Quantum Duality Principle for Quantum Continuous Kac-Moody Algebras

摘要

For the quantized universal enveloping algebra U-(h) over bar(gx) associated with a continuous Kac-Moody algebra gx as in A. Appel, F. Sala, Quantization of continuum Kac-Moody algebras, Pure Appl. Math. Q. 16 (2020), 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) over bar(gx) as a formal QUEA over k(h) over bar - and in a "polynomial" one - i.e., for a suitable polynomial form of U-(h) over bar(gx) over kq, q(-1) . 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 gx.