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By looking at defining relations of standard deformation of $\mathfrak{sl}_2$, which are:

$$ [E,F] = \frac{q^{H}-q^{-H}}{q-q^{-1}}, \quad [H,E] = 2E, \quad \text{ and } \quad [H,F] = -2F, $$

some questions come around.

For example, one can check that Jacobi identity is satisfied, but it would be also satisfied if one considers an arbitrary function $Fun(D)$ instead of the original one $\frac{q^{H}-q^{-H}}{q-q^{-1}}$.

I know that for the algebra to be quantum, the conditions of Hopf algebra have to be enjoyed.

But still, can we imagine another deformation for $\mathfrak{sl}_2$? Or there is a theorem which says that it is the only deformation?

Thanks in advance!

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    $\begingroup$ What exactly is $Fun(D)$? $\endgroup$
    – Alex M.
    Commented Oct 14, 2018 at 13:20
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    $\begingroup$ Perhaps I am missing something, but if you are including Lie element $H$ in the generators along with group-like $q^{\pm H}$ (for which you would also need to specify the multiplicative commutation relations with $E$ and $F$), you are getting not the "standard" deformation, but rather a certain extension. Having said that, there are 2- parameter "quantum groups", where the relations involve $p$ and $q$. $\endgroup$
    – Victor Protsak
    Commented Oct 14, 2018 at 23:54
  • $\begingroup$ To Alex, Under $Fun(D)$ was meant any arbitary function of generator $D$. It is easy to check that Jacobi identity is satisfied as well. (I do realize that it is not enough for algebra to be quantum) $\endgroup$
    – Jedy
    Commented Oct 17, 2018 at 1:59
  • $\begingroup$ To Victor, You are right, but multiplicative commutation relations with $E$ and $F$ are nothing else than rewrting of two of three defining relations. How are you going to add second parameter of deformation $p$? Could you please write it here? $\endgroup$
    – Jedy
    Commented Oct 17, 2018 at 2:07

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Regarding your second question, on other possible deformations of $sl(2)$:

There have been various studies on (multi-parametric) deformations of Lie algebras -as has already been mentioned in the comments to the OP- during the last decades:
An example of a $2$-parameter deformation $sl_{pq}(2)$ which leads to a quantum group is given by: $$ [H,E_{\pm}]=\pm E_{\pm}, \ \ \ [E_+,E_-]=\frac{q^{2H}-p^{-2H}}{q-p^{-1}} $$ where $E_-$ stands for $F$,up to a suitable rescaling and $p$, $q$ are complex parameters.
(setting $p=q$ and rescaling the generators produces the $q$-deformation described in the OP as a special case). You can find more details at arXiv:math/0506539, where this deformation is studied and it is proved that it admits a class of infinite dimensional representations with no analogue in the undeformed or in the $q$-deformed case.
Another -similar- example can be found in the article: A two-parameter deformation of the universal enveloping algebra of $sl(3,C)$, by J.F. Cornwell. A detailed discussion on the hopf structure of the deformed algebra and its implications on the usual hopf structure(s) of the undeformed algebra is also included.
In Introduction to quantum algebras, by M.R. Kibler, two parameter deformations such as $u_{pq}(2)$ and $u_{pq}(1,1)$ are studied: their hopf algebraic structures are investigated and their realizations (that is: homomorphisms or isomorphisms) with two parameter deformations of the Weyl algebras and the angular momentum algebras are used as a tool of investigating their representations.

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    $\begingroup$ Much appreciated! A small research showed that it is indeed the case -- to find appropriate Hopf structure for another deformations. It was the case of q-deformed oscillator, for instance. $\endgroup$
    – Jedy
    Commented Oct 24, 2018 at 6:49

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