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May 25, 2011 at 0:53 comment added Charlie Frohman Finally, this might be pertinent; [38] Non-involutory Hopf algebras and 3-manifold invariants. Duke Math. J. 84 (1996), 83-129, arXiv:q-alg/9712047 , MR 1394749 (97g:57021)
May 25, 2011 at 0:32 comment added Charlie Frohman Here is a way to understand the antipode. The linear maps from the Hopf algebra to itself form an algebra under convolution. That is, $$ L*M(h)=m\circ(L\otimes M)\circ \Delta(h)$$ where $m$ is the multiplication and $\Delta$ is the comultiplication. The anditpode is the multiplicative inverse of the identity under this product.
May 25, 2011 at 0:28 history answered Charlie Frohman CC BY-SA 3.0