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Re: Nested Commutators

• To: mathgroup at smc.vnet.net
• Subject: [mg58117] Re: Nested Commutators
• From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
• Date: Sun, 19 Jun 2005 03:43:20 -0400 (EDT)
• References: <d90tqp$9di$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

This sort of thing will do what you want:

comm[a, b] = b;
comm[a, c] = c;
comm[b, c] = d;
comm[_, _] = 0;

then you can evaluate your nested commutators thus

comm[a, comm[a, comm[a, b]]]

b

comm[a, comm[b, c]]

0

Steve Luttrell

"Shug Boabby" <Shug.Boabby at gmail.com> wrote in message 
news:d90tqp$9di$1 at smc.vnet.net...
> 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.
>