Noncommutative complex structures on quantum homogeneous spaces
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In this article, a framework for noncommutative complex geometry on quantum homogeneous spaces is introduced, based on covariant differential calculi and categorical equivalence for quantum projective spaces.About:
This article is published in Journal of Geometry and Physics.The article was published on 2016-01-01 and is currently open access. It has received 31 citations till now. The article focuses on the topics: Quantum topology & Quantum differential calculus.read more
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Noncommutative Kähler structures on quantum homogeneous spaces
TL;DR: In this article, the concept of noncommutative Kahler complex structures is introduced, which is based on the theory of non-commutativity of complex structures in the quantum projective space and the Heckenberger-Kolb calculus.
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On the Dolbeault–Dirac operator of quantized symmetric spaces
TL;DR: In this article, the authors define a spectral triple quantizing the Dolbeault Dirac operator associated to the canonical spin c structure in terms of the Koszul complex of a braided symmetric algebra.
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A Borel-Weil Theorem for the Quantum Grassmannians
TL;DR: In this paper, a noncommutative generalisation of the Borel-Weil theorem for the Heckenberger-Kolb calculus of the quantum Grassmannians is established.
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Kähler structures on quantum irreducible flag manifolds
TL;DR: In this article, it was shown that all quantum irreducible flag manifolds admit Kahler structures, as defined by O Buachalla, and that the differential calculi defined by Heckenberger and Kolb are differential ∗ -calculi in a natural way.
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Noncommutative K\"ahler Structures on Quantum Homogeneous Spaces
TL;DR: In this paper, the concept of noncommutative Kahler complex structures is introduced for the quantum projective space, endowed with its Heckenberger-Kolb calculus, and the general theory is used to show that the calculus has cohomology groups of at least classical dimension.
References
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Differential calculus on compact matrix pseudogroups (quantum groups)
TL;DR: In this paper, a general theory of non-commutative differential geometry on quantum groups is developed, where bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied.
Journal ArticleDOI
Equivariant $K$-theory
TL;DR: In this paper, the authors present a generalisation of K-theory to non-compact spaces, namely equivariant Ktheory on G-spaces, which is a generalization of the notion of vector-bundles.
Book
Complex Geometry: An Introduction
TL;DR: Local Theory and Applications of Cohomology: Complex Manifolds, Vector Bundles, and Deformations of Complex Structures as discussed by the authors Theoretically, complex manifolds are a type of complex structures.