AN EXPLICIT ISOMORPHISM BETWEEN QUANTUM AND CLASSICAL $$ \mathfrak{s}{\mathfrak{l}}_n $$

Abstract

Let \( \mathfrak{g} \) be a complex semisimple Lie algebra. Although the quantum group \( {U}_{\hslash \mathfrak{g}} \) is known to be isomorphic, as an algebra, to the undeformed enveloping algebra \( U\mathfrak{g}\left\llbracket \hslash \right\rrbracket \), no such isomorphism is known when \( \mathfrak{g}\ne \mathfrak{s}{\mathfrak{l}}_2 \). In this paper, we construct an explicit isomorphism for \( \mathfrak{g}=\mathfrak{s}{\mathfrak{l}}_n \), for every n ≥ 2, which preserves the standard flag of type A. We conjecture that this isomorphism quantizes the Poisson diffeomorphism of Alekseev and Meinrenken [2].