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Réamonn Ó Buachalla

Publications and source records attributed to Réamonn Ó Buachalla.

At least 19 recordsLinked to original sources
arXiv2026

Compact Quantum Homogeneous Kähler Spaces

Noncommutative Kähler structures were recently introduced as an algebraic framework for studying noncommutative complex geometry on quantum homogeneous spaces. In this paper, we introduce the notion of a \emph{compact quantum homogeneous Kähler space} which gives a natural set of compatibility conditions between covariant Kähler structures and Woronowicz's theory of compact quantum groups. Each such object admits a Hilbert space completion possessing a remarkably rich yet tractable structure. The analytic behaviour of the associated Dolbeault-Dirac operators is moulded by the complex geometry of the underlying calculus. In particular, twisting the Dolbeault-Dirac operator by a negative Hermitian holomorphic module is shown to give a Fredholm operator if and only if the top anti-holomorphic cohomology group is finite-dimensional. In this case, the operator's index coincides with the twisted holomorphic Euler characteristic of the underlying noncommutative complex structure. The irreducible quantum flag manifolds, endowed with their Heckenberger-Kolb calculi, are presented as motivating examples.

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arXiv2025

Torsion-Free Bimodule Connections and the Maximal Prolongation of a First-Order Differential Calculus

We give an unexpectedly simple presentation of the maximal prolongation of a first-order differential calculus in terms of the bimodule map of a torsion-free bimodule connection. We then show that in the quantum homogeneous space case this simplifies even further. More explicitly, we show that the bimodule map associated to a bimodule connection, for any relative left Hopf module endowed with its canonical right module structure, admits a concise formula, given in terms of the adjont action of a Hopf algebra on a bimodule. %{\color{red} We also have the dual tangent space formula.} This is then used to derive sufficient conditions, in terms of the first-order differential forms, for the extendability of a first-order almost-complex structure. These results are applied to the quantum Grassmannian Heckenberger--Kolb calculi, yielding a simple uniform presentation of their degree two anti-holomorphic relations.

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arXiv2025

Lusztig's Quantum Root Vectors and a Dolbeault Complex for the A-Series Full Quantum Flag Manifolds

For the Drinfeld-Jimbo quantum enveloping algebra $U_q(\frak{sl}_{n+1})$, we show that the span of Lusztig's positive root vectors, with respect to Littlemann's nice reduced decompositions of the longest element of the Weyl group, form quantum tangent spaces for the full quantum flag manifold $\mathcal{O}_q(\mathrm{F}_{n+1})$. The associated differential calculi are direct $q$-deformations of the anti-holomorphic Dolbeault complex of the classical full flag manifold $\mathrm{F}_{n+1}$. As an application we establish a quantum Borel-Weil theorem for the $A_n$-series full quantum flag manifold, giving a noncommutative differential geometric realisation of all the finite-dimensional type-$1$ irreducible representations of $U_q(\frak{sl}_{n+1})$. Restricting this differential calculus to the quantum Grassmannians is shown to reproduce the celebrated Heckenberger-Kolb anti-holomorphic Dolbeault complex. Lusztig's positive root vectors for non-nice decompositions of the longest element of the Weyl group are examined for low orders, and are exhibited to either not give tangents spaces, or to produce differential calculi of non-classical dimension.

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arXiv2024

Noncommutative Complex Structures for the Full Quantum Flag Manifold of Quantum SU(3)

In recent work, Lusztig's positive root vectors (with respect to a distinguished choice of reduced decomposition of the longest element of the Weyl group) were shown to give a quantum tangent space for every $A$-series Drinfeld--Jimbo full quantum flag manifold $\mathcal{O}_q(\mathrm{F}_n)$. Moreover, the associated differential calculus $Ω^{(0,\bullet)}_q(\mathrm{F}_n)$ was shown to have classical dimension, giving a direct $q$-deformation of the classical anti-holomorphic Dolbeault complex of $\mathrm{F}_n$. Here we examine in detail the rank two case, namely the full quantum flag manifold of $\mathcal{O}_q(\mathrm{SU}_3)$. In particular, we examine the $*$-differential calculus associated to $Ω^{(0,\bullet)}_q(\mathrm{F}_3)$ and its non-commutative complex geometry. We find that the number of almost-complex structures reduces from $8$ (that is $2$ to the power of the number of positive roots of $\frak{sl}_3$) to $4$ (that is $2$ to the power of the number of simple roots of $\frak{sl}_3$). Moreover, we show that each of these almost-complex structures is integrable, which is to say, each of them is a complex structure. Finally, we observe that, due to non-centrality of all the non-degenerate coinvariant $2$-forms, none of these complex structures admits a left $\mathcal{O}_q(\mathrm{SU}_3)$-covariant noncommutative Kähler structure.

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arXiv2024

Levi-Civita connection on the irreducible quantum flag manifolds

We classify covariant metrics (in the sense of Beggs and Majid) on a class of quantum homogeneous spaces. In particular, our classification implies the existence of a unique (up to scalar) quantum symmetric covariant metric on the Heckenberger--Kolb calculi for the quantized irreducible flag manifolds. Moreover, we prove the existence and uniqueness of Levi-Civita connection for any real covariant metric for the Heckenberger--Kolb calculi. This generalizes Matassa's result for the quantum projective spaces.

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arXiv2024

A remark on the Hochschild dimension of liberated quantum groups

Let $A$ be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra $\mathcal{O}(G)$ of a semisimple algebraic group $G$. It is shown that if $A$ admits a suitably non-degenerate comodule $V$ and the induced $G$-module structure of $V$ is non-trivial, then the third Hochschild homology group of $A$ is non-trivial.

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arXiv2023

Nichols Algebras and Quantum Principal Bundles

We introduce a general framework for associating to a homogeneous quantum principal bundle a Yetter-Drinfeld module structure on the cotangent space of the base calculus. The holomorphic and anti-holomorphic Heckenberger-Kolb calculi of the quantum Grassmannians are then presented in this framework. This allows us to express the calculi in terms of the corresponding Nichols algebras. The extension of this result to all irreducible quantum flag manifolds is then conjectured.

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arXiv2022

Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds

It was recently shown (by the second author and Díaz García, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$ admits a unique $\mathcal{O}_q(G)$-covariant connection with respect to the Heckenberger-Kolb differential calculus $Ω^1_q(G/L_S)$. In this paper we show that these connections are bimodule connections with an invertible associated bimodule map. This is proved by applying general results of Beggs and Majid, on principal connections for quantum principal bundles, to the quantum principal bundle presentation of the Heckenberger-Kolb calculi recently constructed by the authors and Díaz García. Explicit presentations of the associated bimodule maps are given first in terms of generalised quantum determinants, then in terms of the FRT presentation of the algebra $\mathcal{O}_q(G)$, and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.

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arXiv2022

Positive Line Bundles Over the Irreducible Quantum Flag Manifolds

Noncommutative Kähler structures were recently introduced by the second author as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant Kähler structures of irreducible type (those having an irreducible space of holomorphic one-forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds $\mathcal{O}_q(G/L_S)$. Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over $\mathcal{O}_q(G/L_S)$ can be identified as positive, negative, or flat, and hence we can conclude that each Kähler structure is of Fano type. Moreover, it proves possible to extend the Borel-Weil theorem for $\mathcal{O}_q(G/L_S)$ to a direct noncommutative generalisation of the classical Bott-Borel-Weil theorem for positive line bundles.

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arXiv2022

Spectral gaps for twisted Dolbeault-Dirac operators over the irreducible quantum flag manifolds

We show that tensoring the Laplace and Dolbeault-Dirac operators of a Kähler structure (with closed integral) by a negative Hermitian holomorphic module, produces operators with spectral gaps around zero. The proof is based on the recently established Akizuki-Nakano identity of a noncommutative Kähler structure. This general framework is then applied to the Heckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it is shown that twisting their Dirac and Laplace operators by negative line bundles produces a spectral gap, for q sufficiently close to 1. The main technical challenge in applying the framework is to establish positivity of the quantum Fubini-Study metric of the quantum flag manifold. Importantly, combining positivity with the noncommutative hard Lefschetz theorem, it is additionally observed that the even degree de Rham cohomology groups of the Heckenberger-Kolb calculi do not vanish.

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arXiv2021

Principal Pairs of Quantum Homogeneous Spaces

We propose a simple but effective framework for producing examples of covariant faithfully flat (generalised) Hopf-Galois extensions from a nested pair of quantum homogeneous spaces. Our construction is modelled on the classical situation of a homogeneous fibration $G/N \to G/M$, for $G$ a group, and $N \subseteq M \subseteq G$ subgroups. Variations on Takeuchi's equivalence and Schneider's descent theorem are presented in this context. Quantum flag manifolds and their associated quantum Poisson homogeneous spaces are taken as motivating examples. Moreover, a large collection of noncommutative fibrations (in the spirit of Brzeziński and Szymański) are constructed.

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arXiv2021

A Borel-Weil theorem for the irreducible quantum flag manifolds

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 of a number of authors (including the first and third authors of this paper) on the quantum Grassmannians $\mathcal{O}_q(\mathrm{Gr}_{n,m})$. 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 $\mathcal{O}_q(G/L^{\mathrm{s}}_S)$.

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arXiv2021

A Dolbeault-Dirac Spectral Triple for the $B_2$-Irreducible Quantum Flag Manifold

The quantum version of the Bernstein-Gelfand-Gelfand resolution is used to construct a Dolbeault-Dirac operator on the anti-holomorphic forms of the Heckenberger-Kolb calculus for the $B_2$-irreducible quantum flag manifold. The spectrum and the multiplicities of the eigenvalues of the Dolbeault-Dirac operator are computed. It is shown that this construction yields an equivariant, even, $0^+$-summable spectral triple.

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arXiv2021

A Borel-Weil Theorem for the Quantum Grassmannians

We establish a noncommutative generalisation of the Borel-Weil theorem for the Heckenberger-Kolb calculi of the quantum Grassmannians. The result is formulated in the framework of quantum principal bundles and noncommutative complex structures, and generalises previous work of a number of authors on quantum projective space. As a direct consequence we get a novel noncommutative differential geometric presentation of the twisted Grassmannian coordinate ring studied in noncommutative projective geometry. A number of applications to the noncommutative Kähler geometry of the quantum Grassmannians are also given.

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arXiv2021

Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds

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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arXiv2020

A Dolbeault-Dirac Spectral Triple for Quantum Projective Space

The notion of a Kähler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kähler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes' axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kähler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.

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arXiv2018

A Kodaira Vanishing Theorem for Noncommutative Kahler Structures

Using the framework of noncommutative Kahler structures, we generalise to the noncommutative setting the celebrated vanishing theorem of Kodaira for positive line bundles. The result is established under the assumption that the associated Dirac-Dolbeault operator of the line bundle is diagonalisable, an assumption that is shown to always hold in the quantum homogeneous space case. The general theory is then applied to the covariant Kahler structure of the Heckenberger-Kolb calculus of the quantum Grassmannians allowing us to prove a direct q-deformation of the classical Grassmannian Bott-Borel-Weil theorem for positive line bundles.

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