# The Quantum Group ${\cal O}_q(SL(n))$, for $q>1$

For the quantum group ${\cal O}_q(SL(n))$, $q\in \mathbb{R}$, I have read, without a proof, that for $p>1$, there exists a $q\in (0,1)$ such that $${\cal O}_p(SL(n)) \simeq {\cal O}_q(SL(n)).$$ What is this isomorphism, i.e. how does it act specifically on the standard generators $x_{ij}$? Moreover, for the compact real form ${\cal O}_q(SU(n))$, does this isomorphism still hold?

• Is this just $q=p^{-1}$? – Theo Johnson-Freyd Apr 16 '16 at 14:19
• Probably, but what are the isomorphism? – Alesandro Levi Apr 16 '16 at 14:32

The isomorphism between $\mathcal{U}_q(\mathfrak{g})$ and $\mathcal{U}_{q^{-1}}(\mathfrak{g})$ of the form presented in Lemma 2.4.2 of the Neshveyev-Tuset book gives you the isomorphism of Hopf *-algebras $\mathcal{O}_q({\rm SU}(n))$ and $\mathcal{O}_{q^{-1}}({\rm SU}(n))$. The elements $x_{ij}$ are matrix coefficients of the defining representation, so the presentation on this side can be worked out too.