Homogeneous right coideal subalgebras of quantized enveloping algebras

TL;DR

This paper classifies all homogeneous right coideal subalgebras of quantized enveloping algebras for complex semisimple Lie algebras, using Weyl group elements and root subsets, extending previous classifications.

Contribution

It provides a complete classification of these subalgebras via triples involving Weyl group elements and root subsets, building on prior work with Lusztig automorphisms.

Findings

Classification is determined by triples of Weyl group elements and root subsets.

Each subalgebra is uniquely identified by these triples.

The classification extends previous results on Borel subalgebras.

Abstract

For a quantized enveloping algebra of a complex semisimple Lie algebra with deformation parameter not a root of unity, we classify all homogeneous right coideal subalgebras. Any such right coideal subalgebra is determined uniquely by a triple consisting of two elements of the Weyl group and a subset of the set of simple roots satisfying some natural conditions. The essential ingredients of the proof are the Lusztig automorphisms and the classification of homogeneous right coideal subalgebras of the Borel Hopf subalgebras of quantized enveloping algebras obtained previously by H.-J. Schneider and the first named author. Key words: Quantum groups, coideal subalgebras, Weyl group, weak order