Braided enveloping algebras associated to quantum parabolic subalgebras

Details Braided enveloping algebras associated to quantum parabolic subalgebras Braided enveloping algebras associated to quantum parabolic subalgebras

2007-06-04

arxiv.org/abs/0706.0455v3

Communications in Algebra, 39 (2011), no. 10, 3491-3514

Mathematics

Quantum Algebra

21 pages. Corrected statement and proof of Proposition 3.3 with subsequent minor changes to the remainder section 3. Revised p

Scientific paper

10.1080/00927872.2010.498394

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.

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