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DEGREE CONES AND MONOMIAL BASES OF LIE ALGEBRAS AND QUANTUM GROUPS

Published online by Cambridge University Press:  20 March 2017

TEODOR BACKHAUS
Affiliation:
Mathematisches Institut, Universität zu Köln, Cologne, North Rhine-Westphalia, Germany e-mails: tbackha@math.uni-koeln.de, xinfang.math@gmail.com
XIN FANG
Affiliation:
Mathematisches Institut, Universität zu Köln, Cologne, North Rhine-Westphalia, Germany e-mails: tbackha@math.uni-koeln.de, xinfang.math@gmail.com
GHISLAIN FOURIER
Affiliation:
School of Mathematics and Statistics, University of Glasgow, United Kingdom Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Hannover, Lower Saxony, Germany e-mail: fourier@math.uni-hannover.de
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Abstract

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We provide ℕ-filtrations on the negative part Uq($\mathfrak{n}$) of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an ℕ-filtration on any finite-dimensional simple $\mathfrak{g}$-module. We prove for type An, Cn, B3, D4 and G2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

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