Dear Bartosz,
currently there is no immediate function in GAP to construct the
Clifford
algebra. If I understood correctly what are you going to do, that
maybe the
first step should be to find out how to do this.
I think you should be able to construct it using the
AlgebraByStructureConstants
function, see http://www.gap-system.org/Manuals/doc/htm/ref/
CHAP060.htm#SSEC003.5
For example, this is how the QuaternionAlgebra function works, and
quaternion
algebras could be regarded as a special case of Clifford algebras.
You can have
a look on the function QuaternionAlgebra in gap4r4/lib/algsc.gi to
get an idea
how to write your code.
When this will work, you should be able to generate subgroups by
elements of
this algebra, for example:
gap> R:=QuaternionAlgebra(Rationals,1,1);
<algebra of dimension 4 over Rationals>
gap> gens:=GeneratorsOfAlgebra(R);
[ e, i, j, k ]
gap> G:=Group(gens);
#I default `IsGeneratorsOfMagmaWithInverses' method returns `true'
for [ e, i, j, k ]
<group with 4 generators>
gap> IdGroup(G);
[ 8, 3 ]
Also there is an undeposited implementation relevant to quaternion
algebras
over the Rationals: http://www.geocities.com/assafwool/Quat/Quat.html.
Best wishes,
Alexander
On 14 Mar 2007, at 08:38, Bartosz Putrycz wrote:
Dear Forum,
I need to have some way to put/represent in GAP finite subgroups of
the
spin group,
given by generators in Clifford algebra,
or given by their image in canonical epimorphism onto SO(n)
For example:
1)
<e_1 e_2, e_2 e_3, e_1 e_3> maped onto C2^2(diagonal +-1) in SO(3)
2)
<1/sqrt{2} (1 - e_3 e_4), 1/sqrt{2} e_1(e_3 - e_4)> -> D8 in SO(4)
Of course I could analyze these groups by myself, first is Q8,
but I have more examples and I want to analyze them automatically
by GAP.
Regards,
Bartosz Putrycz.
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