[GAP Forum] Re: Quaternion vs real algebras

[mim_ at op.pl]

Hello again,

The fourth sentence in my first mail should be 
"It seems that there is no way in GAP to convert algebra over quaternions into algebra over reals (Rationals) !?"

I am sorry !

Regards,
Marek Mitros

mim_ at op.pl napisał:
>Hello,
>
>I want to define real lie algebra using quaternion matrices. How to do this ? See below code with example. It seems that there is no way in GAP to convert algebra over quaternions into algebra over reals (Rationals) !?
>
>The workaround is to define quaternions in M2C or M4R matrices, but this is not so nice.
>Or define my lie algebra using structure constants but this is also annoying.
>
>Regards,
>Marek Mitros
>
>H:=QuaternionAlgebra(Rationals);
>bh:=Basis(H);
>e:=bh[1]; i:=bh[2]; j:=bh[3]; k:=bh[4];
>
>g1:=e*[[i,0],[0,-i]]; g2:=e*[[j,0],[0,-j]]; g3:=e*[[k,0],[0,-k]]; g4:=e*[[0,1],[-1,0]];
>
>ralg:=LieAlgebra(Rationals, [g1,g2,g3,g4]);
>
>halg:=LieAlgebra(H, [g1,g2,g3,g4]);
>
>Dimension(halg);   # returns 3
>
>Dimension(ralg);   # returns error, should be 10 as ralg=sp(2)=C2
>
>hb:=MutableBasis(H, [g1,g2]);   # returns one vector
>
>rb:=MutableBasis(Rationals,  [g1,g2]); # returns error
>