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Braided enveloping algebras associated to quantum parabolic subalgebras

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Braided enveloping algebras associated to quantum parabolic subalgebras. / Grabowski, Jan.
In: Communications in Algebra, Vol. 39, No. 10, 14.10.2011, p. 3491-3514.

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Grabowski J. Braided enveloping algebras associated to quantum parabolic subalgebras. Communications in Algebra. 2011 Oct 14;39(10):3491-3514. doi: 10.1080/00927872.2010.498394

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Grabowski, Jan. / Braided enveloping algebras associated to quantum parabolic subalgebras. In: Communications in Algebra. 2011 ; Vol. 39, No. 10. pp. 3491-3514.

Bibtex

@article{7d4de271ecae44a78ed4104cb8249908,
title = "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. ",
author = "Jan Grabowski",
note = "The final, definitive version of this article has been published in the Journal, Communications in Algebra, 39 (10), 2011, {\textcopyright} Informa Plc",
year = "2011",
month = oct,
day = "14",
doi = "10.1080/00927872.2010.498394",
language = "English",
volume = "39",
pages = "3491--3514",
journal = "Communications in Algebra",
issn = "0092-7872",
publisher = "Taylor and Francis Ltd.",
number = "10",

}

RIS

TY - JOUR

T1 - Braided enveloping algebras associated to quantum parabolic subalgebras

AU - Grabowski, Jan

N1 - The final, definitive version of this article has been published in the Journal, Communications in Algebra, 39 (10), 2011, © Informa Plc

PY - 2011/10/14

Y1 - 2011/10/14

N2 - 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.

AB - 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.

U2 - 10.1080/00927872.2010.498394

DO - 10.1080/00927872.2010.498394

M3 - Journal article

VL - 39

SP - 3491

EP - 3514

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 10

ER -