A Borel-Weil theorem for the irreducible quantum flag manifolds

2021/12/06 by Alessandro Carotenuto, Carotenuto, Alessandro, Fredy Díaz García +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2112.03305

openalex publication_date 2021/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a noncommutative generalisation of the Borel-Weil theorem for the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds O_q(G/L_S), generalising previous work of a number of authors (including the first and third authors of this paper) on the quantum Grassmannians O_q(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 O_q(G/L^s_S).

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