Braided enveloping algebras associated to quantum parabolic subalgebras

TL;DR

This paper generalizes the triangular decomposition of Lie algebras to their quantum counterparts, identifying a braided Hopf algebra that quantizes certain subalgebras and often forms a Nichols algebra, enriching the structure of quantum groups.

Contribution

It introduces a quantum analogue of the triangular decomposition for Lie algebras, constructing a graded braided Hopf algebra that generalizes previous work and often forms a Nichols algebra.

Findings

Identifies a graded braided Hopf algebra quantizing $ abla_{D}^{-}$

Shows this algebra often is a Nichols algebra

Generalizes classical triangular decompositions to quantum groups

Abstract

Associated to each subset $J$ of the nodes $I$ of a Dynkin diagram is a triangular decomposition of the corresponding Lie algebra $\mathfrak{g}$ into three subalgebras $\widetilde{\mathfrak{g}_{J}}$ (generated by $e_{j}$, $f_{j}$ for $j\in J$ and $h_{i}$ for $i\in I$), $\mathfrak{n}^{-}_{D}$ (generated by $f_{d}$, $d\in D=I\setminus J$) and its dual $\mathfrak{n}_{D}^{+}$. We demonstrate a quantum counterpart, generalising work of Majid and Rosso, by exhibiting analogous triangular decompositions of $U_{q}(\mathfrak{g})$ and identifying a graded braided Hopf algebra that quantizes $\mathfrak{n}_{D}^{-}$. This algebra has many similar properties to $U_{q}^{-}(\mathfrak{g})$, in many cases being a Nichols algebra and therefore completely determined by its associated braiding.