We establish a noncommutative generalisation of the Borel–Weil theorem for the Heckenberger–Kolb calculi of the irreducible quantum flag manifolds ${\mathcal {O}}_q(G/L_S)$, generalising previous work for the quantum Grassmannians ${\mathcal {O}}_q(\textrm {Gr}_{n,m})$. As a direct consequence we get a novel noncommutative differential geometric presentation of the quantum coordinate rings $S_q[G/L_S]$ of the irreducible quantum flag manifolds. The proof is formulated in terms of quantum principal bundles, and the recently introduced notion of a principal pair, and uses the Heckenberger and Kolb first-order differential calculus for the quantum Possion homogeneous spaces ${\mathcal {O}}_q(G/L^{\,\textrm {s}}_S)$.

中文翻译：

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不可约量子旗流形的 Borel-Weil 定理

我们为不可约量子标志流形 ${\mathcal {O}}_q(G/L_S)$ 的 Heckenberger-Kolb 演算建立了 Borel-Weil 定理的非交换推广，推广了量子格拉斯曼的先前工作 ${\mathcal {O}}_q(\textrm {Gr}_{n,m})$. 作为直接结果，我们得到了不可约量子标志流形的量子坐标环 $S_q[G/L_S]$ 的新的非交换微分几何表示。证明是根据量子主丛和最近引入的主对概念制定的，并使用 Heckenberger 和 Kolb 一阶微分计算量子 Possion 齐次空间 ${\mathcal {O}}_q(G/ L^{\,\textrm {s}}_S)$。