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Here is one of the figures whose commutativity express the fact that $\Delta$ is a morphism of algebra:

$\require{AMScd}$ $$\begin{CD}H\otimes H @>\Delta\otimes\Delta>> (H\otimes H)\otimes (H\otimes H)\\ @V \mu V V @VV (\mu\otimes\mu)({\rm id}\otimes \tau\otimes{\rm id}) V\\ H @>>\Delta > H\otimes H. \end{CD}$$

(Source: Christian Kassel, Quantum Groups, p.45.)

But I do not quite sure understand the following notation, what is the meaning of $(\mu \otimes \mu) ({\rm id} \otimes \tau \otimes {\rm id}) $? how should I apply it? Could someone explain this to me, please?

MJD
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Emptymind
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  • Is $\Delta$ an abbreviation for $\mathrm{id}\otimes\mathrm{id}$? – MJD Feb 23 '23 at 21:41
  • Also, if you can tell us where you saw the diagram (what is the title of the book or paper, and what page; provide a link if possible) someone might be more likely to give you the right answer. – MJD Feb 23 '23 at 21:42
  • @MJD it is the comultiplication map (or we can say the diagonal map) – Emptymind Feb 23 '23 at 21:42
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    @MJD The meaning of $\Delta$ in a question tagged with "Hopf algebras" and "Quantum groups" is self-explanatory. – J. De Ro Feb 23 '23 at 21:48
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    I think $\tau $ here is the map $\langle a,b\rangle\mapsto \langle b, a\rangle$. – MJD Feb 23 '23 at 21:52
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    @QuantumSpace Thanks, I just wanted to be sure I understood correctly. – MJD Feb 23 '23 at 21:52

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$\tau$ here is the switch map $$H\otimes H \to H \otimes H: h \otimes g \mapsto g \otimes h$$ you then form $$\operatorname{id}\otimes \tau \otimes \operatorname{id}: H \otimes H \otimes H \otimes H \to H \otimes H \otimes H \otimes H: h \otimes g \otimes k \otimes l \mapsto h \otimes k \otimes g \otimes l.$$

$\mu$ is the multiplication map of the algebra $$\mu: H \otimes H \to H: h\otimes g \mapsto hg$$ Thus $$\mu\otimes \mu: H \otimes H \otimes H \otimes H \to H \otimes H: g\otimes h \otimes k\otimes l \mapsto gh \otimes kl$$

You then consider the composition $$(\mu\otimes \mu)\circ (\operatorname{id}\otimes \tau \otimes \operatorname{id}): H \otimes H \otimes H \otimes H \to H\otimes H$$ which maps $$g\otimes h \otimes k \otimes l \mapsto gk \otimes hl.$$

J. De Ro
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  • In my case $h = t^p$ and $g = t^q$ and so on ..... ? – Emptymind Feb 27 '23 at 04:04
  • I am a little bit confused when should I put the summation sign at the very beginning or not ...... like in our case here, should I add summation before the tensor product of $h,g,k$ and $l$ or not? – Emptymind Feb 27 '23 at 04:06
  • I am sorry, I did not state that I am working on $k[t]$ the polynomials in one variable and I am speaking about summation because $\Delta (t^n) := \sum_{p+q = n} t^p \otimes t^q$ – Emptymind Feb 27 '23 at 04:11
  • @Emptymind Every element in a tensor product $V\otimes W$ is a sum of the form $\sum_{i=1}^n v_i\otimes w_i$. Since I am dealing with linear maps, it suffices to show what they do on $v\otimes w$ to know what they do everywhere. Also, look up "Universal property of the tensor product". – J. De Ro Feb 27 '23 at 09:02