# Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid

## TL;DR

This paper explores the structure of graded right coideal subalgebras in Nichols algebras of semisimple Yetter-Drinfeld modules, establishing a correspondence with the Weyl groupoid's morphisms and proving a conjecture in the finite case.

## Contribution

It constructs an order-preserving bijection between morphisms of the Weyl groupoid and right coideal subalgebras, linking algebraic and combinatorial structures in Nichols algebras.

## Key findings

- Established a bijective correspondence for finite Weyl groupoids.
- Proved the Nichols algebra is decomposable under certain conditions.
- Confirmed a conjecture of Kharchenko for the Borel part of quantized enveloping algebras.

## Abstract

We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the above map is bijective. In the special case of the Borel part of quantized enveloping algebras our result implies a conjecture of Kharchenko. Key words: Hopf algebra, quantum group, root system, Weyl group