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!



Is there another quantum deformation of sl(2)?
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](https://arxiv.org/abs/math/0506539v1), 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 $sl(3,C)$](https://link.springer.com/article/10.1007/BF01555521), 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](https://arxiv.org/pdf/hep-th/9409012.pdf), 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. 


