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Nested Commutators

• To: mathgroup at smc.vnet.net
• Subject: [mg58114] Nested Commutators
• From: "Shug Boabby" <Shug.Boabby at gmail.com>
• Date: Sat, 18 Jun 2005 06:08:51 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

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.

• Follow-Ups:
  • Re: Nested Commutators
    • From: Andrzej Kozlowski <andrzej@akikoz.net>
  • Re: Nested Commutators
    • From: Pratik Desai <pdesai1@umbc.edu>