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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Realisation of Lusztig cones
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by Philippe Caldero, Bethany Marsh and Sophie Morier-Genoud PDF
Represent. Theory 8 (2004), 458-478

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
  • Bethany Marsh
  • Affiliation: Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England
  • MR Author ID: 614298
  • ORCID: 0000-0002-4268-8937
  • Sophie Morier-Genoud
  • Affiliation: Département de Mathématiques, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France
  • Email: morier@igd.univ-lyon1.fr
  • Received by editor(s): December 19, 2003
  • Received by editor(s) in revised form: June 21, 2004
  • Published electronically: September 27, 2004
  • Additional Notes: Bethany Marsh was supported by a Leverhulme Fellowship
  • © Copyright 2004 P. Caldero, B.R. Marsh and S. Morier-Genoud
  • Journal: Represent. Theory 8 (2004), 458-478
  • MSC (2000): Primary 17B37
  • DOI: https://doi.org/10.1090/S1088-4165-04-00225-0
  • MathSciNet review: 2110356