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Réamonn Ó Buachalla

Publications and source records attributed to Réamonn Ó Buachalla.

23 records · Page 2Linked to original sources
arXiv2017

Noncommutative Kähler Structures on Quantum Homogeneous Spaces

Building on the theory of noncommutative complex structures, the notion of a noncommutative Kähler structure is introduced. In the quantum homogeneous space case many of the fundamental results of classical Kähler geometry are shown to follow from the existence of such a structure, allowing for the definition of noncommutative Lefschetz, Hodge, Kähler-Dirac, and Laplace operators. Quantum projective space, endowed with its Heckenberger-Kolb calculus, is taken as the motivating example. The general theory is then used to show that the calculus has cohomology groups of at least classical dimension.

math.QA↗
arXiv2015

Noncommutative Complex Structures on Quantum Homogeneous Spaces

A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully flat quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. Throughout, the family of quantum projective spaces, endowed with the Heckenberger--Kolb calculus, is taken as the motivating set of examples.

math.QA↗
arXiv2014

Noncommutative Kahler Geometry of the Standard Podles Sphere

Building on the now established presentation of the standard Podles sphere as an example of a noncommutative complex structure, we investigate how its classical Kahler geometry behaves under $q$-deformation. Discussed are noncommutative versions of Hodge decomposition, Lefschetz decomposition, the Kahler identities, and the refinement of de Rham cohomology by Dolbeault cohomology

math.QA↗
arXiv2012

Quantum Bundle Description of the Quantum Projective Spaces

We realise Heckenberger and Kolb's canonical calculus on quantum projective (n-1)-space as the restriction of a distinguished quotient of the standard bicovariant calculus for Cq[SUn]. We introduce a calculus on the quantum (2n-1)-sphere in the same way. With respect to these choices of calculi, we present quantum projective (N-1)-space as the base space of two different quantum principal bundles, one with total space Cq[SUn], and the other with total space Cq[S^(2n-1)]. We go on to give Cq[CP^n] the structure of a quantum framed manifold. More specifically, we describe the module of one-forms of Heckenberger and Kolb's calculus as an associated vector bundle to the principal bundle with total space Cq[SUn]. Finally, we construct strong connections for both bundles.

math.QA↗