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More precisely, does there exist a Hopf algebra $H$ whose category of (finite-dimensional, complex) representations is generated under direct sum and tensor product by two one-dimensional representations $X, X^{\ast}$ such that $X^{\ast} \otimes X \cong I$ but such that all of the representations $X^{\otimes m} \otimes (X^{\ast})^{\otimes n}$ ($m, n \ge 0$) are non-isomorphic? Does there exist such a Hopf algebra $H$ which in addition fits into a sequence $$U(\mathfrak{gl}_1) \to H \to U(\mathfrak{sl}_2)$$

such that the corresponding restriction map $\text{Rep}(U(\mathfrak{sl}_2)) \to \text{Rep}(H)$ sends the defining representation of $\mathfrak{sl}_2$ to $X \oplus X^{\ast}$ and such that the restriction map $\text{Rep}(H) \to \text{Rep}(U(\mathfrak{gl}_1))$ sends $X$ to the defining representation of $S^1$ and $X^{\ast}$ to its dual?

(I expect the answer to be no, possibly for quite trivial reasons.)


Motivation: let $T$ be the *-algebra generated by the shift operator $S(e_i) \mapsto e_{i+1}$ on $\ell^2(\mathbb{N})$ (the completion of this algebra is the Toeplitz algebra). It is not hard to see that $S^{\ast} S = I$ and that the operators $S^m (S^{\ast})^n, m, n \ge 0$ form a basis of $T$. The operator $S + S^{\ast}$ acting on the basis vectors $e_1, e_2, ... \in \ell^2(\mathbb{N})$ describes the action of tensoring by the defining representation $V$ of $\mathfrak{sl}_2$ on its irreducible representations (more precisely their highest weights, I think); in terms of the corresponding representations, we have $$V \otimes S^n(V) \cong S^{n+1}(V) \oplus S^{n-1}(V)$$

corresponding to the identity $$(S + S^{\ast}) e_n = e_{n+1} + e_{n-1}.$$

Taking powers of $S + S^{\ast}$ corresponds to acting on irreducible representations by tensoring with powers of $V$, and this seems to also correctly describe the weight spaces of the irreducible representations. For example, $$(S + S^{\ast})^2 = 1 + (S^2 + S S^{\ast} + S^{\ast 2})$$

corresponds to $$V^{\otimes 2} \cong 1 \oplus S^2(V).$$

Quotienting $T$ by the relation $S S^{\ast} = I$ we can think of the result as the representation ring of $S^1$, and the corresponding map $\text{Rep}(\text{SU}(2)) \to \text{Rep}(S^1)$ is of course just restriction to some maximal torus. But working in the representation ring of $S^1$ does not allow us to distinguish weight spaces which are the same representation of $S^1$ but which occur in different irreducible representations of $\text{SU}(2)$, while working in $T$ seems to accomplish this. So I am curious whether the appearance of $T$ here can be explained by postulating a sequence of "morphisms" $$S^1 \to ??? \to \text{SU}(2)$$

such that the maps on representation rings going the other way factors through $T$ in the middle. The thing in the middle has a category of representations with a tensor product that is not symmetric so maybe it is a non-cocommutative Hopf algebra. Thus the above question. Again I expect the answer to be no, so my real question is whether anyone can categorify the appearance of $T$ here in any way whatsoever.

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