Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum Groups) that there is an ad-invariant nondegenerate bilinear form on $U_q$. By the standard base-change rigamarole, from an appropriate $\mathbb Z[q,q^{-1}]$-subalgebra of $U_q$ we can obtain the enveloping algebra $\bar U_{\mathbb C}$ of $\mathfrak g$ over $\mathbb C$ and, for a field $k$ of positive characteristic, the hyperalgebra $\bar U_k$ of a linear algebraic group over $k$ associated to $\mathfrak g$.

My question is: can we base-change the bilinear form on $U_q$ to obtain ad-invariant bilinear forms on $\bar U_{\mathbb C}$ and $\bar U_k$? Or is there an easier direct construction that can be made without going through the quantum construction first?

`$q$`

has more than one meaning.) $\endgroup$ – Jim Humphreys Aug 3 '12 at 14:30