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MathGroup Archive 1992

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  • To: mathgroup at yoda.physics.unc.edu
  • From: John Lee <lee at math.washington.edu>
  • Date: Sun, 2 Feb 92 11:35:33 -0800

In <9201311923.AA02621 at fubini.math.ufl.edu>, livio flaminio writes:

   In an attempt to define a non commutative algebra over the complex by  
   using NonCommutativeMultiply I defined

   ... (* some definitions omitted *)

   NonCommutativeMultiply[ Times[a_ ,b_] , Times[d_ ,c_]]:=
   Times[Times[a, d] ,NonCommutativeMultiply[  b ,  c]] /; NumberQ[a] &&   
   NumberQ[d]

   ...

   ... the semplifications x ** (I y) = I x ** y and (-I y) ** x = -I  
   y ** x have not been carried out. Any clue of why that is the case?

The problem is that your pattern requires _both_ factors of
NonCommutativeMultiply to be multiplied by constants.  Try the following:

  In[1]:= Unprotect[NonCommutativeMultiply];

  In[2]:= NonCommutativeMultiply[ a_ + b_ ,c_ ]:=
  NonCommutativeMultiply[ a  ,c] +
  NonCommutativeMultiply[  b ,c] 

  In[3]:= NonCommutativeMultiply[ a_ , b_ + c_]:=
  NonCommutativeMultiply[ a  ,b] +
  NonCommutativeMultiply[  a ,c] 

  In[4]:= NonCommutativeMultiply[ b_ , Times[d_ ,c_]]:=
  Times[ d, NonCommutativeMultiply[ b, c]] /; NumberQ[d]

  In[5]:= NonCommutativeMultiply[ Times[a_ ,b_] , c_]:=
  Times[a, NonCommutativeMultiply[  b ,  c]] /; NumberQ[a]

  In[6]:= brac[a_, b_]:= a**b - b**a

  In[7]:= mplus= (x - I y) /2;

  In[8]:= mminus= (x+ I y)/2;

  In[9]:= brac[mplus, mminus]

           -(x ** x - I x ** y + I (y ** x - I y ** y))
  Out[9]= -------------------------------------------- + 
                                4

       x ** x + I x ** y - I (y ** x + I y ** y)
  >    -----------------------------------------
                           4

  In[10]:= %//Expand

           I          I
  Out[10]= - x ** y - - y ** x
           2          2

Jack Lee
Dept. of Mathematics
University of Washington





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