  
  [1X3 [33X[0;0YThe Database[133X[101X
  
  [33X[0;0YThis  package  gives  access to the database of Lie [22Xp[122X-rings of order at most
  [22Xp^7[122X  as  determined  by Mike Newman, Eamonn O'Brien and Michael Vaughan-Lee,
  see  [NOV03]  and [OV05]. A description of the database can also be found in
  [Vau13].[133X
  
  [33X[0;0YFor  each  [22Xn  ∈ {1, ..., 7}[122X this package contains a (finite) list of generic
  presentations  of Lie [22Xp[122X-rings. For each prime [22Xp ≥ 5[122X, each of the generic Lie
  [22Xp[122X-rings gives rise to a family of Lie [22Xp[122X-rings over the considered prime [22Xp[122X by
  specialising  the  indeterminates to a certain list of values. The resulting
  lists  of  Lie [22Xp[122X-rings provide a complete and irredundant set of isomorphism
  type  representatives  of  the  Lie  [22Xp[122X-rings  of  order [22Xp^n[122X. The generic Lie
  [22Xp[122X-rings  of  [22Xp[122X-class  at most 2 can also be considered for the prime [22Xp=3[122X and
  yield  a  list  of  isomorphism  type representatives for the Lie [22Xp[122X-rings of
  order [22X3^n[122X and [22Xp[122X-class at most [22X2[122X.[133X
  
  [33X[0;0YThe  Lazard  correspondence  has  been  used to check the correctness of the
  database  of  Lie [22Xp[122X-rings: for various small primes it has been checked that
  the Lie [22Xp[122X-rings of this database define non-isomorphic finite [22Xp[122X-groups.[133X
  
  [33X[0;0YIn  the  following  we describe functions to access the database. Throughout
  this chapter, we assume that [22Xdim ∈ {1, ..., 7}[122X and [22XP[122X is a prime with [22XP ≠ 2[122X.[133X
  
  
  [1X3.1 [33X[0;0YAccessing Lie [22Xp[122X[101X[1X-rings[133X[101X
  
  [1X3.1-1 LiePRingsByLibrary[101X
  
  [33X[1;0Y[29X[2XLiePRingsByLibrary[102X( [3Xdim[103X[, [3Xgen[103X][, [3Xcl[103X] ) [32X function[133X
  
  [33X[0;0Yreturns the generic Lie [22Xp[122X-rings of dimension [22Xdim[122X in the database. The second
  form  returns the Lie [22Xp[122X-rings of minimal generator number [22Xgen[122X and [22Xp[122X-class [22Xcl[122X
  only.[133X
  
  [1X3.1-2 LiePRingsByLibrary[101X
  
  [33X[1;0Y[29X[2XLiePRingsByLibrary[102X( [3Xdim[103X, [3XP[103X[, [3Xgen[103X][, [3Xcl[103X] ) [32X function[133X
  
  [33X[0;0Yreturns   isomorphism  type  representatives  of  ordinary  Lie  [22Xp[122X-rings  of
  dimension  [22Xdim[122X  for  the prime [22XP[122X. The second form returns the Lie [22Xp[122X-rings of
  minimal generator number [22Xgen[122X and [22Xp[122X-class [22Xcl[122X only. The function assumes [22XP ≥ 3[122X
  and for [22XP = 3[122X there are only the Lie [22Xp[122X-rings of [22Xp[122X-class at most 2 available.[133X
  
  [33X[0;0YThe first example yields the generic Lie [22Xp[122X-rings of dimension [22X4[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLiePRingsByLibrary(4);[127X[104X
    [4X[28X[ <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>,[128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 4 over prime p> ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  next example yields the isomorphism type representatives of Lie [22Xp[122X-rings
  of dimension [22X3[122X for the prime [22X5[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLiePRingsByLibrary(3, 5);[127X[104X
    [4X[28X[ <LiePRing of dimension 3 over prime 5>, [128X[104X
    [4X[28X  <LiePRing of dimension 3 over prime 5>, [128X[104X
    [4X[28X  <LiePRing of dimension 3 over prime 5>, [128X[104X
    [4X[28X  <LiePRing of dimension 3 over prime 5>, [128X[104X
    [4X[28X  <LiePRing of dimension 3 over prime 5> ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  example extracts the generic Lie [22Xp[122X-rings of dimension [22X5[122X with
  minimal generator number [22X2[122X and [22Xp[122X-class [22X4[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLiePRingsByLibrary(5, 2, 4);[127X[104X
    [4X[28X[ <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p> ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we determine the isomorphism type representatives of Lie [22Xp[122X-rings of
  dimension [22X5[122X, minimal generator number [22X2[122X and [22Xp[122X-class [22X4[122X for the prime [22X7[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLiePRingsByLibrary(5, 7, 2, 4);[127X[104X
    [4X[28X[ <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 7> ][128X[104X
  [4X[32X[104X
  
  
  [1X3.2 [33X[0;0YNumbers of Lie [22Xp[122X[101X[1X-rings[133X[101X
  
  [1X3.2-1 NumberOfLiePRings[101X
  
  [33X[1;0Y[29X[2XNumberOfLiePRings[102X( [3Xdim[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the number of generic Lie [22Xp[122X-rings in the database of the considered
  dimension for [22Xdim ∈ {1, ..., 7}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList([1..7], x -> NumberOfLiePRings(x));[127X[104X
    [4X[28X[ 1, 2, 5, 15, 75, 542, 4773 ][128X[104X
  [4X[32X[104X
  
  [1X3.2-2 NumberOfLiePRings[101X
  
  [33X[1;0Y[29X[2XNumberOfLiePRings[102X( [3Xdim[103X, [3XP[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the  number  of  isomorphism types of ordinary Lie [22Xp[122X-rings of order
  [22XP^dim[122X  in the database. If [22XP ≥ 5[122X, then this is the number of all isomorphism
  types  of Lie [22Xp[122X-rings of order [22XP^dim[122X and if [22XP = 3[122X then this is the number of
  all  isomorphism  types  of Lie [22Xp[122X-rings of [22Xp[122X-class at most [22X2[122X. If [22XP ≥ 7[122X, then
  this number coincides with [10XNumberSmallGroups[110X([22XP^dim[122X).[133X
  
  [1X3.2-3 NumberOfLiePRingsInFamily[101X
  
  [33X[1;0Y[29X[2XNumberOfLiePRingsInFamily[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the  number of Lie [22Xp[122X-rings associated to [22XL[122X as a polynomial in [3Xp[103X and
  possibly some residue classes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := LiePRingsByLibrary(7)[780];[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p with parameters[128X[104X
    [4X[28X[ x, y, z, t, s, u, v ]>[128X[104X
    [4X[25Xgap>[125X [27XNumberOfLiePRingsInFamily(L);[127X[104X
    [4X[28X-1/3*p^5*(p-1,3)+p^5-1/3*p^4*(p-1,3)+p^4-1/3*p^3*(p-1,3)+p^3-1/3*p^2*(p-1,3)[128X[104X
    [4X[28X+p^2-p*(p-1,3)+3*p-3/2*(p-1,3)+9/2[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YSearching the database[133X[101X
  
  [33X[0;0YWe  now  consider  a generic Lie [22Xp[122X-ring [3XL[103X from the database and consider the
  family of ordinary Lie [22Xp[122X-rings that arise from it.[133X
  
  [1X3.3-1 LiePRingsInFamily[101X
  
  [33X[1;0Y[29X[2XLiePRingsInFamily[102X( [3XL[103X, [3XP[103X ) [32X function[133X
  
  [33X[0;0Ytakes  as  input  a generic Lie [22Xp[122X-ring [3XL[103X from the database and a prime [3XP[103X and
  returns  all  Lie  [22Xp[122X-rings  determined  by  [3XL[103X  and [3XP[103X up to isomorphism. This
  function  returns  [9Xfail[109X  if  the  generic  Lie [22Xp[122X-ring does not exist for the
  special  prime [3XP[103X; this may be due to the conditions on the prime or (if [22XP=3[122X)
  to the [22Xp[122X-class of the Lie [22Xp[122X-ring.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := LiePRingsByLibrary(7)[118];[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p with parameters [ x, y ]>[128X[104X
    [4X[25Xgap>[125X [27XLibraryConditions(L);[127X[104X
    [4X[28X[ "[x,y]~[x,-y]", "p=1 mod 4" ][128X[104X
    [4X[25Xgap>[125X [27XLiePRingsInFamily(L, 7);[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XLength(LiePRingsInFamily(L,13));[127X[104X
    [4X[28X91[128X[104X
    [4X[25Xgap>[125X [27X13^2;[127X[104X
    [4X[28X169[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following example shows how to determine all Lie [22Xp[122X-rings of dimension [22X5[122X
  and [22Xp[122X-class [22X4[122X over the prime [22X29[122X up to isomorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := LiePRingsByLibrary(5);;[127X[104X
    [4X[25Xgap>[125X [27XL := Filtered(L, x -> PClassOfLiePRing(x)=4);[127X[104X
    [4X[28X[ <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime p> ][128X[104X
    [4X[25Xgap>[125X [27XK := List(L, x-> LiePRingsInFamily(x, 29));[127X[104X
    [4X[28X[ [ <LiePRing of dimension 5 over prime 29> ], [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], fail, fail, [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], fail, fail, [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ], [128X[104X
    [4X[28X  [ <LiePRing of dimension 5 over prime 29> ] ][128X[104X
    [4X[25Xgap>[125X [27XK := Filtered(Flat(K), x -> x<>fail);[127X[104X
    [4X[28X[ <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29>, [128X[104X
    [4X[28X  <LiePRing of dimension 5 over prime 29> ][128X[104X
  [4X[32X[104X
  
  
  [1X3.4 [33X[0;0YMore details[133X[101X
  
  [33X[0;0YLet  [22XL[122X  be  a  Lie  [22Xp[122X-ring  from the database. Then the following additional
  attributes are available.[133X
  
  [1X3.4-1 LibraryName[101X
  
  [33X[1;0Y[29X[2XLibraryName[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns  a  string  with  the  name  of  [22XL[122X in the database. See [11Xp567.pdf[111X for
  further background.[133X
  
  [1X3.4-2 ShortPresentation[101X
  
  [33X[1;0Y[29X[2XShortPresentation[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns a string exhibiting a short presentation of [22XL[122X.[133X
  
  [1X3.4-3 LibraryConditions[101X
  
  [33X[1;0Y[29X[2XLibraryConditions[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns the conditions on [22XL[122X. This is a list of two strings. The first string
  exhibits  the  conditions  on  the  parameters  of  [22XL[122X,  the second shows the
  conditions on primes.[133X
  
  [1X3.4-4 MinimalGeneratorNumberOfLiePRing[101X
  
  [33X[1;0Y[29X[2XMinimalGeneratorNumberOfLiePRing[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns the minimal generator number of [22XL[122X.[133X
  
  [1X3.4-5 PClassOfLiePRing[101X
  
  [33X[1;0Y[29X[2XPClassOfLiePRing[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0Yreturns the [22Xp[122X-class of [22XL[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := LiePRingsByLibrary(7)[118];[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p with parameters [ x, y ]>[128X[104X
    [4X[25Xgap>[125X [27XLibraryName(L);[127X[104X
    [4X[28X"7.118"[128X[104X
    [4X[25Xgap>[125X [27XLibraryConditions(L);[127X[104X
    [4X[28X[ "[x,y]~[x,-y]", "p=1 mod 4" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAll  of  the  information  listed  in  this  section  is inherited when [22XL[122X is
  specialised.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := LiePRingsByLibrary(7)[118];[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p with parameters [ x, y ]>[128X[104X
    [4X[25Xgap>[125X [27XK := SpecialiseLiePRing(L, 13, ParametersOfLiePRing(L), [0,0]);[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime 13>[128X[104X
    [4X[25Xgap>[125X [27XLibraryName(K);[127X[104X
    [4X[28X"7.118"[128X[104X
    [4X[25Xgap>[125X [27XLibraryConditions(K);[127X[104X
    [4X[28X[ "[x,y]~[x,-y]", "p=1 mod 4" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  example  shows how to find a Lie [22Xp[122X-ring with a given name in
  the database.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := LiePRingsByLibrary(7);;[127X[104X
    [4X[25Xgap>[125X [27XFiltered(L, x -> LibraryName(x) = "7.1010")[1];[127X[104X
    [4X[28X<LiePRing of dimension 7 over prime p> [128X[104X
  [4X[32X[104X
  
  
  [1X3.5 [33X[0;0YSpecial functions for dimension 7[133X[101X
  
  [33X[0;0YThe  database  of  Lie  [22Xp[122X-rings  of  dimension [22X7[122X is very large and it may be
  time-consuming  (or even impossible due to storage problems) to generate all
  Lie [22Xp[122X-rings of dimension [22X7[122X for a given prime [22XP[122X.[133X
  
  [33X[0;0YThus there are some special functions available that can be used to access a
  particular  set  of  Lie  [22Xp[122X-rings  of dimension [22X7[122X only. In particular, it is
  possible  to  consider  the  descendants  of  a single Lie [22Xp[122X-ring of smaller
  dimension by itself. The Lie [22Xp[122X-rings of this type are all stored in one file
  of the library. Thus, equivalently, it is possible to access the Lie [22Xp[122X-rings
  in one single file only.[133X
  
  [33X[0;0YThe  table [10XLIE_TABLE[110X contains a list of all possible files together with the
  number of Lie [22Xp[122X-rings generated by their corresponding Lie [22Xp[122X-rings.[133X
  
  [1X3.5-1 LiePRingsDim7ByFile[101X
  
  [33X[1;0Y[29X[2XLiePRingsDim7ByFile[102X( [3Xnr[103X ) [32X function[133X
  
  [33X[0;0Yreturns the generic Lie [22Xp[122X-rings in file number [22Xnr[122X.[133X
  
  [1X3.5-2 LiePRingsDim7ByFile[101X
  
  [33X[1;0Y[29X[2XLiePRingsDim7ByFile[102X( [3Xnr[103X, [3XP[103X ) [32X function[133X
  
  [33X[0;0Yreturns the isomorphism types of Lie [22Xp[122X-rings in file number [22Xnr[122X for the prime
  [3XP[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLIE_TABLE[100];[127X[104X
    [4X[28X[ "3gen/gapdec6.139", 1/2*p+(p-1,3)+3/2 ][128X[104X
    [4X[25Xgap>[125X [27XLiePRingsDim7ByFile(100);[127X[104X
    [4X[28X[ <LiePRing of dimension 7 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime p>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime p>,[128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime p>,[128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime p with parameters [ x ]> ][128X[104X
    [4X[25Xgap>[125X [27XLiePRingsDim7ByFile(100, 7);[127X[104X
    [4X[28X[ <LiePRing of dimension 7 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime 7>, [128X[104X
    [4X[28X  <LiePRing of dimension 7 over prime 7> ][128X[104X
  [4X[32X[104X
  
  
  [1X3.6 [33X[0;0YDimension 8 and maximal class[133X[101X
  
  [33X[0;0YRecently, Lee and Vaughan-Lee [LV22] determined the Lie [22Xp[122X-rings of dimension
  8  with  maximal  class  up  to isomorphism. This classification is now also
  available in the Lie [22Xp[122X-ring package via the following functions.[133X
  
  [1X3.6-1 LiePRingsByLibraryMC8[101X
  
  [33X[1;0Y[29X[2XLiePRingsByLibraryMC8[102X(  ) [32X function[133X
  
  [33X[0;0Yreturns a list of [22X69[122X generic Lie [22Xp[122X-rings.[133X
  
  [1X3.6-2 LiePRingsInFamilyMC8[101X
  
  [33X[1;0Y[29X[2XLiePRingsInFamilyMC8[102X( [3XL[103X, [3XP[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the  isomorphism  types of Lie [22Xp[122X-rings in the family defined by the
  generic Lie [22Xp[122X-ring [3XL[103X for a fixed prime [3XP[103X with [3XP[103X [22X≥ 5[122X.[133X
  
