Quantum duality in quantum deformations

In accordance with the quantum duality principle, the twisted algebra $U_{\mathcal F}(\mathfrak g)$ is equivalent to the quantum group $\mathrm{Fun}_{\mathrm{def}}( \mathfrak G^{\#})$ and has two preferred bases: one inherited from the universal enveloping algebra $U(\mathfrak g)$ and the other generated by coordinate functions of the dual Lie group $\mathfrak G^{\#}$. We show how the transformation $\mathfrak g\longrightarrow\mathfrak g^{\#}$ can be explicitly obtained for any simple Lie algebra and a factorable chain $\mathcal F$ of extended Jordanian twists. In the algebra $\mathfrak g^{\#}$, we introduce a natural vector grading $\Gamma(\mathfrak g^{\#})$, compatible with the adjoint representation of the algebra. Passing to the dual-group coordinates allows essentially simplifying the costructure of the deformed Hopf algebra $U_{\mathcal F}(\mathfrak g)$, considered as a quantum group $\mathrm{Fun}_{\mathrm{def}}(\mathfrak G^{\#})$. The transformation $\mathfrak g\longrightarrow\mathfrak g^{\#}$ can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations $U_{\mathcal{EJ}}\bigl(\mathfrak{sl}(3)\bigr)$ and study it in terms of $\mathcal{SL}(3)^{\#}$; we find new realizations of the parabolic twist.