Do representations of the universal enveloping algebra $\mathrm{U}\mathfrak{su}_2$ retain the Hopf algebra structure?

Question

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 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?

Answer 1

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.