Talk:Commutator/Archive 1

Archive 1

commutator

It seems the element "g\-1 h\-1 gh" would be more clearly stated as "h\-1 g\-1 gh" Or am I missing something?

Ken Krechmer

The second expression you write is the identity always, since the middle two g's cancel. Thus it is trivial, and doesn't say anything about whehter g and h commute. -Lethe | Talk 01:05, 15 June 2005 (UTC)

other convention

Indeed the order must of course be g,h,g,h. However, the convention differ, and often [g,h] is rather defined as g h g¯¹ h¯¹ which I personally prefer, e.g. in view of the "other" ("algebra") commutator, [A,B]=A B - B A (so the "plain" part remains and what is "minus reverse order" in the latter becomes "times the inverses" in the former). — MFH: Talk 20:42, 21 June 2005 (UTC)

commutator in algebras

How does this correspond to the functional analysis commutator, such as that in the Uncertainty Principle defined by [A,B] = AB-BA

They are certainly connected: the common spirit is to give a "measure" of how much A and B (or whatever) do not commute. The final remark about commutators being defined also in structures different from groups should definitely be expanded. --Goochelaar

The connection is given by going to infinitesimal generators of Lie group elements. This is explained (rather alluded to....) in Lie algebra#Examples point 5 (currently *sic*). In some sense, consider e.g.

g = exp( x ) , h = exp( y )

then

g h = exp(x + y + 1/2[x,y] +...), thus g h g¯¹ h¯¹ = exp( [x,y] +...) (where ... are higher order terms).

It can also be formulated in terms of the derivative of the elements of a Lie group at the origin. — MFH: Talk 21:09, 21 June 2005 (UTC)

another property

Looking at the ring theory definition of commutator, the property AB = BA + [A,B] seems obvious in retrospect, but a usage (Griffiths, Quantum Mechanics) of that property had me scratching my head for a good ten minutes. *shrug* In case anyone else is scratching their head about it at any time. Popefelix 00:15, 14 December 2005 (UTC)