Dual affine quantum groups
Identifieur interne : 001458 ( Main/Merge ); précédent : 001457; suivant : 001459Dual affine quantum groups
Auteurs : Fabio GavariniSource :
- Mathematische Zeitschrift [ 0025-5874 ] ; 2000-05-01.
English descriptors
Abstract
Abstract.: Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction – via formal Hopf algebras – we construct a new quantum group $U_q (\hat\mathfrak{h}})$ , dual of $U_q (\hat\mathfrak{g}})$ . Studying its specializations at roots of 1 (in particular, its semi-classical limits), we prove that it yields quantizations of $\hat{\mathfrak{g}}$ and $\widehat{G}^\infty$ (the formal Poisson group attached to $\hat{\mathfrak{g}}$ ), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.
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DOI: 10.1007/s002090050502
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<front><div type="abstract" xml:lang="en">Abstract.: Let $\hat{\mathfrak{g}}$ be an untwisted affine Kac-Moody algebra, with its Sklyanin-Drinfel'd structure of Lie bialgebra, and let $\hat{\mathfrak{h}}$ be the dual Lie bialgebra. By dualizing the quantum double construction – via formal Hopf algebras – we construct a new quantum group $U_q (\hat\mathfrak{h}})$ , dual of $U_q (\hat\mathfrak{g}})$ . Studying its specializations at roots of 1 (in particular, its semi-classical limits), we prove that it yields quantizations of $\hat{\mathfrak{g}}$ and $\widehat{G}^\infty$ (the formal Poisson group attached to $\hat{\mathfrak{g}}$ ), and we construct new quantum Frobenius morphisms. The whole picture extends to the untwisted affine case the results known for quantum groups of finite type.</div>
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