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Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds

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Abstract

We construct covariant q-deformed holomorphic structures for all finitely generated relative Hopf modules over the irreducible quantum flag manifolds endowed with their Heckenberger–Kolb calculi. In the classical limit, these reduce to modules of sections of holomorphic homogeneous vector bundles over irreducible flag manifolds. For the case of simple relative Hopf modules, we show that this covariant holomorphic structure is unique. This generalises earlier work of Majid, Khalkhali, Landi, and van Suijlekom for line modules of the Podleś sphere, and subsequent work of Khalkhali and Moatadelro for general quantum projective space.

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Authors and Affiliations

  1. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Antigua Carretera a Pátzcuaro 8701 Col. Ex Hacienda San José de la Huerta, C.P. 58089, Morelia, Mexico

    Fredy Díaz García

  2. Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    Andrey Krutov

  3. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Prague, Czech Republic

    Andrey Krutov & Karen R. Strung

  4. Independent University of Moscow, Bolshoj Vlasyevskij Per. 11, 119002, Moscow, Russia

    Andrey Krutov

  5. Département de Mathématiques, Faculté des sciences, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050, Bruxells, Belgium

    Réamonn Ó Buachalla

  6. Mathematical Institute of Charles University, Sokolovská 83, 186 75, Prague, Czech Republic

    Réamonn Ó Buachalla & Petr Somberg

Authors
  1. Fredy Díaz García
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  2. Andrey Krutov
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  3. Réamonn Ó Buachalla
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  4. Petr Somberg
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  5. Karen R. Strung
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Correspondence to Réamonn Ó Buachalla.

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FDG is partially funded by Conacyt (Consejo Nacional de Ciencia y Tecnología, México). AK was supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund — the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004). RÓB acknowledges FNRS support through a postdoctoral fellowship within the framework of the MIS Grant “Antipode” grant number F.4502.18. RÓB and PS are partially supported from the Eduard Čech Institute within the framework of the grant GAČR \(19-28628X\), and by the grant GA19-06357S. Research of KRS and AK is supported by the GAČR project 20-17488Y and RVO: 67985840.

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Díaz García, F., Krutov, A., Ó Buachalla, R. et al. Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds. Lett Math Phys 111, 10 (2021). https://doi.org/10.1007/s11005-020-01340-7

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  • DOI: https://doi.org/10.1007/s11005-020-01340-7

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