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 $U_q(\mathfrak{g})$?

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!


1 Answer 1


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.

  • $\begingroup$ Thanks, Uwe! The paper by Twietmeyer was exactly what I was looking for. $\endgroup$
    – MTS
    Feb 26, 2013 at 18:03

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