Quantum orbits of R-matrix type

TL;DR

This paper studies special orbits in Lie algebra duals characterized by R-matrix structures, providing explicit descriptions and quantizations, and discusses related q-deformed algebraic notions.

Contribution

It explicitly describes R-matrix type orbits and constructs their quantizations, advancing understanding of quantum Lie algebra structures.

Findings

Explicit descriptions of R-matrix type orbits

Construction of quantizations for these orbits

Discussion of q-deformed Lie brackets and braided structures

Abstract

Given a simple Lie algebra $\gggg$, we consider the orbits in $\gggg^*$ which are of R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of R-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions of q-deformed Lie brackets, braided coadjoint vector fields and tangent vector fields are discussed as well.