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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.
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which I personally prefer, e.g. in view of the "other" ("algebra") commutator, =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). —
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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. -
196:) 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.
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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?
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Indeed the order must of course be g,h,g,h. However, the convention differ, and often is rather defined as
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definition of commutator, the property AB = BA + seems obvious in retrospect, but a usage (Griffiths,
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g h = exp(x + y + 1/2 +...), thus g h g¯¹ h¯¹ = exp( +...) (where ... are higher order terms).
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If you wish to start a new discussion or revive an old one, please do so on the
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It can also be formulated in terms of the derivative of the elements of a
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The connection is given by going to infinitesimal generators of
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point 5 (currently *sic*). In some sense, consider e.g.
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elements. This is explained (rather alluded to....) in
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151:g = exp( x ) , h = exp( y )
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