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I have a list of expressions like the following:

{{X00 ** X10 + X01 ** X10 - X10 ** X01 - X10 ** X00 }, 
 {X01 ** X10 - X10 ** X01 - X02 ** X10}, 
 {X00 ** X12 -  X12 ** X00}, {X01 ** X12 - X12 ** X01}, {X02 ** X13}}

where all Xij are just symbols. Now suppose we define an expression like

[X0j,X1k] := X0j ** X1k - X1k ** X0j.

Is it possible to let Mathematica substitute all terms in the list that look like the right side of the last expression by its left side?

If necessary I can change the Xij's in the list to something else.

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[X0j,X1k] is not syntactically correct Mathematica. Could you clarify what you mean by it? Could you please give an exampe of a possible substitution (I'm a little confused about what you need exactly)? –  Szabolcs May 10 '12 at 7:19
    
Ok the "[" and "]" symbols are occupied by mathematica itself. But that was just an example. We could use anything. terms like X0j ** X1k - X1k ** X0j should just be rephrased by something visual more meaning full. –  Mark Neuhaus May 10 '12 at 7:24
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2 Answers 2

up vote 6 down vote accepted

Let's represent your bracket expression using the head bb, so

bb[x, y] == x ** y - y ** x

Then we can just use a simple replace rule:

{{X00 ** X10 + X01 ** X10 - X10 ** X01 - X10 ** X00 }, 
 {X01 ** X10 - X10 ** X01 - X02 ** X10}, 
 {X00 ** X12 -  X12 ** X00}, {X01 ** X12 - X12 ** X01}, {X02 ** X13}}
   //. x_ ** y_ - y_ ** x_ :> bb[x, y]

(* ==>
  {{bb[X00, X10] + bb[X01, X10]}, 
   {bb[X01, X10] - X02 ** X10}, 
   {bb[X00, X12]}, {bb[X01, X12]}, {X02 ** X13}}
*)

We can automate the conversion between the two representations using

toBracket[expr_] := expr //. x_ ** y_ - y_ ** x_ :> bb[x, y]
fromBracket[expr_] := expr /. bb[x_, y_] :> x ** y - y ** x

If you wish to have a prettier notiation, you could for example use AngleBracket instead of bb. It is formatted like this:

Mathematica graphics

You can enter the brackets using the key sequence Esc<Esc.

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Ok thanks! That solved my problem. –  Mark Neuhaus May 10 '12 at 7:46
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Mark,

This is not an answer as much as it is a recommendation. I had to manipulate long expressions of non-commutative variables for my PhD going beyond defining a Killing form. If this will be the case for you I really highly recommend the NCAlgebra package someone wrote:

http://www.math.ucsd.edu/~ncalg/

the manuals are pretty much self explanatory and it will solve many problems (like series expansion etc).

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