The key point here is to understand how the parameter $q$ is fixed to a complex number. Usually $U_q(\mathfrak g)$ is introduced as a $\mathbb C(q)$--Hopf algebra with generators and relations, and relations are singular at $q=1$. This means that to obtain a $\mathbb C$--Hopf algebra from $U_q(\mathfrak g)$ one has first to fix a so-called integer form, i.e. a $\mathbb C[q,q^{-1}]$--subalgebra $\cal U$ such that ${\cal U}\otimes\mathbb C(q)=U_q(\mathfrak g)$.

What you are writing is one possible choice of this integer form. However the choice of integer form is, in general, not unique. Basically, if I remember correctly, for the semisimple case any lattice in between the weight lattic and the root lattice should provide you a different integer form, this should be explained in older papers by De Concini (back in the 90ies) on Qg's at roots of unity (i.e. the one contained in here http://link.springer.com/content/pdf/10.1007/BFb0073466).

Your choice should correspond to what is called the simply connected integer form. The key (not so easy) part is to show that indeed the subalegbra generated by the elements you show is a $\mathbb C[q,q^{-1}]$--integer form. Detalis on such fact should be given in the cited ref.

The key point here is to understand how the parameter $q$ is fixed to a complex number. Usually $U_q(\mathfrak g)$ is introduced as a $\mathbb C(q)$-Hopf algebra with generators and relations, and relations are singular at $q=1$. This means that to obtain a $\mathbb C$-Hopf algebra from $U_q(\mathfrak g)$ one has first to fix a so-called integer form, i.e. a $\mathbb C[q,q^{-1}]$--subalgebra $\cal U$ such that ${\cal U}\otimes\mathbb C(q)=U_q(\mathfrak g)$.

What you are writing is one possible choice of this integer form. However the choice of integer form is, in general, not unique. Basically, if I remember correctly, for the semisimple case any lattice in between the weight lattice and the root lattice should provide you a different integer form, this should be explained in older papers by De Concini (back in the 90ies) on Qg's at roots of unity (i.e., the chapter De Concini and Procesi - Quantum groups in D-modules, representation theory, and quantum groups).

Your choice should correspond to what is called the simply connected integer form. The key (not so easy) part is to show that indeed the subalegbra generated by the elements you show is a $\mathbb C[q,q^{-1}]$-integer form. Details on such fact should be given in the cited reference.