caa a22 4500 467910685 CHVBK 20180406152848.0 cr unu---uuuuu 170328e20060701xx s 000 0 eng 10.1007/s11232-006-0093-6 doi (NATIONALLICENCE)springer-10.1007/s11232-006-0093-6 Lyakhovsky V. St. Petersburg State University, St. Petersburg, Russia aut Quantum duality in quantum deformations [Elektronische Daten] [V. Lyakhovsky] In accordance with the quantum duality principle, the twisted algebra $$U_\mathcal{F} (\mathfrak{g})$$ is equivalent to the quantum group $$Fun_{def} (\mathfrak{G}^\# )$$ 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 howthe transformation $$\mathfrak{g} \to \mathfrak{g}^\# $$ 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 $$Fun_{def} (\mathfrak{G}^\# )$$ . The transformation $$\mathfrak{g} \to \mathfrak{g}^\# $$ can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations $$U_{\varepsilon \mathcal{J}} (\mathfrak{s}\mathfrak{l}(3))$$ and study it in terms of $$\mathcal{S}\mathcal{L}(3)^\# $$ ; we find new realizations of the parabolic twist. Springer Science+Business Media, Inc., 2006 Lie—Poisson structures nationallicence quantum deformations of symmetry nationallicence quantum duality nationallicence Theoretical and Mathematical Physics Kluwer Academic Publishers-Consultants Bureau 148/1(2006-07-01), 968-979 0040-5779 148:1<968 2006 148 11232 https://doi.org/10.1007/s11232-006-0093-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11232-006-0093-6 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Lyakhovsky V. St. Petersburg State University, St. Petersburg, Russia aut NATIONALLICENCE 773 0- Theoretical and Mathematical Physics Kluwer Academic Publishers-Consultants Bureau 148/1(2006-07-01), 968-979 0040-5779 148:1<968 2006 148 11232 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer