GAP (ModIsom) - Chapter 5: Nilpotent Quotients

This chapter contains a description of the nilpotent quotient algorithm for associative finitely presented algebras. We refer to [Eic11] for background on the algorithms used in this chapter.

5.1 Computing nilpotent quotients

Let \(A\) be a finitely presented algebra in the GAP sense. The following function can be used to determine the class-\(c\) nilpotent quotient of \(A\). The quotient is described by a nilpotent table.

5.1-1 NilpotentQuotientOfFpAlgebra

The output of this function is a nilpotent table with some additional entries. In particular, there is the additional entry img which describes the images of the generators of \(A\) in the nilpotent table.

5.2 Example of nilpotent quotient computation

gap> F := FreeAssociativeAlgebra(GF(2), 2);;
gap> g := GeneratorsOfAlgebra(F);;
gap> r := [g[1]^2, g[2]^2];;
gap> A := F/r;;
gap> NilpotentQuotientOfFpAlgebra(A,3);
rec( def := [ 1, 2 ], dim := 8, fld := GF(2), 
  img := [ <a GF2 vector of length 8>, <a GF2 vector of length 8> ], 
  mat := [ [  ], [  ] ], rnk := 2, 
  tab := 
    [ 
      [ <a GF2 vector of length 8>, <a GF2 vector of length 8>, 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ]
        , 
      [ <a GF2 vector of length 8>, <a GF2 vector of length 8>, 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ]
        , 
      [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ]
        , 
      [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ], 
          [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] 
         ] ], 
  wds := [ ,, [ 2, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 4 ], [ 2, 5 ], [ 1, 6 ] ], 
  wgs := [ 1, 1, 2, 2, 3, 3, 4, 4 ] )

5 Nilpotent Quotients

5.1 Computing nilpotent quotients

5.1-1 NilpotentQuotientOfFpAlgebra

5.2 Example of nilpotent quotient computation