4 problems

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody Lie algebra, let $W_{\mathfrak{g}}$ be its Weyl group, and let $w\in W_{\mathfrak{g}}$. Write ${\mathcal U}_q^\pm[w]$ for the quantu…

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $r>1$, let $J$ be a nonempty subset of simple roots, and let $w=w_o^Jw_o$ be the corresponding parabolic eleme…

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, let $U_q^-(\mathfrak{g})$ and $U_q^+(\mathfrak{g})$ be its negative and positive quantum enveloping algebras, and let $w$ b…

Conjecture on $A[\widehat{w}]$. The vectors $\Omega_1,\ldots,\Omega_{13}$ are highest weight vectors, and the monomials $\{\Omega_1^{r_1}\cdots\Omega_{13}^{r_{13}}\}$ satisfying…