Quantum Duality Principle for quantum continuous Kac-Moody algebras

TL;DR

This paper proves a version of the Quantum Duality Principle for the quantized universal enveloping algebra of a continuous Kac-Moody algebra, establishing a duality between quantum groups and Poisson groups in both formal and polynomial settings.

Contribution

It extends the Quantum Duality Principle to continuous Kac-Moody algebras, connecting their quantizations with dual Poisson groups in new formal and polynomial frameworks.

Findings

QDP holds for formal quantizations of g_X

QDP holds for polynomial quantizations of g_X

Establishes duality between quantized algebras and Poisson groups

Abstract

For the quantized universal enveloping algebra U_h(g_X) associated with a continuous Kac-Moody algebra g_X as in [A. Appel, F. Sala, "Quantization of continuum Kac-Moody algebras", Pure Appl. Math. Q. 16 (2020), no. 3, 439-493], we prove that a suitable formulation of the Quantum Duality Principle holds true, both in a "formal" version - i.e., applying to the original definition of U_h(g_X) as a formal QUEA over the algebra of formal series in h - and in a "polynomial" one - i.e., for a suitable polynomial form of U_h(g_X) over the algebra of Laurent polynomials in q. In both cases, the QDP states that a suitable 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 .