  
  [1X1 [33X[0;0YThe FPLSA Package[133X[101X
  
  [33X[0;0YThis  chapter  describes  the  [5XGAP[105X package [5XFPLSA[105X , an interface to the [10Xfplsa[110X
  program  by  V. Gerdt  and  V. Kornyak  (version 4) for the computation with
  finitely presented Lie superalgebras.[133X
  
  [33X[0;0YAt present [5XGAP[105X uses only the facility to compute a structure constants table
  of  a  finite  dimensional Lie algebra over the rationals that is given by a
  finite presentation.[133X
  
  [33X[0;0YThe  package  uses  an  external  binary, probably it will only work on UNIX
  platforms.[133X
  
  
  [1X1.1 [33X[0;0YMain Functions[133X[101X
  
  [33X[0;0YA  finitely  presented Lie algebra is a quotient of a free Lie algebra by an
  ideal  generated  by  a finite number of elements. In [5XGAP[105X a free Lie algebra
  can  be created by the command [10XFreeLieAlgebra[110X; we refer to the [5XGAP[105X Reference
  Manual  for  more  details.  A  finitely  presented  Lie  algebra  [3XK[103X  can be
  constructed by [10X[3XK[103X[10X := [3XL[103X[10X/[3Xrels[103X[10X[110X, where [3XL[103X is a free Lie algebra and [3Xrels[103X a list of
  elements of [3XL[103X that constitute the relations that hold in [3XK[103X. Given a finitely
  presented  Lie  algebra  we  want  to calculate a basis and a multiplication
  table  of  it.  The  interface  to  the [5XFPLSA[105X package comes with two related
  functions for doing that.[133X
  
  [1X1.1-1 SCAlgebraInfoOfFpLieAlgebra[101X
  
  [33X[1;0Y[29X[2XSCAlgebraInfoOfFpLieAlgebra[102X( [3XL[103X, [3Xrels[103X, [3Xlimit[103X, [3Xwords[103X, [3Xrels[103X ) [32X function[133X
  
  [33X[0;0YLet  [3XL[103X  be a free Lie algebra over the rationals, [3Xrels[103X a list of elements in
  [3XL[103X, [3Xlimit[103X a positive integer and [3Xwords[103X and [3Xrels[103X two booleans.[133X
  
  [33X[0;0YIf the algebra [10X[3XL[103X[10X / [3Xrels[103X[10X[110X is finite-dimensional and if a basis of this algebra
  can  be constructed using elements in [3XL[103X that involve only words of length at
  most [3Xlimit[103X then [10XSCAlgebraInfoOfFpLieAlgebra[110X returns a record whose component
  [10Xsc[110X  contains  an algebra that is isomorphic with [10X[3XL[103X[10X / [3Xrels[103X[10X[110X. Otherwise [9Xfail[109X is
  returned.[133X
  
  [33X[0;0YThe function calls the [10Xfplsa[110X standalone.[133X
  
  [33X[0;0YIf  [3Xwords[103X  is  [9Xtrue[109X then the component [10Xwords[110X of the result record contains a
  list of elements in [3XL[103X that correspond to the basis elements.[133X
  
  [33X[0;0YIf [3Xrels[103X is [9Xtrue[109X then the component [10Xrels[110X of the result record contains a list
  of  reduced  relators  in  [3XL[103X  that describes how algebra generators of [3XL[103X are
  expressed in terms of the basis elements.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "fplsa" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieAlgebra( Rationals, "x", "y" );[127X[104X
    [4X[28X<Lie algebra over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xg:= GeneratorsOfAlgebra( L );; x:= g[1];; y:= g[2];;[127X[104X
    [4X[25Xgap>[125X [27Xrels:= [ x*(x*y) - x*y, y*(y*(x*y)) ];[127X[104X
    [4X[28X[ (-1)*(x*y)+(1)*(x*(x*y)), (1)*(((x*y)*y)*y) ][128X[104X
    [4X[25Xgap>[125X [27XSCAlgebraInfoOfFpLieAlgebra( L, rels, 100, true, true );[127X[104X
    [4X[28Xrec( sc := <Lie algebra of dimension 4 over Rationals>, [128X[104X
    [4X[28X  words := [ (1)*x, (1)*y, (1)*(y*x), (1)*((y*x)*y) ], [128X[104X
    [4X[28X  rels := [ (1)*((y*x)*x)+(1)*(y*x), (1)*(((y*x)*y)*y), (1)*(((y*x)*y)*(y*x)) [128X[104X
    [4X[28X     ] )[128X[104X
  [4X[32X[104X
  
  [1X1.1-2  IsomorphicSCAlgebra[101X
  
  [33X[1;0Y[29X[2X IsomorphicSCAlgebra[102X( [3XK[103X[, [3Xbound[103X] ) [32X function[133X
  
  [33X[0;0Ycomputes  a  Lie  algebra  given by a multiplication table isomorphic to the
  finitely  presented  Lie  algebra  [3XK[103X.  If  the  optional  parameter [3Xbound[103X is
  specified  the  computation will be carried out using monomials of degree at
  most  [3Xbound[103X.  If  [3Xbound[103X  is  not specified, then it will initially be set to
  10000.  If  this does not suffice to calculate a multiplication table of the
  algebra,  then  the  bound will be increased until a multiplication table is
  found.  If the computation was successful then a structure constants algebra
  will be returned isomorphic to [3XK[103X. Otherwise [9Xfail[109X will be returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "fplsa" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieAlgebra( Rationals, "x", "y" );[127X[104X
    [4X[28X<Lie algebra over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xg:= GeneratorsOfAlgebra( L );; x:= g[1];; y:= g[2];;[127X[104X
    [4X[25Xgap>[125X [27Xrels:= [ x*(x*y) - x*y, y*(y*(x*y)) ];[127X[104X
    [4X[28X[ (-1)*(x*y)+(1)*(x*(x*y)), (1)*(((x*y)*y)*y) ][128X[104X
    [4X[25Xgap>[125X [27XK:= L/rels;[127X[104X
    [4X[28X<Lie algebra over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsomorphicSCAlgebra( K );[127X[104X
    [4X[28X<Lie algebra of dimension 4 over Rationals>[128X[104X
  [4X[32X[104X
  
  
  [1X1.2 [33X[0;0YAuxiliary Variables of FPLSA[133X[101X
  
  [1X1.2-1 FPLSA[101X
  
  [33X[1;0Y[29X[2XFPLSA[102X [32X global variable[133X
  
  [33X[0;0Yis  the  global  record  used  by  the  functions  in  the  package. Besides
  components  that  describe  parameters  for  the  standalone,  the following
  components are used.[133X
  
  [8X[10XRelation_size[110X[8X [108X
        [33X[0;6Yparameter that controls the memory usage by the fplsa program,[133X
  
  [8X[10XLie_monomial_table_size[110X[8X [108X
        [33X[0;6Yparameter that controls the memory usage by the fplsa program,[133X
  
  [8X[10XNode_Lie_term_size[110X[8X [108X
        [33X[0;6Yparameter that controls the memory usage by the fplsa program,[133X
  
  [8X[10XNode_scalar_factor_size[110X[8X [108X
        [33X[0;6Yparameter that controls the memory usage by the fplsa program,[133X
  
  [8X[10XNode_scalar_term_size[110X[8X [108X
        [33X[0;6Yparameter that controls the memory usage by the fplsa program,[133X
  
  [8X[10Xprogname[110X[8X [108X
        [33X[0;6Ythe file name of the executable,[133X
  
  [8X[10XT[110X[8X [108X
        [33X[0;6Ystructure constants table of the algebra under consideration,[133X
  
  [8X[10Xwords[110X[8X [108X
        [33X[0;6Ylist  of elements in the free Lie algebra that correspond to the basis
        elements,[133X
  
  [8X[10Xrels[110X[8X [108X
        [33X[0;6Ylist  of  relators  in  the  free Lie algebra that are used to express
        redundant algebra generators in terms of the basis.[133X
  
  [33X[0;0YIn  order  to  be  able  to  run the [10Xfplsa[110X program successfully, it might be
  necessary  to  customize  the  parameters  that  control  the memory the the
  program  uses, to suit the computer the user is working on. In particular if
  the  program exits with an error message of the form: [10XError, the process did
  not succeed[110X, then it may be necessary to change these parameters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "fplsa" );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieAlgebra( Rationals, "x", "y" );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= GeneratorsOfAlgebra( L );; x:= g[1];; y:= g[2];;[127X[104X
    [4X[25Xgap>[125X [27Xrels:= [ x*(x*y) - x*y, y*(y*(x*y)) ];;[127X[104X
    [4X[25Xgap>[125X [27XSCAlgebraInfoOfFpLieAlgebra( L, rels, 100, true, true );;[127X[104X
    [4X[25Xgap>[125X [27XFPLSA;[127X[104X
    [4X[28Xrec( Relation_size := 2500000, Lie_monomial_table_size := 1000000, [128X[104X
    [4X[28X  Node_Lie_term_size := 2000000, Node_scalar_factor_size := 2000, [128X[104X
    [4X[28X  Node_scalar_term_size := 20000, progname := "fplsa4", [128X[104X
    [4X[28X  T := [ [ [ [  ], [  ] ], [ [ 3 ], [ -1 ] ], [ [ 3 ], [ 1 ] ], [128X[104X
    [4X[28X          [ [ 4 ], [ 1 ] ] ], [128X[104X
    [4X[28X      [ [ [ 3 ], [ 1 ] ], [ [  ], [  ] ], [ [ 4 ], [ -1 ] ], [ [  ], [  ] ] ],[128X[104X
    [4X[28X      [ [ [ 3 ], [ -1 ] ], [ [ 4 ], [ 1 ] ], [ [  ], [  ] ], [ [  ], [  ] ] ],[128X[104X
    [4X[28X      [ [ [ 4 ], [ -1 ] ], [ [  ], [  ] ], [ [  ], [  ] ], [ [  ], [  ] ] ], [128X[104X
    [4X[28X      -1, 0 ], words := [ 1, 2, [ 2, 1 ], [ [ 2, 1 ], 2 ] ], [128X[104X
    [4X[28X  rels := [ [ [ [ 2, 1 ], 1 ], 1, [ 2, 1 ], 1 ], [128X[104X
    [4X[28X      [ [ [ [ 2, 1 ], 2 ], 2 ], 1 ], [ [ [ [ 2, 1 ], 2 ], [ 2, 1 ] ], 1 ] ] )[128X[104X
  [4X[32X[104X
  
  
  [1X1.3 [33X[0;0YInstalling the FPLSA Package[133X[101X
  
  [33X[0;0YTo  install  unpack  the archive file in a directory in the [10Xpkg[110X hierarchy of
  your  version  of  [5XGAP[105X.  (This  might  be  the [10Xpkg[110X directory of the [5XGAP[105X home
  directory;  it  is however also possible to keep an additional [10Xpkg[110X directory
  in  your  private  directories,  see  Section [14X'Reference:  Installing  a GAP
  Package'[114X  of  the [5XGAP[105X Reference Manual for details on how to do this.) Go to
  the  newly  created  [10Xfplsa[110X directory and call [10X./configure [3Xpath[103X[10X[110X where [3Xpath[103X is
  the  path  to  the  [5XGAP[105X  home  directory.  So for example if you install the
  package in the main [10Xpkg[110X directory call[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X./configure ../..[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  will  fetch the architecture type for which [5XGAP[105X has been compiled last
  and create a [10XMakefile[110X. Now simply call[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xmake[128X[104X
  [4X[32X[104X
  
  [33X[0;0Yto compile the binary and to install it in the appropriate place.[133X
  
  [33X[0;0YIf you use this installation of [5XGAP[105X on different hardware platforms you will
  have  to  compile  the  binary for each platform separately. This is done by
  calling  [10Xconfigure[110X and [10Xmake[110X for the package anew immediately after compiling
  [5XGAP[105X  itself  for  the  respective  architecture.  If  your version of [5XGAP[105X is
  already  compiled  (and has last been compiled on the same architecture) you
  do  not  need  to  compile [5XGAP[105X again, it is sufficient to call the [10Xconfigure[110X
  script in the [5XGAP[105X home directory.[133X
  
