A real form of a Hopf algebra $H$ over $\mathbb{C}$ is defined to be a $\ast$-structure on $H$ which is compatible with the coproduct. Compatibility of the $\ast$-structure with the counit and antipode then follows.


The real forms of the Drinfeld-Jimbo quantum groups $U\_q(\mathfrak{g})$ have been classified. This result is stated as Theorem 20 in Chapter 6 of *Quantum Groups and their Representations*, by Klimyk and Schmudgen, and also Proposition 9.4.2 of *A Guide to Quantum Groups*, by Chari and Pressley.


For a given $\mathfrak{g}$, there are $\ast$-structures when $q \in \mathbb{R}$ or when $|q|=1$. These $\ast$-structures depend on a diagram automorphism of the Dynkin diagram of $\mathfrak{g}$, plus some extra parameters, and it is understood which sets of parameters give equivalent $\ast$-structures.


There is also an exceptional case, but the two sources I have cited differ on what the exceptional case is. Klimyk and Schmudgen say that the exceptional case is when $\mathfrak{g} = \mathfrak{sp}\_{2n}$ and $q \in i \mathbb{R}$, while Chari and Pressley say that the exceptional case is $\mathfrak{g} = \mathfrak{so}\_{2n+1}$ and $q \in i \mathbb{R}$.


Neither book contains a proof, nor cites a source, although Chari-Pressley gives a sketch of the idea of the proof. So I would be interested in knowing the following things:


1. Which is correct?
2. What is the original reference for the classification of the real forms of $U\_q(\mathfrak{g})$?




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### To set the record straight


It appears that the paper *Real forms of $U\_q(\mathfrak{g})$*, by Eric Twietmeyer, is the original reference. According to that paper, it is $\mathfrak{sp}\_{2n}$ that has real forms for $q \in i \mathbb{R}$.


Many thanks to Uwe Franz for digging up that reference!



Real forms of Drinfeld-Jimbo quantum groups
Two references I recall are


E. Twietmeyer, Real forms of Uq (g), Lett. Math. Phys. 24, 49-58, 1992.


V. Lyubashenko, Real and imaginary forms of quantum groups, Lecture Notes in Math. 1510, 1992, pp 67-78.


