[Reduce-algebra-developers] About unexpected behavior operator sum and commutation rules.

[abpetrov <abp...@uf...>]

Hi,
I wanted calculate some commutators, but I got next unexpectedbehavior  of operator sum.
The program is below. Aspreliminary note I understand that i <> i1 always.
So, I defined anticommuting operators a, ac with corresponding commutation rules.
Then I defined commutator com and operator H, after that two rules for ~a and sum
- left rule and right rule. After that I tried to calculate commutator com( H, a(i1) )
using rule r1. And I got unexpected result, in which left rule works, but right rule not works.
I expectedsome  symmetry  between left and right.
Why is it not true?

The program is:

off lower$
on intstr$

load_package noncom2;

operator a,ac;
noncom a,a;
noncom ac,ac;
noncom a,ac;

for all i let a(i)*ac(i) = 1 - ac(i)*a(i);
for all i,j such that i neq j let a(i)*ac(j) = -ac(j)*a(i);
for all i,j such that ordp(i,j) let a(i)*a(j) = -a(j)*a(i);
for all i,j such that ordp(i,j) let ac(i)*ac(j) = -ac(j)*ac(i);

operator com;
for all x,y let com(x,y) = x*y - y*x;

r1 := {~a*sum(~k, ~i)=>sum(a*k,i),
           sum(~k, ~i)*~a =>sum(k*a,i)};

operator H;
H := sum( J1*ac(i)*a(i+1), i  );
H1 := com( H, a(i1) );
H2 := (H1 where r1);

end;


  Best regards, Petrov Alexander.