Realisation of Lusztig cones

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.