GAP (LieRing) - Chapter 1: Introduction

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1 Introduction
1.1 Preliminaries
1.2 The free Lie ring
1.3 The Lazard correspondence

### 1 Introduction

#### 1.1 Preliminaries

A Lie ring \(L\) is a \(\mathbb{Z}\)-module equipped with a multiplication, denoted by a bracket \([~,~]\) with

- \([x,x]=0\) for all \(x\) in \(L\),

- \([x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0\) for all \(x,y,z\) in \(L\).

Contrary to Lie algebras (which are defined over a field), Lie rings may have torsion elements, i.e., elements \(x \neq 0\) such that \(mx=0\) for some \(m\in \mathbb{Z}\).

We say that a Lie ring is finite-dimensional if it is finitely-generated as abelian group. All functions of this package deal with finite-dimensional Lie rings.

Here is an example of a Lie ring \(L\) of order \(5^6\). As abelian group \(L\) is generated by \(x_1,x_2,x_3,x_4,x_5\). We have \(5x_i=0\) for \(i=1,\ldots,4\), and \(25x_5=0\). Furthermore,

\[ [x_1,x_4] = 4x_2+5x_5,~ [x_3,x_4] = 4x_1,~ [x_3,x_5]=4x_2,~ [x_4,x_5]=4x_3. \]

One of the main functions of this package constructs a Lie ring given by a multiplication table (as above) from a finite presentation. The Lie ring above can be obtained as follows.

    gap> L:= FreeLieRing( Integers, ["a","b"] );
    <Free algebra over Integers generators: a, b >
    gap> a:= L.1; b:= L.2;
    a
    b
    gap> S:= [ 5*a-(b*a)*a-((b*a)*b)*b,5*b];
    [ (5)*a+(-1)*(a,(a,b))+(b,(b,(a,b))), (5)*b ]
    gap> K:= FpLieRing( L, S : maxdeg:= 4 );
    <Lie ring with 5 generators>
    gap> v:=BasisVectors( Basis(K) );
    [ v_1, v_2, v_3, v_4, v_5 ]
    gap> v[1]*v[4];
    4*v_2+5*v_5
    gap> Torsion( Basis(K) );
    [ 5, 5, 5, 5, 25 ]

#### 1.2 The free Lie ring

Let \(X\) be a set of letters, which we denote by \( x_1,\ldots,x_n\). Then the free magma \(M(X)\) on \(X\) is defined to be the set of all bracketed expressions in the elements of \(X\). More precisely, we have that \(X\) is a subset of \(M(X)\) and if \(a,b\in M(X)\), then also \((a,b)\in M(X)\). The free magma has a natural binary operation \(m\) with \(m(a,b) = (a,b)\).

The elements of the free magma have a degree which is defined as \(\deg(a,b) = \deg(a)+\deg(b)\). The degree of the elements of \(X\) can be set to be any positive integer. (Usually this is 1, but it is possible to use different degrees for the elements of \(X\).)

Let \(R\) be a ring; then the free algebra \(A_R(X)\) on \(X\) over \(R\) is the \(R\)-span of \(M(X)\). The product on \(A_R(X)\) is obtained by bilinearly extending the map \(m\).

The elements of \(M(X)\) are called monomials of \(A_R(X)\). We use the following ordering on them. The elements of \(X\) are ordered arbitrarily. Then \( (a,b) < (c,d)\) if \(\deg(a,b) < \deg(c,d)\). If these two numbers are equal, then \( (a,b) < (c,d)\) if \(a < c\), and in case \(a=c\), if \(b < d\). Using this ordering we can speak of leading monomial, and leading coefficient of an element of \(A_R(X)\). Using these notions one can develop a Groebner basis theory for ideals in \(A_R(X)\) (see [CdG07] and [CdG09]).

Let \(J\) be the ideal of \(A_R(X)\) generated by all elements

- \((a,a)\),

- \((a,b)+(b,a)\),

- \((a,(b,c))+(c,(a,b))+(b,(c,a))\),

for \(a,b,c\in M(X)\). Set \(L_R(X) = A_R(X)/J\), which is called the free Lie ring over \(R\) generated by \(X\).

The free Lie ring is one of the central objects of this package. It can be defined over the integers, or over a field. The free Lie rings that can be constructed using this package rewrite their elements using anticommutativity. The Jacobi identity is not used for rewriting; this is because that would lead to expression swell, and sometimes tedious rewriting of elements to a form in which that can no longer be recognised. So, strictly speaking, we work with the free anticommutative algebra.

#### 1.3 The Lazard correspondence

Using the Baker-Campbell-Hausdorff (or BCH) formula one can define an associative multiplication on a nilpotent Lie ring of order \(p^n\) and nilpotency class \( < p \). This makes the Lie ring into a \(p\)-group of the same order and nilpotency class. The BCH-formula also has inverses, which can be used to define an addition and a Lie bracket on a \(p\)-group of class \( < p \). These make the group into a Lie ring of the same order and nilpotency class.

These two operations are mutually inverse, and so define an equivalence of the categories of \(p\)-groups of class \( < p \) and nilpotent Lie rings of the same order and nilpotency class. This equivalence is known as the Lazard correspondence (see [Khu98]). This package has functions for performing this correspondence, i.e., to make a \(p\)-group into a Lie ring and vice versa. For the algorithms used we refer to [CdGVL11].

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