72.1 Operation for Finitely Presented Algebras

Operation( F, Q )

This is the default application of VE. Finitely Presented Algebras), Q is a quotient of a free F-module, and the result is a matrix algebra representing a faithful action on Q.

If Q is the zero module then the matrices have dimension zero, so the result is a null algebra (see NullAlgebra) consisting only of a zero element.

The algebra homomorphism, the isomorphic module for the matrix algebra, and the module homomorphism can be constructed as described in chapters Algebras and Modules.

    gap> a:= FreeAlgebra( GF(2), 2 );
    UnitalAlgebra( GF(2), [ a.1, a.2 ] )
    gap> a:= a / [ a.1^2 - a.one, # group algebra of $V_4$ over $GF(2)$
    >              a.2^2 - a.one,
    >              a.1*a.2 - a.2*a.1 ];
    UnitalAlgebra( GF(2), [ a.1, a.2 ] )
    gap> op:= Operation( a, a^1 );
    UnitalAlgebra( GF(2),
    [ [ [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2),
              Z(2)^0 ], [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
          [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ],
      [ [ 0*Z(2), Z(2)^0, 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), Z(2)^0, 0*Z(2) ] ] ] )
    gap> Size( op );
    16 

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