Realisation of Lusztig cones

Details Realisation of Lusztig cones Realisation of Lusztig cones

2003-12-10

arxiv.org/abs/math/0312208v1

Mathematics

Representation Theory

17 pages, no figures

Scientific paper

The negative part $U^-$ of a quantised enveloping algebra associated to a simple Lie algebra 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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