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Braided enveloping algebras associated to quantum parabolic subalgebras

Research output: Contribution to journalJournal article

Published

Journal publication date14/10/2011
JournalCommunications in Algebra
Journal number10
Volume39
Number of pages24
Pages3491-3514
Original languageEnglish

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.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Communications in Algebra, 39 (10), 2011, © Informa Plc