arXiv subjects

Alessandro Carotenuto

Publications and source records attributed to Alessandro Carotenuto.

9 recordsLinked to original sources

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.

math.QA↗

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.

math.QA↗

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.

math.QA↗

arXiv2021

The Weyl-Mellin quantization map for $κ$-Minkowski Noncommutative Spacetime

We present a quantization of the functions of spacetime, i.e.\ a map, analog to Weyl map, which reproduces the $κ$-Minkowski commutation relations, and it has the desirable properties of mapping square integrable funcions into Hilbert-Schmidt operators, as well as real functions into self-adjoint operators. The map is based on Mellin transform on radial and time coordinates. The map also define a deformed $*$ product which we discuss with examples.

hep-th↗

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.

math.QA↗

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 $\mathcal{O}_q(G/L^{\mathrm{s}}_S)$.

math.QA↗

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.

math.QA↗

arXiv2018

Spin geometry of the rational noncommutative torus

The twined almost commutative structure of the standard spectral triple on the noncommutative torus with rational parameter is exhibited, by showing isomorphisms with a spectral triple on the algebra of sections of certain bundle of algebras, and a spectral triple on a certain invariant subalgebra of the product algebra. These isomorphisms intertwine also the grading and real structure. This holds for all four inequivalent spin structures, which are explicitly constructed in terms of double coverings of the noncommutative torus (with arbitrary real parameter). These results are extended also to a class of curved (non flat) spectral triples, obtained as a perturbation of the standard one by eight central elements.

math.QA↗

arXiv2018

Differential calculus on Jordan algebra and Jordan modules

Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra and provide a complete characterization of the theory of connections for free Jordan modules.

math.QA↗