Braided enveloping algebras associated to quantum parabolic subalgebras

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.