[GAP Forum] AdjointBasis() and IndicesOfAdjointBasis()

[Kursat Aker]

Dear GAP Forum,

I now understood that the answer to my question is yes.

My confusion was a result of me looking at

gap> Elements(B);
[k, j, i, e]

I should have look at
gap> BasisVectors(B);
[e,i,j,k]

Sorry for bothering the Forum.

Best wishes,

Kursat


On 6/22/10 8:37 PM, Kursat Aker wrote:
> Dear GAP Forum,
>
> I would like to ask a question about AdjointBasis().
>
> I have the following code, which I essentially took from GAP Manual.
>
> gap> A:= QuaternionAlgebra( Rationals );;
> gap> B := Basis( A );
> Basis( <algebra-with-one of dimension 4 over Rationals>, ... )
> gap> Elements( B );
> [ k, j, i, e ]
> gap> adjB := AdjointBasis( B );
> Basis( <vector space over Rationals, with 4 generators>,
> [ [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ],
>   [ [ 0, -1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ],
>   [ [ 0, 0, -1, 0 ], [ 0, 0, 0, 1 ], [ 1, 0, 0, 0 ], [ 0, -1, 0, 0 ] ],
>   [ [ 0, 0, 0, -1 ], [ 0, 0, -1, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ] ] 
> ] )
> gap> IndicesOfAdjointBasis( adjB );
> [ 1, 2, 3, 4 ]
>
> My question is this:
>
> Is the matrix adjB[i] supposed to the left action of the element B[i] 
> for i=1,2,3,4?
>
> Thanks,
>
> Kursat
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