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Saturday, 28 October 2006

Are supervector spaces the representations of a Hopf algebra?

As a monoidal category, supervector spaces are the same as mathbbZ/2mathbbZ representations, so you can think of them as representations of the Hopf algebra mathbbC[mathbbZ/2mathbbZ].



However, the symmetric structure is different, so the thing you need to do is make mathbbC[mathbbZ/2mathbbZ] into a quasi-triangular Hopf algebra in an interesting way, that is, change things so that the map from VotimesWtoWotimesV isn't just flipping, it's flipping followed by an element of mathbbC[mathbbZ/2mathbbZ]otimesmathbbC[mathbbZ/2mathbbZ] which is called the R-matrix. I'll leave actually writing this element as an exercise to the reader, but it's uniquely determined by acting by -1 on sign tensor sign and by 1 on everything else.



You may want to have a look at the relevant Wikipedia article.

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