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For an infinite dimensional Hopf algebra $H$, a non-degenerate dually pairing Hopf algebra $H'$, and a choice of basis $e_i$ of $H$, is the dual basis $e^i$ (defined of course by $e^i(e_j) = \delta_{ij}$) contained in $H'$?

I am interested in the specific case of $SU_q(N)$ and the dually paired Hopf algebra $\mathfrak{sl}_N$.

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No. Let $k$ be a field of characteristic $0$. Consider the symmetric algebra on one generator $k[x]$, with comultiplication $x \mapsto x\otimes 1 + 1\otimes x$. It has a Hopf pairing with itself, given by $\langle x^m,x^n\rangle = n! \hspace{.5ex} \delta_{m=n}$. Then consider the bases of $k[x]$ given by expansion around $1$, i.e. $e_n = (x-1)^n$. The dual basis, if it exists, includes $e^0$ such that $\langle e^0, (x-1)^n\rangle = \delta_{0,n}$. So suppose that $e^0 = \sum a_n x^n$; then: $$ \begin{aligned} 1 & = a_0 \\ 0 & = a_1 - a_0 \\ 0 & = a_2 - 2a_1 + a_0 \\ \dots & \phantom= \dots \\ 0 & = \sum_{k=0}^n (-1)^{n-k} \hspace{.5ex} n!\hspace{.5ex} a_n \hspace{.5ex} \binom{n}{k} \\ \dots & \phantom= \dots \\ \end{aligned} $$ The solution is that all $a_n = 1/n!$, i.e. $e^0 = \sum x^n/n! = \exp(x)$. But this is not a polynomial.

In the case you ask about, you will similarly have some bases with dual bases, and some without.

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