. [7, 8, 11, 12, 15, 16]) but it has some distinctions.
2. Preliminaries
In this paper, we assume that is a finite set. By denote the free Lie ring with the set of free generators and by denote the free metabelian Lie ring with the same set of generators, i.e. free and free metabelian Lie algebras respectively. By and denote the derived Lie rings of and respectively.
Let be a Lie ring with generators . By denote the universal enveloping algebra of .
We denote Lie monomials by bracketed lowercase Latin characters and associative monomials by lowercase Latin characters without brackets. Moreover, we use brackets to denote the images of Lie monomials by the natural map from to . Namely, in is a homogeneous associative polynomial obtained by complete removal of brackets by the rule .
Let be a Lie monomial. By denote the length of this monomial i.e. the total number of entries of all generators in .
Any element in can be considered as a Lie polynomial in the variables . So, we denote it by . Although a representation of an element of is not unique, we will see below that one can use any such representation.
Definition 2.1.
A system of elements () is said to be a primitive system of elements of the metabelian Lie ring , if it can be included in a system of free generators of this ring.
Let be the direct sum of copies of , where is an arbitrary Lie ring. Namely . By denote a tuple of elements in (then ). Similarly, if be a module then by denote the module .
Let be a finite Lie ring of some variety . For an arbitrary system of elements in the free Lie ring with generators define the map such that
Namely, to obtain one should substitute instead of in any representation of as Lie polynomials and take the obtained elements in the same order to form tuple. Since any map of generators of a free Lie ring in a variety to an arbitrary Lie ring of this variety can be extended to a homomorphism uniquely the values of depend on the elements in but do not depend on a polynomials representing these elements. Consider the uniform distribution on . Namely, suppose that each element is chosen independently with probability . Then for any probability to choose it is equal to .
Definition 2.2.
Let be an arbitrary variety of Lie rings and let be a finite Lie ring in this variety. The system of elements () of a free Lie ring in a variety is called uniformly distributed on , if for any probability that is equal to , where is chosen at random. It means that if runs over then any is the image of exactly elements of .
Definition 2.3.
A system of elements () of a free Lie ring of a variety is uniformly distributed on if it is uniformly distributed on any finite Lie ring of this variety.
Clearly, the property of a system of elements to be uniformly distributed on the variety of metabelian Lie algebras does not depend on a set of free generators chosen in . Indeed, let be some other system of free generators of . Then we have
(1) 
Here and are Lie polynomials. Let
and
for . Then, obviously, in .
Let take each to , where . Let us show that different ’s correspond to different ’s. If it is not the case then the images of some two elements under coincide. Since the images of are in and is finite there exists an tuple not lying in the image of . Let . Then (1) implies , we get a contradiction. Consequently, is a bijection. Thus for any the number of tuples such that in is equal to the number of tuples , such that in .
The group analogues of uniformly distributed elements are called measure preserving elements (see [13]). Later on we will need the lemma from [13].
Lemma 2.4.
Let be a free abelian group of rank . A system of elements () of this group preserves measure on the variety of abelian groups iff it is primitive.
Definition 2.5.
For any associative commutative ring a vector is called unimodular if the ideal generated by the coordinates of this vector coincides with .
The following definition generalizes Definition 2.5.
Definition 2.6.
Let be an associative commutative ring and let be an ideal in this ring. A vector is called modular if the ideal generated by the coordinates of this vector coincides with .
Let us formulate one more statement we will need in this paper.
Theorem 2.7.
[2] Let be a chain of commutative rings satisfying the following properties.

For any the unit of lies in .

Any ring is a retract of , the kernel of this retract is generated by an element , and is not a zero divisor of the ring .

For any the group of invertible matrices of order acts transitively on the set of unimodular vectors in .

If is the ideal in generated by , then is a free module of rank .
Then for all the group acts transitively on the set of modular vectors in .
Let us remind the definition of partial derivatives in free and free metabelian Lie rings. Consider the image of in under the natural embedding. For the sake of simplicity let us denote this image also by . It is clear that there are unique elements such that
(2) 
These elements are called partial (right) derivatives of . Evidently, we can say that the maps are derivations. Namely, these maps have the following properties
(more precisely the second property should be written as
where is the natural embedding of the free Lie ring to , i.e. and by induction ).
Let us define partial derivatives on a free metabelian Lie ring. By denote the set of commutative associative polynomials in the variables . Let be the natural homomorphism i.e. the homomorphism taking each to itself and let also be the natural homomorphism (that is ). Consider the maps . It is easy to see that are well defined. Indeed, the only difficulty in defining this map is that is not an isomorphism if . Therefore, each element has more than one preimage under in . However, if are such that then , where is the derivative ring of . Therefore, we have , where are monomials in , are monomials in , and . So, we obtain
(3) 
Next, let be in . Then . Indeed, for some Lie monomials in consequently . If lies in then . Since we obtain as above that . Therefore, the value of (3) is . So, the value of does not depend on the element in we are taking.
Let be an element of a free (metabelian) Lie algebra. It follows from the definition of derivatives that the value of a partial derivative of depends on the choice of the system of free generators.
Let be a metabelian Lie ring and let , , , be Lie polynomials. Substitute for in . By denote the obtained expression considered as an element of .
Given the system of elements of by denote the Jacobi matrix of this system, i.e. the matrix
Let
be Lie polynomials. Substitute these polynomials in for the corresponding and denote by the obtained matrix.
Let be the free right module with a basis . Consider the set of square matrices of the second order
where is a linear polynomial in and . Define the multiplication on by the rule . It is easy to show that is a metabelian Lie ring with respect to this multiplication.
Let be the homomorphism from to taking each generator to the matrix
It is wellknown (see, for example, [1, 2]) that the analogue of for Lie algebras over a field (for instance, over ) is an embedding, . Since , this map is obviously an embedding of into .
By denote the set of linear polynomials (without a free term) in variables from the set . Next, let be positive integers and . By denote the ideal in generated by and , where . Besides, by denote the commutative associative ring , by the set of linear polynomials in , and by the free right generated module. To denote the elements of a basis of we use the characters as well as to denote the elements of the basis of . Finally, by we denote the Lie ring of matrices of the form
where , , and the Lie multiplication is defined in the natural way. Obviously, is a finite metabelian Lie ring.
Let be the natural homomorphism. By denote the image of under the action of . Respectively, by let us denote the Jacobi matrix for a system of elements over , i.e. the matrix
The following diagram shows the relationship among the rings described above:
3. Property of primitivity and uniform distribution of elements
Let . Later on, by we denote the element of such that is a linear combination of the elements in . This linear combination itself will be denoted by . The identities of a free metabelian Lie ring are homogeneous, therefore, for any the elements and are defined uniquely and . By denote the ideal in generated by the set . Finally, for any matrix we will use the following notation
Lemma 3.1.
Let be an invertible matrix such that its coefficients are in . Then generates the ideal .
Proof.
The statement of this lemma follows from the following chain of equalities.
∎
Lemma 3.2.
Let be an automorphism of such that . Then the Jacobi matrix is invertible in the ring of matrices over .
Proof.
Let and be endomorphisms of the free metabelian Lie ring defined as follows: . Then we have
By denote . The following “chainrule” formulas hold:
These formulas imply
(4) 
Let be an automorphism of . Then is also an automorphism. Let . Using (4) for the identity automorphism we obtain
(5) 
Note that the elements of are polynomials with integer coefficients. It implies that the matrix is invertible from the right in the ring of matrices over . By [16] the Jacobi matrix is invertible in the ring of matrices over . Consequently, the right and left inverses of this matrix coincide. Thus, is invertible in the ring of matrices over . ∎
Theorem 3.3.
A system of elements () of the free metabelian Lie ring is primitive iff the ideal generated by all minors of coincides with .
Proof.
Suppose that the ideal generated by all minors of coincides with . Then each column of is a unimodular vector. Consequently, [10] implies that there exists an invertible matrix with entries in and such that . Hence
It can be shown that minors of are linear combinations of the minors of . So, the ideal generated by minors of is contained in the ideal generated by the minors of . Since is invertible, these ideals coincide.
Let be obtained from by deletion the first row and the first column. Then the set of minors of coincides with . Therefore the first column of is unimodular. Thus there exist matrix such that the product of by the first column of is equal to the column of the length . extend up to an matrix as follows:
We obtain
And so on. Finally we obtain a matrix of the form
Extend this matrix up to an matrix as follows:
Clearly, is of the form
for some matrix .
It is obvious that is invertible. Moreover, let us notice that for .
Consider the homomorphism that takes each polynomial in to its free term. The ideal generated by all minors of coincides with . Therefore applying to the minors of we obtain that the ideal generated by all minors of coincides with . It is well known that in this case the system can be included in a set of generators of the free abelian Lie algebra with a basis . Therefore, this system can be included in a system of generators of the free metabelian Lie algebra as well.
Let be a subring of such that it is generated by . Consider the elements . they can be represented as polynomials in . Thus, one can find linear combinations of the elements , with coefficients in such that these combinations satisfy following property. If we represent () as polynomials in then these polynomials do not depend on . For all subtract the linear combinations corresponding to from the last th columns of . Let be the obtained matrix.
Denote by the identity matrix and by the matrix unit corresponding to the th row and the th column. Then we have is a product of and the matrices of the form for and . Since the matrices and are invertible so is . The elements , , , generate the same ideal in as the elements do. At it was shown above, this ideal is . Consequently, , , , generate the same ideal of as do.
Consider the following chain of the subrings of : . Easy to see, that this chain satisfies the conditions of Theorem 2.7. Indeed, the first two conditions are obvious (in the second condition we take for ). The third condition follows from [10]. The fourth condition is also obvious because , therefore is isomorphic to the set of linear combinations of . This means that is a module of rank . Consequently, the set of all matrices with coefficients in acts transitively on the set of all modular vectors. In other words, there exists an invertible matrix (in which the elements are written as expressions in ) such that