The GAP Package LiePRing

Abstract

A symbolic Lie p-ring defines a family of Lie rings with elements for infinitely many different primes p and a fixed positive integer n. Symbolic Lie p-rings are used to describe the classification of isomorphism types of nilpotent Lie rings of order for all primes p and all . This classification is available as the LiePRing package of the computer algebra system GAP. We give a brief description of this package, including an approach towards computing the automorphism group of a symbolic Lie p-ring.

Introduction

A Lie ring is an additive abelian group with a multiplication, denoted by [., .], that is bilinear, alternating and satisfies the Jacobi identity. A Lie p-ring is a nilpotent Lie ring with elements for some prime power . Such a Lie p-ring of order can be described by a presentation P(A) on n generators with coefficients , so that and are integers in the range and the following relations hold:

We generalize this type of presentation so that it defines a family of Lie p-rings for various different primes. For this purpose let p be an indeterminate, let be a polynomial ring in commuting variables and let and in R. In some (rare) cases it is convenient to allow some of the coefficients and to be rational functions over R; note that we use this only for coefficients or if so that is an element of order p.

If a fixed prime P and integers are given, then we specify the a polynomial at these values by choosing W to be the smallest primitive root mod P and evaluating in . We specify a rational function a/b with by specifying the polynomials a and b to and in , and then we determine where satisfies . Note that only choices of with are valid.

Let be an infinite set of primes, let and for let

Then the presentation P(A) defines a symbolic Lie p-ring with respect to and if for each and each the presentation P(A) specified at these points is a finite Lie p-ring of order .

A symbolic Lie p-ring describes a family of finite Lie p-rings: for each this contains members. Symbolic Lie p-rings are used to describe the complete classification up to isomorphism of all Lie p-rings of order dividing for as obtained by Newman, O’Brien and Vaughan-Lee [6, 7]. This is available in computational form in the LiePRing package [4] of the computer algebra system GAP [9]. The following exhibits an example.

Example 1

We consider the symbolic Lie p-ring with generators and the (non-trivial) relations

Let be the set of all primes and let

Then defines a family of Lie p-rings of order for each .

The LiePRing package allows symbolic computations with symbolic Lie p-rings . “Symbolic computations” means that it computes with as if computing with all Lie p-rings L in the family defined by simultaneously. For example, it allows us

• to compute series of ideals such as the lower central series of L,

• to describe the automorphism group of L, and

• to determine the Schur multiplier of L, see [3].

Let P be a prime and let with . The Lazard correspondence [5] associates to each Lie p-ring L of order a group G(L) of order . This correspondence translates Lie ring isomorphisms to group isomorphisms and vice versa. Cicalo, de Graaf and Vaughan-Lee [2] determined an effective version of the Lazard correspondence and implemented this in the LieRing package [1] of GAP.

The following sections give a brief overview of some of the algorithms in the LiePRing package and they exhibit how the Lazard correspondence can be evaluated in GAP in this setting.

Elementary Computations

In this section we investigate computations with elements, subrings and ideals. Throughout, let be a symbolic Lie p-ring with respect to , let L be a finite Lie p-ring in the family defined by and let P be the prime of L. We write P(A) for the defining presentation in the finite and in the symbolic case. Thus depending on the context A is an integer matrix or a matrix over the ring Quot(R) of rational functions over the polynomial ring R.

Ring Invariants

The definition of can often be used for computations with . For example, if , then specifies to an invertible element in L and specifies to 0. Hence we can treat as a unit and as zero. The following example illustrates this for .

The Word Problem

Consider the case of a finite Lie p-ring L and let a be an arbitrary word in the generators of P(A). Then the relations in P(A) readily allow us to rewrite a to a unique equivalent normal form

Now consider the case of a symbolic Lie p-ring and let a be a word in the generators of P(A). Then the relations and the zeros of allow us to translate this to an equivalent reduced form; that is, a linear combination of the form

where are reduced modulo the polynomials in zeros; that is, the polynomial division algorithm dividing by the polynomials in zeros yields only trivial quotients. If and , then is the depth of this reduced form and is its leading coefficient. We say that represents the element a.

Example 2

We continue Example 1.

1. Consider the element . Using the relations of this reduces to . Note that a can be zero and non-zero in the Lie p-rings in the family defined by , depending on the choice of .

2. Consider the element . Then and hence, since in , it follows that a is a non-zero element in each Lie p-ring in the family defined by .

Subrings, Ideals and Series

Let be a symbolic Lie p-ring, let be words in the generators of P(A) and let U be the subring of generated by these words. Our aim is to determine an echelon generating set for U; that is, a generating set so that each is a reduced form in the generators with leading coefficient 1, the depths satisfy and each element in U is a linear combination in with coefficients in Quot(R). This may require the distinction of finitely many cases, as the following example indicates.

Example 3

We continue Example 1.

1. Let . As and with , it follows that in each Lie ring in the family defined by .

2. Let . Then using the relations of it follows that . Hence if and otherwise. Thus a case distinction is necessary to determine an echelon generating set for U.

Ideals are subrings that are closed under multiplication and hence they can also be described via echelon generating sets (subject to a case distinction). In turn, this then allows us to determine series such as the lower central series and the derived series of . The following example illustrates the handling of case distinctions in GAP.

Here the LiePRing package returns two new symbolic Lie p-rings S[1] and S[2]. These have different ring invariants and different bases:

In particular, in S[2] the polynomial t is a unit and the rational function x/t turns up as coefficient for the basis element .

Automorphism Groups

Given a symbolic Lie p-ring , we show how to determine a generic description for Aut(L) for each finite Lie p-ring L in the family defined by . The following gives a first illustration.

Example 4

We continue Example 1.

We note that is generated by . This allows us to describe each automorphism of via its images of and the same holds for each finite Lie p-ring in the family defined by . Write for the image of . Then for certain integers . We say that the automorphism is represented by the matrix . Note that different matrices may represent the same automorphism for a finite Lie p-ring L; for example, if P is the prime of L, then has order P and and give the same automorphism. We expand on this below.

Our algorithm determines that each automorphism of corresponds to a matrix of the form

with and arbitrary otherwise. If P is prime and L is a finite Lie p-ring over P, then we can choose for and thus Aut(L) has order .

Given a finite Lie p-ring L with prime P, we define its radical R(L) as the ideal of L generated by . The additive group of L/R(L) is an elementary abelian group of order , say, and the Lie ring multiplication of L/R(L) is trivial. Burnside’s Basis theorem (for example, see [8, page 140]) for finite p-groups translates readily to the following.

Lemma 1

Let L be a finite Lie p-ring and let the natural ring homomorphism.

1. R(L) is the intersection of all maximal Lie subrings of L.

2. Each minimal generating set of L has d elements and maps under onto a minimal generating set of L/R(L).

3. Each list of preimages under of a minimal generating set of L/R(L) is a minimal generating set of L.

Next, let P(A) be the presentation for the finite Lie p-ring L with generators so that . Then is a minimal generating set of L. Thus each automorphism of L is defined by its images on . These have the general form

with integer coefficients . For we note that . This allows us to write as a word in the ideal generators and of R(L) and that, in turn, allows us to determine the image in the form

where is a word in .

Theorem 1

The matrix defines an automorphism of L if and only if

1. , where , and

2. the images satisfy the relations of L.

Proof

First recall that a map for with extends to a Lie ring homomorphism if and only if satisfy the defining relations of L. This is von Dyck’s theorem (for example, see [8, page 51]) in the case of finitely presented groups and it translates readily to other algebraic objects such as Lie rings.

: Suppose that the coefficients define an automorphism . Then induces an automorphism . As with trivial multiplication, it follows that . Hence so (a) follows. (b) follows from von Dyck’s theorem.

: Suppose that (a) and (b) hold. As (b) holds, von Dyck’s theorem asserts that is a Lie ring homomorphism. As . it follows that the images of generate L as Lie ring. Hence is surjective. Since L is finite, it follows that is also injective and hence an automorphism.

This allows us to determine a generic description for Aut(L). Suppose that we have an automorphism given by indeterminates and write for . For write as a word in the generators and use this to determine . Evaluate the defining relations of L in . For each relation this leads to an expression

with a polynomial in the indeterminates .

Lemma 2

Let P be a prime and k minimal with for . If for all i, j and if , then the matrix defines an automorphism.

Proof

The generators that appear in the relations all lie in the radical, and so ensures that for all i, j. Hence the conditions of Theorem 1 are satisfied and the matrix defines an automorphism.

The integer in Lemma 2 is called the characteristic of R(L). If , then the conditions in Lemma 2 clearly determine all automorphisms of L. If , then the conditions in Lemma 2 may miss some automorphisms and there are examples where

but some of the summands are non-zero. So it seems possible that restricting our search to integer matrices which satisfy the equations could miss some automorphisms in some cases. In practice, we have not found a case where this happens.

Example 5

We continue Example 1 for a specific prime P.

Since the radical has characteristic P our method shows that the matrix

gives an automorphism if and only if . Let . Then

gives an automorphism. There was no need in this case to look for solutions to , but it is easy to “lift” B to a matrix which gives the same automorphism as B, but where . The first row of the matrix B represents the element . Now and so the vector also represents . Similarly the vector represents the same element of L as the third row of B. So

gives the same automorphism as B, but the (1, 1) entry in C satisfies the equation . Note that B gives an automorphism, but does not have the form specified in Example 4, whereas C gives the same automorphism as B, but does have the form specified.

More generally, in every case of Lie p-rings from our database that we have examined, we can show that if B is an integer matrix which gives an automorphism of L for some prime P, and if k is any positive integer, then B can be “lifted” to an integer matrix which gives the same automorphism as B but where the entries satisfy all the equations . So in every case that we have examined our method finds the full automorphism group.

We do not have a proof that our method always finds the full automorphism group. But there are several general criteria (such as the radical having characteristic P) which imply that our method does not miss any automorphisms. So in most cases our program is able to issue a “certificate of correctness”. In some cases it may be necessary to examine the output from our program to prove that it has found the full automorphism group.

Example 6

We consider the symbolic Lie p-ring on 7 generators with the non-trivial relations

Then and each Lie p-ring L in the family of is generated by . We define

Next, we write as words in . It can be read off from the defining relations that . Using this, we expand the mapping defined by to the remaining generators . For example, for this yields

We now evaluate the defining relations of in . For example evaluates to

Note that the coefficient of in this relation is zero. More generally, if is any of the relations then

and all have order p. So we obtain an automorphism at the prime P if and only if () for all relations .

Now let L be a finite Lie p-ring in the family defined by and let P be its prime. If the integer coefficients define an automorphism of L, then det(G) is coprime to P. Hence, examining the coefficient of in the relation above we see that

is equivalent to . In turn, this can now be used to simplify the remaining coefficients. Using now yields

As via , it follows that is coprime to P and . We now iterate this approach. Introducing another indeterminate D with we finally obtain that

evaluate to 0 modulo P. We use this to eliminate indeterminates in the descriptions of ; for example, we can replace by 0. We obtain

subject to the additional condition that the polynomials

must evaluate to . This is the resulting description of the automorphism groups of the Lie p-rings L in the family defined by . It implies that , where . The precise value of k depends on the two parameters x, y. When , if then ; if and then ; if and then ; finally if then . When then or 2.

The Lazard Correspondence

The final example of this abstract illustrates how the Lazard correspondent G(L) to a finite Lie p-ring L can be determined using the LieRing package [1].