Feigin's quotient-ring conjecture for quantum nilpotent algebras

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra, let $w_0=r_{i_1}\cdots r_{i_N}$ be a reduced decomposition of the longest Weyl-group element, and define

$$\Phi(i_1,\ldots,i_N):U_q^-(\mathfrak{g})\longrightarrow \mathbb{C}[X_{1i_1}\ldots X_{Ni_N}]$$

using the iterated map constructed from the maps $\rho_i$ and the modified coproduct. The target is the skew-polynomial algebra with relations $X_{si_s}X_{ti_t}=q^{(\alpha_{i_s},\alpha_{i_t})}X_{ti_t}X_{si_s}$ for $s>t$.

Feigin's conjecture. The map $\Phi(i_1,\ldots,i_N)$ is an embedding, and, for at least one special reduced decomposition of $w_0$, it extends to an isomorphism

$$Q(U_q^-(\mathfrak{g}))\cong Q(\mathbb{C}[X_{1i_1}\ldots X_{Ni_N}]).$$

This conjecture connects quantum nilpotent enveloping algebras with skew-polynomial algebras and their quotient rings. The source treats the embedding and the quotient-ring isomorphism as the key structural input for the subsequent quantum Gel'fand–Kirillov discussion.