A Lie algebra $\mathfrak{g}$ generates its universal enveloping algebra $\mathrm{U}\mathfrak{g}$, which has the structure of a Hopf algebra. Modules of $\mathrm{U}\mathfrak{g}$ are exactly the of modules of $\mathfrak{g}$ itself.


In the case $\mathfrak{g} = \mathfrak{su}\_2$, the simple modules are labeled uniquely (up to isomorphism) by their dimension, and we label them $V\_d$ for $d = 1, 2,$ etc. Note that $\mathrm{U}\mathfrak{su}\_2$ is itself a $\mathfrak{su}\_2$ module and decomposes into simple modules as $\bigoplus\_{j = 0}^\infty V\_{2j+1}$, i.e. only odd-dimensional modules occur. We can consider truncations of this sum too:
$$\mathrm{U}\_J\mathfrak{su}\_2 \equiv \mathrm{End}\,V\_{2J+1} \simeq V\_{2J+1}^\ast \otimes V\_{2J+1} \simeq \bigoplus\_{j = 0}^{2J} V\_{2j+1}$$
In particular, $\dim \mathrm{U}\_J\mathfrak{su}\_2 = 1 + 3 + \cdots + (4J+1) = (2J+1)^2$, which is $\dim \mathrm{End}\,V\_{2J+1}$. An explicit relationship between the basis adapted to the decomposition of $\mathrm{U}\_J \mathfrak{su}\_2$ given above and the basis $\{E\_{ij}\}$ (with components 1 at $(i,j)$ and 0 elsewhere) on $\mathrm{End}\,V\_{2J}$ can be given in terms of the $3jm$ symbols. The result shows that $\mathrm{U}\_J \mathfrak{su}\_2$ is isomorphism to $\mathrm{End}\,V\_{2J+1}$ as associative algebras.


W. Smoke's paper [Invariant Differential Operators](http://www.ams.org/journals/tran/1967-127-03/S0002-9947-1967-0210835-1/S0002-9947-1967-0210835-1.pdf) uses the relationship between $\mathrm{U}\mathfrak{su}\_2$ and differential operators on the $\mathrm{SU}(2)$ group manifold. (He actually works for a generic Lie algebra $\mathfrak{g}$ but my focus is only on $\mathfrak{su}\_2$ for now.) The filtration of $\mathrm{U}\mathfrak{su}\_2$ is the filtration of differential operators by degree, where $\mathrm{U}\_J\mathfrak{su}\_2$ gives differential operators with degree at most $J$. At the top of his eighth page (p. 467 in the journal), it is proved that $\mathrm{U}\_J \mathfrak{su}\_2$ is a subcoalgebra of $\mathrm{U}\mathfrak{su}\_2$.


My understanding is that there is no Hopf algebra structure on matrix algebras like $\mathrm{End}\,V\_{2J+1} \simeq \mathrm{End}\,\mathbb{C}^{2J+1}$ because they are simple. There is an algebra structure and a coalgebra structure, but these must not be compatible in the sense of bialgebras. But in the limit $J \to \infty$ the compatibility seems to be restored, since we know $\mathrm{U}\mathfrak{su}\_2$ is a Hopf algebra. How does this work?



Do representations of the universal enveloping algebra $\mathrm{U}\mathfrak{su}_2$ retain the Hopf algebra structure?
The problem is that $U\_J\frak{su}\_2$ as you define it, that is $\bigoplus\limits\_{j=0}^{2J}V\_{2j+1}$, is NOT the space of all differential operators of order at most $J$. For the latter, Formula (2) in the paper suggests that it is the sum $\bigoplus\limits\_{j=0}^J S^j(\mathfrak{su}\_2)$, so its dimension is equal to the sum $\sum\limits\_{j=0}^{J}\frac{(j+1)(j+2)}{2}=\frac{(J+1)(J+2)(J+3)}{6}\ne(2J+1)^2$. 


In particular, a natural way to view $U\_J\frak{su}\_2$ as you define it is as a quotient space of $U\frak{su}\_2$, not a subspace, so there is a natural algebra structure but no natural coalgebra structure.


