[GAP Forum] RadicalOfAlgebra

[Willem De Graaf]

Dear Neha,

You asked:

> Could you please tell me as to where can I find the code for the standard
> RadicalOfAlgebra method in GAP?? I wanted to see the theory behind how it
> computes the radical as I could not find a detailed documentation for it.

The code for matrix algebras is in lib/algmat.gi; I also append it
below. It contains a reference.
For non-matrix algebras first a faithful representation is computed,
and then the code for matrix algebras is used.

Best wishes,

Willem

#############################################################################
##
#M  RadicalOfAlgebra( <A> ) . . . . . . . . for an associative matrix algebra
##
##  The radical of an associative algebra <A>  defined as the ideal of all
##  elements $x$ of <A> such that $x y$ is nilpotent for all $y$ in <A>.
##
##  For Gaussian matrix algebras this is easy to calculate,
##  for arbitrary associative algebras the task is reduced to the
##  Gaussian matrix algebra case.
##
##  The implementation of the characterisitc p>0 part is by Craig Struble.
##
InstallMethod( RadicalOfAlgebra,
    "for associative Gaussian matrix algebra",
    true,
    [ IsAlgebra and IsGaussianMatrixSpace and IsMatrixFLMLOR ], 0,
    function( A )

    local F,           # the field of A
          p,           # the characteristic of F
          q,           # the size of F
          n,           # the dimension of A
          ident,       # the identity matrix
          bas,         # a list of basis vectors of A
          minusOne,    # -1 in F
          eqs,         # equation set
          i,j,         # loop variables
          G,           # Gram matrix
          I,           # a list of basis vectors of an ideal of A
          I_prime,     # I \cup ident
          changed,     # flag denoted if $I_i <> I_{i-1}$ in ideal sequence
          pexp,        # a power of the prime p
          dim,         # the dimension of the vector space where A acts on
          charPoly,    # characteristic polynomials of elements in A
          invFrob,     # inverse of the Frobenius map of F
          invFrobexp,  # a power of the invFrob
          r, r_prime;  # the length of I and I_prime

    # Check associativity.
    if not IsAssociative( A ) then
      TryNextMethod();
    fi;

    if Dimension( A ) = 0 then return A; fi;

    F:= LeftActingDomain( A );
    p:= Characteristic( F );
    n:= Dimension( A );
    bas:= BasisVectors( Basis( A ) );

    if p = 0 then

      # First we treat the characteristic 0 case.
      # According to Dickson's theorem we have that in this case
      # the radical of `A' is $\{ x\in A \mid Tr(xy) = 0 \forall y \in A \}$,
      # so it can be computed by solving a system of linear equations.

      eqs:= List( [ 1 .. n ], x -> [] );

      for i in [1..n] do
        for j in [i..n] do
          eqs[i][j]:= TraceMat( bas[i] * bas[j] );
          eqs[j][i]:= eqs[i][j];
        od;
      od;

      return SubalgebraNC( A, List( NullspaceMat( eqs ),
                                    x -> LinearCombination( bas, x ) ),
                           "basis" );

    else

      # If `p' is greater than 0, then the situation is more difficult.
      # We implement the algorithm presented in
      # "Cohen, Arjeh M, G\'{a}bor Ivanyos, and David B. Wales,
      # 'Finding the radical of an algebra of linear tranformations,'
      # Journal of Pure and Applied Algebra 117 & 118 (1997), 177-193".

      q := Size( F );
      dim := Length( bas[1] );
      pexp := 1;
      invFrob := InverseGeneralMapping(FrobeniusAutomorphism(F));
      invFrobexp := invFrob;
      minusOne := -One(F);
      ident := IdentityMat( dim, F );
      changed := true;
      I := ShallowCopy( bas );

      # Compute the sequence of ideals $I_i$ (see the paper by Cohen, et.al.)
      while pexp <= dim do
          # These values need recomputation only when $I_i <> I_{i-1}$
          if changed then
              I_prime := ShallowCopy( I );
              if not ident in I_prime then
                  Add( I_prime, ident );
              fi;
              r := Length( I );
              r_prime := Length( I_prime );
              eqs := NullMat( r, r_prime, F );
              charPoly := List( [1..r], x -> [] );
              for i in [1..r] do
                  for j in [1..r_prime] do
                      charPoly[i][j] :=
                          CoefficientsOfUnivariatePolynomial(
CharacteristicPolynomial( F, F, I[i]*I_prime[j]
) );
                  od;
              od;
              changed := false;
          fi;

          for i in [1..r] do
              for j in [1..r_prime] do
                  eqs[i][j] := minusOne^pexp * charPoly[i][j][dim-pexp+1];
              od;
          od;

          G := NullspaceMat( eqs );

          if Length( G ) = 0 then
              return TrivialSubalgebra( A );
          elif Length( G ) <> r then
              # $I_i <> I_{i-1}$, so compute the basis for $I_i$
              changed := true;
              if 1 < pexp and pexp < q then
                  G := List( G, x -> List( x, y -> y^invFrobexp ) );
              fi;
              I := List( G, x -> LinearCombination( I, x ) );
          fi;

          # prepare for next step

          invFrobexp := invFrobexp * invFrob;
          pexp := pexp*p;
      od;

      return SubalgebraNC( A, I, "basis" );
    fi;
    end );