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Realisation of Lusztig cones


Authors: Philippe Caldero, Robert Marsh and Sophie Morier-Genoud
Journal: Represent. Theory 8 (2004), 458-478
MSC (2000): Primary 17B37
DOI: https://doi.org/10.1090/S1088-4165-04-00225-0
Published electronically: September 27, 2004
MathSciNet review: 2110356
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Abstract: Let $U_q(\mathfrak{g})$ be the quantised enveloping algebra associated to a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. The negative part $U^-$ of $U_q(\mathfrak{g})$ possesses a canonical basis $\mathcal{B}$ with favourable properties. Lusztig has associated a cone to a reduced expression $\mathbf{i}$ for the longest element $w_0$ in the Weyl group of $\mathfrak{g}$, with good properties with respect to monomial elements of $\mathcal{B}$. The first author has associated a subalgebra $A_{\mathbf{i}}$ of $U^-$, compatible with the dual basis $\mathcal{B}^*$, to each reduced expression $\mathbf{i}$. We show that, after a certain twisting, the string parametrisation of the adapted basis of this subalgebra coincides with the corresponding Lusztig cone. As an application, we give explicit expressions for the generators of the Lusztig cones.


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Additional Information

Philippe Caldero
Affiliation: Département de Mathématiques, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France
Email: caldero@igd.univ-lyon1.fr

Robert Marsh
Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England
Email: R.Marsh@mcs.le.ac.uk

Sophie Morier-Genoud
Affiliation: Département de Mathématiques, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France
Email: morier@igd.univ-lyon1.fr

DOI: https://doi.org/10.1090/S1088-4165-04-00225-0
Received by editor(s): December 19, 2003
Received by editor(s) in revised form: June 21, 2004
Published electronically: September 27, 2004
Additional Notes: Robert Marsh was supported by a Leverhulme Fellowship
Article copyright: © Copyright 2004 P. Caldero, R.J. Marsh and S. Morier-Genoud

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