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Journal of Lie Theory 32 (2022), No. 3, 839--862
Copyright Heldermann Verlag 2022

Quantum Duality Principle for Quantum Continuous Kac-Moody Algebras

Fabio Gavarini
Department of Mathematics, University of Rome "Tor Vergata", Rome, Italy


\newcommand{\uhgx}{U_\hbar(\,\g_X)} \newcommand{\kh}{{\Bbbk[[\hbar]]}} \newcommand{\kqqm}{{\Bbbk\big[\,q\,,q^{-1}\big]}} \newcommand {\g}{\mathfrak{g}} For the quantized universal enveloping algebra $\uhgx$ associated with a continuous Kac-Moody algebra $\g_X$ as in [A.\ Appel, F.\ Sala, \emph{Quantization of continuum {K}ac-{M}oody algebras}, Pure Appl.\ Math.\ Q.\ \textbf{16} (2020), 439--493], we prove that a suitable formulation of the \textsl{Quantum Duality Principle\/} holds true, both in a ``formal'' version -- i.e., applying to the original definition of $\uhgx$ as a \textsl{formal\/} QUEA over $\kh$ -- and in a ``polynomial'' one -- i.e., for a suitable polynomial form of $\uhgx$ over $\kqqm$. In both cases, the QDP states that a suitable Hopf subalgebra of the given quantization of the Lie bialgebra $\g_X$ is in fact a suitable quantization (in formal or in polynomial sense) of a connected Poisson group $G_X^*$ dual to $\g_X$.

Keywords: Continuous Kac-Moody algebras, continuous quantum groups, quantization of Lie bialgebras, quantization of Poisson groups.

MSC: 17B37, 20G42; 17B65, 17B62.

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