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Dual affine quantum groups. (English) Zbl 1015.17014

Author’s summary: Let \(\widehat{\mathfrak g}\) be an untwisted affine Kac-Moody algebra, with the Sklyanin-Drinfel’d structure of a Lie bialgebra, and let \(\widehat{\mathfrak h}\) be the dual Lie bialgebra. By dualizing the quantum double construction – via formal Hopf algebras – we construct a new quantum group \(U_q(\widehat{{\mathfrak h}})\), the dual of \(U_q(\widehat{{\mathfrak g}})\). We introduce two integer forms of it and study their specializations at roots of 1 (in particular, their semiclassical limits). So we prove that \(U_q(\widehat{{\mathfrak h}})\) yields – via its integer forms – quantizations of \(\widehat{\mathfrak h}\) and of \(\widehat{G}^\infty\) (the formal group attached to \(\widehat{\mathfrak g}\)). In addition, we construct new quantum Frobenius morphisms, linking the specializations at roots of 1 with the semiclassical limit (= specialization at \(q = 1\)). The whole picture extends the results known for quantum groups of finite type to the untwisted affine case.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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