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MathGroup Archive 2005

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Re: Nested Commutators

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58119] Re: [mg58114] Nested Commutators
  • From: Andrzej Kozlowski <andrzej at akikoz.net>
  • Date: Sun, 19 Jun 2005 03:43:21 -0400 (EDT)
  • References: <200506181008.GAA08906@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On 18 Jun 2005, at 19:08, Shug Boabby wrote:

> hi there,
>
> i have searched the archive for theads on nested commutators in
> mathematica, but i found no answer to my problem and i was  
> wondering if
> anyone had any helpful advise for me.
>
> i wish to be able to define a commutator[A, B]
>   http://mathworld.wolfram.com/Commutator.html
> between 2 operators.
>
> for example i would like to be able to define an algebra by presenting
> the commutators, such as
>   [A, B] = B
>   [A, C] = C
>   [B, C] = D
> (also explicitly defining the rest to be zero) and then be able to ask
> mathematica to return the solution when i nest the operators like so
>   [A, [A, [A, B]]]
> which should return B
>   [A, [B, C]]
> should return 0.
>
> it would also be nice if the Jacobi identities are used in
> simplification routines, so that terms such as
>   [X, [Y, [X, Y]]]
>   -[Y, [X, [Y, X]]]
>   [[X, [Y, X]], Y]
> are identified as being the same as
>   [Y, [X, [X, Y]]]
> right ordering the output when the commutators cannot be calculated
> explicitly.
>
> apologies if you have seen this question recently... i stupidly asked
> it at the end of my last posting where it is surely to be missed. this
> question really warrants its own thread.
>
>


It might perhaps be useful to take a look at my response (in 1999) to  
Melih Sener in response to his question: "How to do Lie Algebras in  
Mathematica":

http://forums.wolfram.com/mathgroup/archive/1999/Nov/msg00213.html

Andrzej Kozlowski


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