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2 papers
Quantum duality in quantum deformations
V. D. Lyakhovsky Saint-Petersburg State University
Abstract:
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.
Keywords:
Lie–Poisson structures, quantum deformations of symmetry, quantum duality. Received: 30.10.2005
Revised: 24.11.2005
DOI:
10.4213/tmf2062
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