Antiautomorphism

I’ve been wanting to ask this for awhile.  What, precisely, is an “antiautomorphism?”  I think understanding that would help with problem 3.  

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One comment

  1. Depends on the context.

    An antiautomorphism of an algebra is the same as an automorphism of this algebra, except that instead of satisfying f\left(ab\right) = f\left(a\right) f\left(b\right) (or equivalently m\circ\left(f\otimes f\right) = f\circ m) it must satisfy f\left(ab\right) = f\left(b\right) f\left(a\right) (or equivalently m\circ\left(f\otimes f\right) = f\circ m\circ\tau, where \tau is the twist/flip).

    An antiautomorphism of a coalgebra is the same as an automorphism of this coalgebra, except that instead of satisfying \left(f\otimes f\right)\circ\Delta = \Delta\circ f it must satisfy \left(f\otimes f\right)\circ\Delta = \tau\circ \Delta\circ f, where \tau is the twist/flip.

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