Hello,
Have you found the way to represent Clifford algebras in GAP ?
Personally I have represented monomial from Clifford algebra as list [cf, ind]
where <cf> is coefficient and <ind> are list of indexes e.g. e12=[1, [1,2]] and
-e12=[-1,[1,2]].
See below functions which I have created. In my functions I have assumed that
either ei*ei=1 or ei*ei=-1 for given algebra. See below functions
<cliff_mult_plus> and <cliff_mult_minus>.
Regards,
Marek
# 2009-05-08 Create function which generate abstract clifford algebra
# elements are represented as list with coefficients e123= [1,2,3] or [1,
[1,2,3]] etc
# Define first function for multiplying list elements.
# Phase 1: Apply bubble sort to the concatenated list, each exchange of element
will change the sign
# Phase 2: Multiply all neighbour identical elements. The result will depend on
the signature of Clifford algebra
# i.e. whether e1 * e1 = 1 or -1.
# This function should return new element (sorted list of integers) and sign 1
or -1.
# Note: for sorting we can use SignPerm(Sortex(list)) functions which return
permutation and calculates its sign.
# In the same time <list> is sorted.
# Examples:
# SignPerm(Sortex([1,3,2])) = -1 ; [2,3,1] -> 1; [1,2,1] -> -1;
#
# Such a solution does not require writing function for bubble sort.
# Case when all squares of generators are equal to 1
cliff_mult_plus:=function(a,b)
local res, sign, ind;
res:=Concatenation(a,b);
sign:=SignPerm(Sortex(res)); # sort
# now we remove all repeats
ind:=First([1..Length(res)-1], s->res[s]=res[s+1]);
while (ind <> fail) do
res:=Concatenation(res{[1..ind-1]}, res{[ind+2..Length(res)]});
ind:=First([1..Length(res)-1], s->res[s]=res[s+1]);
od;
return [sign, res];
end;
lie_bracket_plus:=function(a,b)
local c,d;
c:=cliff_mult_plus(a,b);
d:=cliff_mult_plus(b,a);
if c=d then return [0,[]];
else return [c[1]-d[1], c[2]];
fi;
end;
# Case when all squares of generators are equal to -1
cliff_mult_minus:=function(a,b)
local res, sign, ind;
res:=Concatenation(a,b);
sign:=SignPerm(Sortex(res)); # sort
# now we have to remove all repeats and change sign for each removal
ind:=First([1..Length(res)-1], s->res[s]=res[s+1]);
while (ind <> fail) do
res:=Concatenation(res{[1..ind-1]}, res{[ind+2..Length(res)]});
sign:=-sign;
ind:=First([1..Length(res)-1], s->res[s]=res[s+1]);
od;
return [sign, res];
end;
# 2009-05-22 function return SCT table for Lie algebra with cllifford algebra
elements
# parameters: bas - basis of clifford Lie algebra in form [[1,2], [2,3],...]
sct_lie_alg_cliff_plus:=function(bas)
local T, dim, s,t,res, pos;
dim:=Length(bas);
T:=EmptySCTable(dim, 0, "antisymmetric");
for s in [1..dim-1] do
for t in [s+1..dim] do
res:=lie_bracket_plus(bas[s], bas[t]);
if res[1]<>0 then
pos:=Position(bas, res[2]);
SetEntrySCTable(T,s,t, [res[1],pos]);
fi;
od;
od;
return T;
end;;
_______________________________________________
Forum mailing list
Forum@mail.gap-system.org
http://mail.gap-system.org/mailman/listinfo/forum
