Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

TL;DR

This paper investigates how Etingof-Kazhdan quantization functors for Lie bialgebras behave under duality and doubling operations, establishing their compatibility and applications to affine Lie superalgebras.

Contribution

It proves that these quantization functors commute with duality and doubling operations, extending their applicability to affine Lie superalgebras.

Findings

Quantization functors are compatible with duality operations.

Quantization functors commute with taking doubles.

Etingof-Kazhdan quantizations match Drinfeld-Jimbo quantizations for certain superalgebras.

Abstract

We study the behavior of the Etingof-Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are "compatible with duality", i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof-Kazhdan quantization of some affine Lie superalgebras coincide with their Drinfeld-Jimbo-type quantizations.