Braided enveloping algebras associated to quantum parabolic subalgebras

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.

21 pages. Corrected statement and proof of Proposition 3.3 with subsequent minor changes to the remainder section 3. Revised presentation in introduction and added concluding remarks

References in corpus (3)

• Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
• New Quantum Groups by Double-Bosonisation
• On Lie induction and the exceptional series

Cited by in corpus (2)

• Double-bosonization and Majid's conjecture, (I): rank-induction of $ABCD$
• Double-bosonization and Majid's Conjecture, (IV): Type-Crossings from $A$ to $BCD$