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Let $I_1(L)$ be a Reshetikhine-Turaev link invariant coming from the (quantum Lie) group $G_1$ and representation $\lambda_1$ having the S matrix $S_1$. Let $I_2(L)$ be a Reshetikhine-Turaev link invariant coming from the (quantum Lie) group $G_2$ and representation $\lambda_2$ having the S matrix $S_2$ (hooray for copy'n'paste!). Then, for now just by my experiments, there always exists an invariant $I(L)=I_1(L)*I_2(L)$ (good old multiplication) with the S matrix $S_1\otimes{S_2}$ (here the trouble already begins: you can have S in tensor from, or condensed to matrix form, or condensed with index swap - and "$\otimes$" may be correct only in one case) which surely comes from the group $G_1°G_2$ (Tensor product? Direct product? Product of my imagination? :-) and the representation $\lambda_1°\lambda_2$ (even worse - if I remember correctly, now there are even 3 kind of products to be considered).

Please help me out with the correct signs $*,\otimes,\times$ which I confuse all the time. I also like to know about sort of a converse: If $G=G_1°G_2$ (Tensor? Direct?), are the representations $\lambda$ of $G$ always of the form $\lambda_1°\lambda_2$?

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 If you are viewing the quantum groups as Hopf algebras, then you should use tensor product on both the quantum groups and the representations, where the Hopf algebras act component-wise on the representations. – S. Carnahan♦ Aug 25 2011 at 9:57
@Hauke: you have asked many questions on similar topics before, and to be honest the answer to most of them is "it would be a good idea for you to learn abstract algebra in general at the undergraduate and beginning graduate level before thinking about such things." (Not that I blame you for being interested in this material; it is of course very interesting! But patience is a virtue in these matters.) – Qiaochu Yuan Aug 25 2011 at 22:45
@Qiaochu: I surely would do MO a favor, yes :-) And I'm not even especially stupid, I always had an A in school math. But, and this is a but of cosmic dimensions. I am 50. Each time I try myself on Harris&Fuller or comparable stuff I see I have no, NO talent for abstract math at all. I probably need 20 years of intensive study to be able to answer such questions myself. Which is a complete waste of my brilliancy :-) (I could never compete with a math pro, my talents are conjectures drawn from the hat, generalizing a pattern of 3 items to infinity and doing all with diagrams.) – Hauke Reddmann Aug 26 2011 at 10:57
@Hauke: forgive me my impertinency, but I don't necessarily think that being 50 is a barrier. It's all a matter of starting at the right level (Fulton and Harris, if that is what you were referring to, being absolutely the wrong level to start at). You should start by looking at books on general abstract algebra (I like Artin's Algebra), then moving on to books on the representation theory of finite groups (I like James and Liebeck's Representations and Characters of Groups) before tackling the harder stuff. – Qiaochu Yuan Aug 28 2011 at 4:42
Book tips are duely noted (in the lib, check.) I'm surely not the guy giving up fast. :-) (It's just that math is a building and if I run in yet another notion I'm unfamiliar with I know I still didn't start low enough.) For triangle geometry the situation is almost ideal, there exists "Hyacinthos" on Yahoo where you can discuss everything informally and it doesn't hurt much when you lack a proof or your result is aniticipated, about 1800. :-) I wish there were the same for knot theory, and in lack of that I tend to abuse MO... – Hauke Reddmann Aug 28 2011 at 17:22
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