  
  [1X1 [33X[0;0YThe theory[133X[101X
  
  [33X[0;0YThe  [5XSophus[105X  package  was  originally designed to aid the author to classify
  some  small-dimensional  nilpotent  Lie  algebras  over  small  fields.  The
  classification  follows  the ideas that were used to classify small [23Xp[123X-groups
  by  O'Brien [O'B90]. The theory developed by O'Brien could easily be adopted
  to  Lie  algebras, and the details of this new theory can be found in [Sch].
  Here  we  only summarise the main ideas, so that the user can understand the
  procedures implemented in this package. In this section [23XL[123X denotes a finitely
  generated, and hence finite-dimensional, nilpotent Lie algebra. Suppose that
  [23XL[123X has nilpotency class [23Xc[123X, and hence the lower central series is as follows:[133X
  
  
  [24X[33X[0;6YL=                             \gamma_1(L)>\gamma_2(L)=L'>\gamma_3(L)>\cdots
  >\gamma_c(L)>\gamma_{c+1}(L)=0.[133X
  
  [124X
  
  [33X[0;0YWe  say  that a basis [23X\mathcal B=\{b_1,\ldots,b_n\}[123X for [23XL[123X is [13Xcompatible with
  the  lower central series[113X if there are indices [23X1=i_1<i_2<\cdots<i_{c}<n[123X such
  that    [23X\{b_{i_k},\ldots,b_n\}[123X    is    a    basis    of   [23X\gamma_k(L)[123X   for
  [23Xk\in\{1,\ldots,c\}[123X. We compute the structure constants table with respect to
  this  basis,  that  is,  we  compute  coefficients  [23X\alpha_{i,j}^k[123X for [23X1\leq
  i<j<k\leq n[123X such that[133X
  
  
  [24X[33X[0;6Y[b_i,b_j]=\sum_{k=j+1}^n\alpha_{i,j}^k b_k.[133X
  
  [124X
  
  [33X[0;0YSuppose that [23Xb_i\in\gamma_j(L)\setminus\gamma_{j+1}(L)[123X. Then we say that the
  number [23Xj[123X is the [13Xweight[113X of the basis element [23Xb_i[123X.[133X
  
  [33X[0;0YNote  that  in  the nilpotent Lie algebra [23XL[123X minimal generating sets have the
  same  size,  namely  the  dimension of [23XL/L'[123X. If [23X\dim L/L'=d[123X then we call [23XL[123X a
  [23Xd[123X[13X-generator  algebra[113X.  We  call  a basis [23X\mathcal B[123X a [13Xnilpotent basis[113X if the
  following hold.[133X
  
  [30X    [33X[0;6YThe basis [23X\mathcal B[123X is compatible with the lower central series.[133X
  
  [30X    [33X[0;6YFor  each  [23Xb_i\in  \mathcal  B[123X with weight [23Xw\geq 2[123X there are [23Xb_{j_1},\
        b_{j_2}\in\mathcal  B[123X  with  weight  1 and [23Xw-1[123X, respectively such that
        [23Xb_i=[b_{j_1},b_{j_2}][123X.  The  product  [23X[b_{j_1},b_{j_2}][123X  is called the
        definition of [23Xb_i[123X.[133X
  
  [33X[0;0YA  Lie  algebra  [23XK[123X  is said to be a [13Xcentral extension[113X of [23XL[123X if [23XL\cong K/I[123X for
  some  ideal  [23XI[123X  such  that  [23XI\leq  Z(K)\cap  K'[123X.  Suppose that [23Xc[123X denotes the
  nilpotency  class of [23XL[123X. Then a Lie algebra [23XK[123X is an [13Ximmediate descendant[113X of [23XL[123X
  if   [23XK[123X  has  class  [23Xc+1[123X  and  [23XK/\gamma_{c+1}(K)\cong  L[123X.  As  in  this  case
  [23X\gamma_{c+1}(K)\leq  Z(K)\cap  K'[123X, it follows that an immediate descendant [23XK[123X
  is  a central extension of [23XL[123X. If [23Xs=\dim \gamma_{c+1}(K)[123X then [23XK[123X is said to be
  a [13Xstep-[23Xs[123X[113X immediate descendant of [23XL[123X.[133X
  
  [33X[0;0YLet  [23XL[123X  be  a [23Xd[123X-generator nilpotent Lie algebra with class [23Xc[123X, and let [23XF[123X be a
  free  Lie  algebra  of  rank [23Xd[123X. Choose an ideal [23XI[123X of [23XF[123X such that [23XL\cong F/I[123X.
  Then  the  Lie  algebra  [23XL^*=F/[I,F][123X  is  called the [13XLie cover[113X of [23XL[123X. The [13XLie
  multiplicator[113X  in  [23XL^*[123X  is  the  subspace  [23XI/[I,F][123X  and  the  [13XLie nucleus[113X is
  [23X\gamma_c(L^*)[123X.  It  clear  from  the  definition  that  [23XL^*/M\cong  L[123X. It is
  verified  in  [Sch]  that,  up  to  isomorphism,  the  Lie  cover,  the  Lie
  multiplicator  and the Lie nucleus are determined by the isomorphism type of
  [23XL[123X.  Further,  each  central  extension  of  the nilpotent Lie algebra [23XL[123X is a
  quotient  of  the  Lie  cover  [23XL^*[123X.  Thus  it is possible to obtain all such
  descendants by first computing the Lie cover; this procedure is explained in
  [Sch].  Similar  ideas  can  be  used to compute the automorphism group of a
  nilpotent  Lie  algebra, and to verify isomorphism between two nilpotent Lie
  algebras; see [Sch] for details.[133X
  
  [33X[0;0YThe main functions in [5XSophus[105X are thus able to compute[133X
  
  [30X    [33X[0;6Ya nilpotent basis for a nilpotent Lie algebra;[133X
  
  [30X    [33X[0;6Ythe cover of a nilpotent Lie algebra;[133X
  
  [30X    [33X[0;6Ythe immediate descendants of a nilpotent Lie algebra;[133X
  
  [30X    [33X[0;6Ythe full automorphism group of a nilpotent Lie algebra.[133X
  
  [33X[0;0YThere  is  also  a  function  in  the  package to check if two nilpotent Lie
  algebras  are  isomorphic.  After  repeated  applications  of  the immediate
  descendants  algorithm, it is, in theory, possible to list all nilpotent Lie
  algebras  of  a  given  dimension over a prime field [23X\mathbb F_p[123X. Of course,
  this  computation  requires  relatively  large  computational resources, and
  quickly becomes unfeasible as the dimension or the characteristic [23Xp[123X grows.[133X
  
  [33X[0;0YThe  [5XSophus[105X  package  was  written for the GAP~4 computer algebra system. In
  many procedures it is very important that we can compute the stabiliser of a
  subspace  under  some  matrix  group  action.  This is carried out using the
  procedures  implemented  in  the [3Xautpgrp[103X package [EO]. Hence this package is
  required to run [5XSophus[105X.[133X
  
  [33X[0;0YThe current version of [5XSophus[105X deals with general nilpotent Lie algebras over
  finite  prime  fields. If you are to compute with Lie algebras obtained from
  group  algebras  via  the bracket operation, then another GAP package LAGUNA
  [RS] may also offer some very efficient methods.[133X
  
