Quantum flag manifolds as quotients of degenerate quantized universal enveloping algebras

Abstract

Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the defining relations for $U_q(\mathfrak{g})$ can be slightly modified in such a way that the resulting algebra $U_q(\mathfrak{g};\mathcal{S})$ allows a homomorphism onto (an extension of) the algebra $\mathrm{Pol}(\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q})$ of functions on the quantum flag manifold $\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q}$ corresponding to $\mathcal{S}$. Moreover, this homomorphism is equivariant with respect to a natural adjoint action of $U_q(\mathfrak{g})$ on $U_q(\mathfrak{g};\mathcal{S})$ and the standard action of $U_q(\mathfrak{g})$ on $Pol(\mathbb{G}_q/\mathbb{K}_{\mathcal{S},q})$.