  
  [1X2 [33X[0;0YThe functions[133X[101X
  
  
  [1X2.1 [33X[0;0YThe free Lie ring[133X[101X
  
  [1X2.1-1 FreeLieRing[101X
  
  [29X[2XFreeLieRing[102X( [3XR[103X, [3Xnames[103X ) [32X method
  [29X[2XFreeLieRing[102X( [3XR[103X, [3Xnames[103X, [3Xdeg[103X ) [32X method
  [29X[2XFreeLieRing[102X( [3XR[103X, [3Xk[103X ) [32X method
  [29X[2XFreeLieRing[102X( [3XR[103X, [3Xk[103X, [3Xdeg[103X ) [32X method
  
  [33X[0;0YHere  [3XR[103X is a ring, which has to be either the integers, or a field. [3Xnames[103X is
  a  list of strings, which will be the names of the generators. This function
  returns  the  free Lie ring over [3XR[103X, with generators named as in [3Xnames [103X. If [3XL[103X
  denotes the output, then [3XL.i[103X will be the i-th generator. If a third argument
  [3Xdeg[103X  is  given  then  this  must  be  a list of positive integers. Then each
  generator  will have a degree equal to the corresponding element of the list
  [3Xdeg[103X.[133X
  
  [33X[0;0YMonomials  in the free Lie ring of the form [22X(a,b)[122X with [22Xa>b[122X are automatically
  rewritten  as  [22X-(b,a)[122X.  Monomials  of  the form [22X(a,a)[122X are rewritten as zero.
  There  is  no  other  rewriting done. Therefore, the object returned by this
  function  is  strictly speaking not the same as the free Lie ring, it rather
  is the free anticommutative algebra.[133X
  
  [33X[0;0YMonomials  in  the  free Lie ring are printed as bracketed expressions. In a
  printed  element the monomials appear in increasing order; in particular the
  last monomial is the leading monomial.[133X
  
  [33X[0;0YIf  instead  of  the list [3Xnames[103X a positive integer [3Xk[103X is given, then the free
  Lie  ring  on  that number of generators is returned. Again we can give each
  generator a degree different from 1 by adding a third argument [3Xdeg[103X.[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 [27X(a*b)*b+2*a*b;       [127X[104X
    [4X[28X(2)*(a,b)+(-1)*(b,(a,b))[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 Degree[101X
  
  [29X[2XDegree[102X( [3Xf[103X ) [32X operation
  
  [33X[0;0YHere [3Xf[103X is an element of a free Lie ring. Its degree is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["a","b"] );;[127X[104X
    [4X[25Xgap>[125X [27Xa:= L.1;; b:= L.2;;[127X[104X
    [4X[25Xgap>[125X [27Xf:=(a*b)*b+2*a*b;[127X[104X
    [4X[28X(2)*(a,b)+(-1)*(b,(a,b))[128X[104X
    [4X[25Xgap>[125X [27XDegree(f);[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  
  [1X2.2 [33X[0;0YCreating Lie rings[133X[101X
  
  [33X[0;0YThe   package  can  deal  with  finite-dimensional  Lie  rings  given  by  a
  multiplication  table  (which follow the format for multiplication tables in
  the  [5XGAP[105X  library),  and  a  list  of moduli. This list has to have the same
  length  as the number of basis elements of the Lie ring. If the i-th element
  of  this  list is [22Xm[122X then the additive order of the i-th basis edlement if [22Xm[122X.
  If [22Xm=0[122X then the additive order is infinite.[133X
  
  [1X2.2-1 IsLieRing[101X
  
  [29X[2XIsLieRing[102X[32X filter
  
  [33X[0;0YThis is the category of finite-dimensional Lie rings.[133X
  
  [1X2.2-2 LieRingByStructureConstants[101X
  
  [29X[2XLieRingByStructureConstants[102X( [3Xtor[103X, [3XT[103X ) [32X operation
  
  [33X[0;0YHere [3XT[103X is a multiplication table, and [3Xtor[103X is a list of moduli. This function
  returns  the  corresponding Lie ring. In the example below we create the Lie
  ring with basis elements [22Xx,y,z[122X, with [22X[x,y]=z[122X, [22X3x=6y=3z=0[122X.[133X
  
  [33X[0;0YThe  multiplication  table  has  to  be  created using the [5XGAP[105X functions for
  constructing  multiplication tables of Lie algebras. In particular, we refer
  to  the  [5XGAP[105X reference manual for descriptions of the functions [2XEmptySCTable[102X
  ([14XReference: EmptySCTable[114X) [2XSetEntrySCTable[102X ([14XReference: SetEntrySCTable[114X)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 3, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [1,3] );[127X[104X
    [4X[25Xgap>[125X [27XLieRingByStructureConstants( [3,6,3], T );[127X[104X
    [4X[28X<Lie ring with 3 generators>[128X[104X
  [4X[32X[104X
  
  [1X2.2-3 FpLieRing[101X
  
  [29X[2XFpLieRing[102X( [3XL[103X, [3XR[103X ) [32X function
  
  [33X[0;0YHere  [3XL[103X  is  a  free  Lie  ring defined over the integers, and [3XR[103X is a set of
  elements  of  [3XL[103X.  This  function  returns  the  Lie  ring given by structure
  constants, that is isomorphic to [3XL[103X modulo the ideal generated by [3XR[103X.[133X
  
  [33X[0;0YIt  is  possible  to  set  the  option  [3Xmaxdeg[103X to a positive value [3Xd[103X. Then a
  nilpotent  quotient  is computed, i.e., all elements of [3XL[103X of degree strictly
  greater than [3Xd[103X will be treated as relations.[133X
  
  [33X[0;0YThe   algebra   that   is   output   by  this  function  has  an  attribute,
  [3XCanonicalProjection[103X,  which  is  a function mapping elements of the free Lie
  ring [3XL[103X to their projections in the output algebra.[133X
  
  [33X[0;0YThe algorithm behind this function has been described in [CG07] and [CG09].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["x","y"], [1,2] );                   [127X[104X
    [4X[28X<Free algebra over Integers generators: x, y >[128X[104X
    [4X[25Xgap>[125X [27Xx:= L.1;; y:= L.2;;[127X[104X
    [4X[25Xgap>[125X [27XR:= [((y*x)*x)*x-6*(y*x)*y, 3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ];[127X[104X
    [4X[28X[ (-1)*(x,(x,(x,y)))+(-6)*(y,(x,y)), [128X[104X
    [4X[28X  (-3)*(x,(x,(x,(x,(x,y)))))+(-20)*(y,(x,(x,(x,y)))) ][128X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, R : maxdeg:= 15 ); time;[127X[104X
    [4X[28X<Lie ring with 75 generators>[128X[104X
    [4X[28X944[128X[104X
    [4X[25Xgap>[125X [27Xf:=CanonicalProjection(K);[127X[104X
    [4X[28Xfunction( elm ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xf(R[1]);[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27Xf(x);[127X[104X
    [4X[28Xv_1[128X[104X
  [4X[32X[104X
  
  [1X2.2-4 FpLieAlgebra[101X
  
  [29X[2XFpLieAlgebra[102X( [3XL[103X, [3XR[103X ) [32X function
  
  [33X[0;0YThis  is  similar to [3XFpLieRing[103X, with the difference that the free Lie ring [3XL[103X
  must  be  defined  over a field. Then the algorithms become a lot faster (in
  most cases). The result however is a Lie algebra, and not a Lie ring.[133X
  
  
  [1X2.3 [33X[0;0YWorking with Lie rings[133X[101X
  
  [1X2.3-1 Basis[101X
  
  [29X[2XBasis[102X( [3XL[103X ) [32X operation
  
  [33X[0;0YHere [3XL[103X a Lie ring. Its basis is returned.[133X
  
  [33X[0;0YWe  note  that  in  [5XLieRing[105X Lie rings have one basis that is computed by the
  system; one should not try to set a basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 3, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [1,3] );[127X[104X
    [4X[25Xgap>[125X [27XK:= LieRingByStructureConstants( [3,6,3], T );[127X[104X
    [4X[28X<Lie ring with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XBasis(K); [127X[104X
    [4X[28XBasis( <Lie ring with 3 generators>, [ v_1, v_2, v_3 ] )[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( Basis(K) );[127X[104X
    [4X[28X[ v_1, v_2, v_3 ][128X[104X
  [4X[32X[104X
  
  [1X2.3-2 StructureConstantsTable[101X
  
  [29X[2XStructureConstantsTable[102X( [3XB[103X ) [32X operation
  
  [33X[0;0YHere  [3XB[103X  is  the  basis  of  a  Lie  ring.  Its structure constants table is
  returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 3, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [1,3] );[127X[104X
    [4X[25Xgap>[125X [27XK:= LieRingByStructureConstants( [3,6,3], T );[127X[104X
    [4X[28X<Lie ring with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XStructureConstantsTable( Basis(K) );[127X[104X
    [4X[28X[ [ [ [  ], [  ] ], [ [ 3 ], [ 1 ] ], [ [  ], [  ] ] ], [128X[104X
    [4X[28X  [ [ [ 3 ], [ -1 ] ], [ [  ], [  ] ], [ [  ], [  ] ] ], [128X[104X
    [4X[28X  [ [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ] ], -1, 0 ][128X[104X
  [4X[32X[104X
  
  [1X2.3-3 Torsion[101X
  
  [29X[2XTorsion[102X( [3XB[103X ) [32X operation
  
  [33X[0;0YHere  [3XB[103X  is the basis of a Lie ring. The list of torsion moduli of its basis
  elements is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 3, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [1,3] );[127X[104X
    [4X[25Xgap>[125X [27XK:= LieRingByStructureConstants( [3,6,3], T );[127X[104X
    [4X[28X<Lie ring with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XTorsion( Basis(K) );[127X[104X
    [4X[28X[ 3, 6, 3 ][128X[104X
  [4X[32X[104X
  
  [1X2.3-4 Coefficients[101X
  
  [29X[2XCoefficients[102X( [3XB[103X, [3Xelm[103X ) [32X operation
  
  [33X[0;0YHere  [3XB[103X  is  the  basis of a Lie ring, and [3Xelm[103X is an element of the same Lie
  ring. The coefficients of [3Xelm[103X with respect to [3XB[103X are returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["x","y"] );; x:= L.1;; y:= L.2;;[127X[104X
    [4X[25Xgap>[125X [27Xrr:=[((y*x)*x)*x-6*(y*x)*y, 3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ];;[127X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, rr : maxdeg:= 6 );;[127X[104X
    [4X[25Xgap>[125X [27XC:=LieCentre(K);[127X[104X
    [4X[28X<Lie ring with 9 generators>[128X[104X
    [4X[25Xgap>[125X [27XCoefficients( Basis(K), Basis(C)[6] );[127X[104X
    [4X[28X[ 5, 5, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XCoefficients( Basis(C), Basis(C)[6] );[127X[104X
    [4X[28X[ 0, 0, 0, 0, 0, 1, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [1X2.3-5 SubLieRing[101X
  
  [29X[2XSubLieRing[102X( [3XL[103X, [3Xgens[103X[, [3Xstring[103X] ) [32X operation
  
  [33X[0;0YHere  [3XL[103X  is  a  Lie  ring,  and  [3Xgens[103X a list of elements of [3XL[103X. This function
  constructs  the subring generated by the elements in [3Xgens[103X. If these elements
  are  known  to  form a basis of the subalgebra, then as a third argument the
  string  [3X"basis"[103X can be added. That makes the execution of the function a lot
  faster.[133X
  
  [33X[0;0YThis  function  depends  on  hermite  and  Smith  normal  form computations.
  Therefore in practice, for bigger inputs, it can be slow.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["x","y"] );;                         [127X[104X
    [4X[25Xgap>[125X [27Xx:= L.1;; y:= L.2;;[127X[104X
    [4X[25Xgap>[125X [27Xrr:=[((y*x)*x)*x-6*(y*x)*y, 3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ];; [127X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, rr : maxdeg:= 8 );[127X[104X
    [4X[28X<Lie ring with 41 generators>[128X[104X
    [4X[25Xgap>[125X [27Xb:= Basis(K);;[127X[104X
    [4X[25Xgap>[125X [27XM:= SubLieRing( K, [ b[30], b[40] ] );[127X[104X
    [4X[28X<Lie ring with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XTorsion(Basis(M));[127X[104X
    [4X[28X[ 3, 6, 6, 12, 360, 0 ][128X[104X
    [4X[25Xgap>[125X [27XBasis(M)[2];[127X[104X
    [4X[28X3*v_2+2*v_3+2*v_10+4*v_12+4*v_13+5*v_14+v_15+3*v_17+3*v_18+6*v_20+10*v_22+6*v_[128X[104X
    [4X[28X24+6*v_25+10*v_26+4*v_27+18*v_28+30*v_29+60*v_30+360*v_31+5040*v_32[128X[104X
  [4X[32X[104X
  
  [1X2.3-6 LieRingIdeal[101X
  
  [29X[2XLieRingIdeal[102X( [3XL[103X, [3Xgens[103X[, [3Xstring[103X] ) [32X operation
  
  [33X[0;0YThis  is  the  same as [3XSubLieRing[103X except that the output is an ideal (on the
  level of data structures that is the same as a Lie subring).[133X
  
  [1X2.3-7 NaturalHomomorphismByIdeal[101X
  
  [29X[2XNaturalHomomorphismByIdeal[102X( [3XL[103X, [3XI[103X ) [32X operation
  
  [33X[0;0YHere  [3XL[103X  is  a  Lie  ring, and [3XI[103X an ideal of [3XL[103X. This function constructs the
  canonical projection of [3XL[103X on the quotient of [3XL[103X by [3XI[103X.[133X
  
  [33X[0;0YWe  remark that it is [13Xnot checked[113X whether [3XI[103X is an ideal or not. if [3XI[103X is just
  a subalgebra, then nothing is guaranteed about the result of this function.[133X
  
  [33X[0;0YAlso  this  function depends on Smith normal form computations; therefore it
  can be slow on bigger inputs.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["x","y"] );;                         [127X[104X
    [4X[25Xgap>[125X [27Xx:= L.1;; y:= L.2;;[127X[104X
    [4X[25Xgap>[125X [27Xrr:=[((y*x)*x)*x-6*(y*x)*y, 3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ];; [127X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, rr : maxdeg:= 8 );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= Basis(K);;[127X[104X
    [4X[25Xgap>[125X [27XI:= LieRingIdeal( K, [ b[29] ] );[127X[104X
    [4X[28X<Lie ring with 23 generators>[128X[104X
    [4X[25Xgap>[125X [27Xf:= NaturalHomomorphismByIdeal( K, I );;[127X[104X
    [4X[25Xgap>[125X [27XM:= Range(f);[127X[104X
    [4X[28X<Lie ring with 27 generators>[128X[104X
    [4X[25Xgap>[125X [27XTorsion(Basis(M));[127X[104X
    [4X[28X[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 12, 12, 12, 120, 720, 10080, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XImage( f, b[30] );[127X[104X
    [4X[28Xv_16+716*v_17[128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative( f, Basis(M)[10] );[127X[104X
    [4X[28X4*v_2+4*v_3+4*v_4+4*v_5+5*v_6+v_7+5*v_8+v_9+5*v_10+v_11+5*v_12+v_13+5*v_14+v_[128X[104X
    [4X[28X24+v_25+11*v_26+v_29+10*v_30+100*v_31[128X[104X
  [4X[32X[104X
  
  [1X2.3-8 LieLowerCentralSeries[101X
  
  [29X[2XLieLowerCentralSeries[102X( [3XL[103X ) [32X operation
  
  [33X[0;0YHere [3XL[103X is a Lie ring. Its lower central series is returned.[133X
  
  [33X[0;0YThis  repeatedly constructs ideals of [3XL[103X; therefore also this function can be
  rather slow on bigger inputs.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["x","y"] );; x:= L.1;; y:= L.2;;[127X[104X
    [4X[25Xgap>[125X [27Xrr:=[((y*x)*x)*x-6*(y*x)*y, 3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ];;[127X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, rr : maxdeg:= 7 );;[127X[104X
    [4X[25Xgap>[125X [27XLieLowerCentralSeries(K);[127X[104X
    [4X[28X[ <Lie ring with 26 generators>, <Lie ring with 24 generators>, [128X[104X
    [4X[28X  <Lie ring with 23 generators>, <Lie ring with 22 generators>, [128X[104X
    [4X[28X  <Lie ring with 21 generators>, <Lie ring with 19 generators>, [128X[104X
    [4X[28X  <Lie ring with 16 generators>, <Lie ring with 0 generators> ][128X[104X
  [4X[32X[104X
  
  [1X2.3-9 LieLowerPCentralSeries[101X
  
  [29X[2XLieLowerPCentralSeries[102X( [3XL[103X, [3Xp[103X ) [32X operation
  
  [33X[0;0YHere  [3XL[103X  is a Lie ring, and [3Xp[103X is a prime. The lower [3Xp[103X-central series of [3XL[103X is
  returned.  This  is  the  series where the [22XL^k+1[122X is generated by [22X[L,L^k][122X and
  [22XpL^k[122X. Note that this may not be a finite series, if [22XL[122X is not of exponent [22Xp^n[122X
  (as  abelian  group).  The  function  does  not check this; if the series is
  infinite, then it will loop forever.[133X
  
  [33X[0;0YThis  repeatedly constructs ideals of [3XL[103X; therefore also this function can be
  rather slow on bigger inputs.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["x","y"] );; x:= L.1;; y:= L.2;;[127X[104X
    [4X[25Xgap>[125X [27Xrr:=[((y*x)*x)*x-7*(y*x)*y, 7*((((y*x)*x)*x)*x)*x-49*(((y*x)*x)*x)*y, [127X[104X
    [4X[25X>[125X [27X7*x, 49*y ];;[127X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, rr : maxdeg:= 5 );;[127X[104X
    [4X[25Xgap>[125X [27XLieLowerPCentralSeries(K,7);[127X[104X
    [4X[28X[ <Lie ring with 11 generators>, <Lie ring with 10 generators>, [128X[104X
    [4X[28X  <Lie ring with 8 generators>, <Lie ring with 6 generators>,[128X[104X
    [4X[28X  <Lie ring with 4 generators>, <Lie ring with 0 generators> ][128X[104X
  [4X[32X[104X
  
  [1X2.3-10 LieCentre[101X
  
  [29X[2XLieCentre[102X( [3XL[103X ) [32X operation
  
  [33X[0;0YHere [3XL[103X is a Lie ring. Its centre is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["x","y"] );; x:= L.1;; y:= L.2;;[127X[104X
    [4X[25Xgap>[125X [27Xrr:=[((y*x)*x)*x-6*(y*x)*y, 3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ];;[127X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, rr : maxdeg:= 7 );;[127X[104X
    [4X[25Xgap>[125X [27XLieCentre(K);[127X[104X
    [4X[28X<Lie ring with 16 generators>[128X[104X
    [4X[25Xgap>[125X [27XTorsion( Basis(K) );[127X[104X
    [4X[28X[ 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 360, 5040, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [1X2.3-11 TensorWithField[101X
  
  [29X[2XTensorWithField[102X( [3XL[103X, [3XF[103X ) [32X operation
  
  [33X[0;0YHere  [3XL[103X  is  a  Lie  ring,  and  [3XF[103X is a field. This function returns the Lie
  algebra that is obtained by tensoring [3XL[103X with [3XF[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 3, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [1,3] );[127X[104X
    [4X[25Xgap>[125X [27XK:= LieRingByStructureConstants( [3,6,3], T );;[127X[104X
    [4X[25Xgap>[125X [27XTensorWithField( K, GF(3) );[127X[104X
    [4X[28X<Lie algebra of dimension 3 over GF(3)>[128X[104X
    [4X[25Xgap>[125X [27XTensorWithField( K, GF(2) );[127X[104X
    [4X[28X<Lie algebra of dimension 1 over GF(2)>[128X[104X
    [4X[25Xgap>[125X [27XTensorWithField( K, GF(5) );[127X[104X
    [4X[28X<Lie algebra over GF(5), with 0 generators>[128X[104X
  [4X[32X[104X
  
  
  [1X2.4 [33X[0;0YThe Lazard correspondence[133X[101X
  
  [33X[0;0YBy the Lazard correspondence we can put a Lie ring structure on a [22Xp[122X-group of
  class  [22X<p[122X.  Conversely,  we  can define a group structure on a nilpotent Lie
  ring  of  order  [22Xp^n[122X  and class [22X<p[122X. The package contains functions for doing
  this  effectively.  However, we do not work with a single object having both
  the  structure  of a [22Xp[122X-group and a Lie ring. Rather we define two objects, a
  [22Xp[122X-group  and a Lie ring, along with bijections between the two. Our programs
  use  the  BCH-formula  and  its  inverses,  that have been stored in a file,
  truncated  at  weight  14.  This means that currently the package is able to
  deal with groups and algebras up to class 14. The underlying algorithms have
  been described in [CGV11][133X
  
  [1X2.4-1 PGroupToLieRing[101X
  
  [29X[2XPGroupToLieRing[102X( [3XG[103X ) [32X attribute
  
  [33X[0;0YHere  [3XG[103X  is  a [22Xp[122X-group of class [22X<p[122X. This function returns a record with four
  components: [3Xpgroup[103X (the group [3XG[103X), [3Xliering[103X (the corresponding Lie ring), [3XGtoL[103X
  (a function mapping elements of the group to elements of the Lie ring), [3XLtoG[103X
  (a function mapping elements of the Lie ring to elements of the group).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF := FreeGroup(IsSyllableWordsFamily,"a","b","c","d", "e", "f", "g");;[127X[104X
    [4X[25Xgap>[125X [27Xa := F.1;; b := F.2;; c := F.3;; d := F.4;; e := F.5;; f := F.6;; g:=F.7;;[127X[104X
    [4X[25Xgap>[125X [27Xrels := [ a^13, b^13/g, c^13, d^13, e^13, f^13, g^13,  [127X[104X
    [4X[25X>[125X [27XComm(b,a)/c, Comm(c,a)/d, Comm(d,a)/e, Comm(e,a)/f, Comm(f,a), Comm(g,a),[127X[104X
    [4X[25X>[125X [27XComm(c,b)/(g^11), Comm(d,b)/g, Comm(e,b)/g, Comm(g,b), Comm(d,c)/(g^12),[127X[104X
    [4X[25X>[125X [27XComm(e,c), Comm(f,c), Comm(g,c), Comm(e,d), Comm(f,d), Comm(g,d), Comm(f,e), [127X[104X
    [4X[25X>[125X [27XComm(g,e), Comm(g,f)];;[127X[104X
    [4X[25Xgap>[125X [27XG := PcGroupFpGroup( F/rels );[127X[104X
    [4X[28X<pc group of size 62748517 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27Xr:= PGroupToLieRing(G);[127X[104X
    [4X[28Xrec( pgroup := <pc group of size 62748517 with 7 generators>, [128X[104X
    [4X[28Xliering := <Lie ring with 6 generators>, [128X[104X
    [4X[28XGtoL := function( g0 ) ... end, LtoG := function( x0 ) ... end )[128X[104X
    [4X[25Xgap>[125X [27Xf:= r.GtoL; h:= r.LtoG;[127X[104X
    [4X[28Xfunction( g0 ) ... end[128X[104X
    [4X[28Xfunction( x0 ) ... end[128X[104X
    [4X[25Xgap>[125X [27XL:= r.liering;[127X[104X
    [4X[28X<Lie ring with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27Xb:= Basis(L);[127X[104X
    [4X[28XBasis( <Lie ring with 6 generators>, [ v_1, v_2, v_3, v_4, v_5, v_6 ] )[128X[104X
    [4X[25Xgap>[125X [27Xh(b[1]);[127X[104X
    [4X[28Xa^12*c*d^5*e^3*f^8*g^7[128X[104X
    [4X[25Xgap>[125X [27Xf(h(b[1]));[127X[104X
    [4X[28Xv_1[128X[104X
  [4X[32X[104X
  
  [1X2.4-2 LieRingToPGroup[101X
  
  [29X[2XLieRingToPGroup[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YHere  [3XL[103X  is  a  nilpotent  Lie ring of class [22X<p[122X and order [22Xp^n[122X. This function
  returns  a record with four components: [3Xpgroup[103X (the [22Xp[122X-group corresponding to
  [3XL[103X), [3Xliering[103X (the Lie ring [3XL[103X), [3XGtoL[103X (a function mapping elements of the group
  to  elements  of the Lie ring), [3XLtoG[103X (a function mapping elements of the Lie
  ring to elements of the group).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieRing( Integers, ["a","b","c"] );; [127X[104X
    [4X[25Xgap>[125X [27Xa:= L.1;; b:= L.2;; c:= L.3;;[127X[104X
    [4X[25Xgap>[125X [27Xrels:= [ (b*a)*b, c*a, c*b-(b*a)*a, 7^2*a, 7*b-((b*a)*a)*a, [127X[104X
    [4X[25X>[125X [27X7*c-((b*a)*a)*a];;[127X[104X
    [4X[25Xgap>[125X [27XK:= FpLieRing( L, rels );[127X[104X
    [4X[28X<Lie ring with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27Xr:= LieRingToPGroup(K);[127X[104X
    [4X[28Xrec( pgroup := <pc group of size 823543 with 7 generators>, [128X[104X
    [4X[28Xliering := <Lie ring with 5 generators>, [128X[104X
    [4X[28XLtoG := function( x0 ) ... end, GtoL := function( g0 ) ... end )[128X[104X
    [4X[25Xgap>[125X [27XG:= r.pgroup;; f:= r.LtoG;; h:= r.GtoL;;[127X[104X
    [4X[25Xgap>[125X [27Xu:= 5*Basis(K)[2]+9*Basis(K)[5];[127X[104X
    [4X[28X5*v_2+9*v_5[128X[104X
    [4X[25Xgap>[125X [27Xf(u);[127X[104X
    [4X[28Xf3^2*f4^2*f5^6*f7^3[128X[104X
    [4X[25Xgap>[125X [27Xh(f(u));[127X[104X
    [4X[28X5*v_2+9*v_5[128X[104X
  [4X[32X[104X
  
  
  [1X2.5 [33X[0;0YThe database of [22Xn[122X[101X[1X-Engel Lie rings[133X[101X
  
  [33X[0;0YA  Lie ring [22XL[122X is said to satisfy the [22Xn[122X-Engel condition if for all [22Xx,y ∈ L[122X we
  have  [22X(ad x)^n (y) = 0[122X. The package [5XLieRing[105X contains a small database of Lie
  rings  that  satisfy  an [22Xn[122X-Engel condition. They have been computed with the
  algorithms described in [CG07] and [CG09].[133X
  
  [33X[0;0YCurrently  the  database  contains  the "freeest" (or "largest") [22Xn[122X-Engel Lie
  rings with [22Xk[122X generators for [22X(n,k) = (3,2), (3,3), (3,4), (4,2), (4,3)[122X.[133X
  
  [1X2.5-1 SmallNEngelLieRing[101X
  
  [29X[2XSmallNEngelLieRing[102X( [3Xn[103X, [3Xk[103X ) [32X operation
  
  [33X[0;0YThis  returns the biggest [3Xn[103X-Engel Lie ring with [3Xk[103X generators, for the values
  of [3Xn,k[103X indicated above. For other values an error is raised.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SmallNEngelLieRing( 4, 3 );[127X[104X
    [4X[28X<Lie ring with 133 generators>[128X[104X
    [4X[25Xgap>[125X [27Xx:= 10*Basis(L)[1]+7*Basis(L)[10]+19*Basis(L)[89];[127X[104X
    [4X[28X7*v_10+19*v_89[128X[104X
    [4X[25Xgap>[125X [27XForAll( Basis(L), y -> IsZero( x*(x*(x*(x*y))) ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XK:= TensorWithField( L, GF(3) );[127X[104X
    [4X[28X<Lie algebra of dimension 83 over GF(3)>[128X[104X
    [4X[25Xgap>[125X [27Xx:= Random(K);;[127X[104X
    [4X[25Xgap>[125X [27XForAll( Basis(K), y -> IsZero( x*(x*(x*(x*y))) ) );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
