  
  [1X1 [33X[0;0YIntroduction[133X[101X
  
  
  [1X1.1 [33X[0;0YPreliminaries[133X[101X
  
  [33X[0;0YA  Lie  ring  [22XL[122X  is  a [22XZ[122X-module equipped with a multiplication, denoted by a
  bracket [22X[~,~][122X with[133X
  
  [30X    [33X[0;6Y[22X[x,x]=0[122X for all [22Xx[122X in [22XL[122X,[133X
  
  [30X    [33X[0;6Y[22X[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0[122X for all [22Xx,y,z[122X in [22XL[122X.[133X
  
  [33X[0;0YContrary  to  Lie  algebras  (which are defined over a field), Lie rings may
  have torsion elements, i.e., elements [22Xx ≠ 0[122X such that [22Xmx=0[122X for some [22Xm∈ Z[122X.[133X
  
  [33X[0;0YWe  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.[133X
  
  [33X[0;0YHere  is  an  example  of  a  Lie ring [22XL[122X of order [22X5^6[122X. As abelian group [22XL[122X is
  generated by [22Xx_1,x_2,x_3,x_4,x_5[122X. We have [22X5x_i=0[122X for [22Xi=1,...,4[122X, and [22X25x_5=0[122X.
  Furthermore,[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[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.[133X [124X[133X
  
  
  [33X[0;0YOne  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.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["a","b"] );[127X[104X
    [4X[28X<Free algebra over Integers generators: a, b >[128X[104X
    [4X[25Xgap>[125X [27Xa:= L.1; b:= L.2;[127X[104X
    [4X[28Xa[128X[104X
    [4X[28Xb[128X[104X
    [4X[25Xgap>[125X [27XS:= [ 5*a-(b*a)*a-((b*a)*b)*b,5*b];[127X[104X
    [4X[28X[ (5)*a+(-1)*(a,(a,b))+(b,(b,(a,b))), (5)*b ][128X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, S : maxdeg:= 4 );[127X[104X
    [4X[28X<Lie ring with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27Xv:=BasisVectors( Basis(K) );[127X[104X
    [4X[28X[ v_1, v_2, v_3, v_4, v_5 ][128X[104X
    [4X[25Xgap>[125X [27Xv[1]*v[4];[127X[104X
    [4X[28X4*v_2+5*v_5[128X[104X
    [4X[25Xgap>[125X [27XTorsion( Basis(K) );[127X[104X
    [4X[28X[ 5, 5, 5, 5, 25 ][128X[104X
  [4X[32X[104X
  
  
  [1X1.2 [33X[0;0YThe free Lie ring[133X[101X
  
  [33X[0;0YLet  [22XX[122X  be  a  set of letters, which we denote by [22Xx_1,...,x_n[122X. Then the free
  magma [22XM(X)[122X on [22XX[122X is defined to be the set of all bracketed expressions in the
  elements  of  [22XX[122X.  More  precisely, we have that [22XX[122X is a subset of [22XM(X)[122X and if
  [22Xa,b∈  M(X)[122X,  then  also  [22X(a,b)∈  M(X)[122X.  The  free magma has a natural binary
  operation [22Xm[122X with [22Xm(a,b) = (a,b)[122X.[133X
  
  [33X[0;0YThe  elements of the free magma have a degree which is defined as [22Xdeg(a,b) =
  deg(a)+deg(b)[122X. The degree of the elements of [22XX[122X can be set to be any positive
  integer. (Usually this is 1, but it is possible to use different degrees for
  the elements of [22XX[122X.)[133X
  
  [33X[0;0YLet  [22XR[122X  be a ring; then the free algebra [22XA_R(X)[122X on [22XX[122X over [22XR[122X is the [22XR[122X-span of
  [22XM(X)[122X. The product on [22XA_R(X)[122X is obtained by bilinearly extending the map [22Xm[122X.[133X
  
  [33X[0;0YThe  elements  of  [22XM(X)[122X are called monomials of [22XA_R(X)[122X. We use the following
  ordering  on  them.  The elements of [22XX[122X are ordered arbitrarily. Then [22X(a,b) <
  (c,d)[122X  if  [22Xdeg(a,b) < deg(c,d)[122X. If these two numbers are equal, then [22X(a,b) <
  (c,d)[122X  if [22Xa < c[122X, and in case [22Xa=c[122X, if [22Xb < d[122X. Using this ordering we can speak
  of  leading monomial, and leading coefficient of an element of [22XA_R(X)[122X. Using
  these  notions  one can develop a Groebner basis theory for ideals in [22XA_R(X)[122X
  (see [CG07] and [CG09]).[133X
  
  [33X[0;0YLet [22XJ[122X be the ideal of [22XA_R(X)[122X generated by all elements[133X
  
  [30X    [33X[0;6Y[22X(a,a)[122X,[133X
  
  [30X    [33X[0;6Y[22X(a,b)+(b,a)[122X,[133X
  
  [30X    [33X[0;6Y[22X(a,(b,c))+(c,(a,b))+(b,(c,a))[122X,[133X
  
  [33X[0;0Yfor  [22Xa,b,c∈  M(X)[122X.  Set [22XL_R(X) = A_R(X)/J[122X, which is called the free Lie ring
  over [22XR[122X generated by [22XX[122X.[133X
  
  [33X[0;0YThe  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.[133X
  
  
  [1X1.3 [33X[0;0YThe Lazard correspondence[133X[101X
  
  [33X[0;0YUsing  the  Baker-Campbell-Hausdorff  (or  BCH)  formula  one  can define an
  associative  multiplication  on  a  nilpotent  Lie  ring  of  order  [22Xp^n[122X and
  nilpotency  class  [22X<  p[122X.  This makes the Lie ring into a [22Xp[122X-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 [22Xp[122X-group of class [22X< p[122X.
  These make the group into a Lie ring of the same order and nilpotency class.[133X
  
  [33X[0;0YThese  two  operations are mutually inverse, and so define an equivalence of
  the  categories of [22Xp[122X-groups of class [22X< p[122X and nilpotent Lie rings of the same
  order  and  nilpotency  class.  This  equivalence  is  known  as  the [13XLazard
  correspondence[113X (see [Khu98]). This package has functions for performing this
  correspondence,  i.e., to make a [22Xp[122X-group into a Lie ring and vice versa. For
  the algorithms used we refer to [CGV11].[133X
  
