Realisation of Lusztig cones

2003/12/10 by Philippe Caldero, Caldero, Philippe, Bethany Marsh +3
Mathematics · #17B37 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:17B37

paper · pdf · doi:10.48550/arxiv.math/0312208

17 pages, no figures

arxiv created 2003/12/10 · openalex publication_date 2003/12/10 · arxiv updated 2020/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The negative part U^- of a quantised enveloping algebra associated to a simple Lie algebra possesses a canonical basis B with favourable properties. Lusztig has associated a cone to a reduced expression i for the longest element w₀ in the Weyl group of \mathfrakg, with good properties with respect to monomial elements of B. The first author has associated a subalgebra A_i of U^-, compatible with the dual basis B^*, to each reduced expression 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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