arXiv subjects

Chiara Pagani

Publications and source records attributed to Chiara Pagani.

17 recordsLinked to original sources

arXiv2025

Push-forward of Hopf--Galois extensions: the non central case

We study the push-forward of Hopf--Galois extensions as the algebraic counterpart of the pullback of principal bundles. We apply the theory of twisted tensor product algebras to endow covariant extensions of modules along a map $\mathsf{F}$ with an algebra structure, under compatibility conditions between $\mathsf{F}$ and the twisting map. The push-forward of an $H$-Galois extension $B \subset A$ along a map $\mathsf{F} : B \to C$ is an $H$-Galois extension of $C$. The corresponding Ehresmann--Schauenburg algebroids are compared.

math.QA↗

arXiv2025

On characteristic classes of vector bundles over quantum spheres

We study the quantization of spaces whose K-theory in the classical limit is the ring of dual numbers $\mathbb{Z}[t]/(t^2)$. For a compact Hausdorff space we recall necessary and sufficient conditions for this to hold. For a compact quantum space, we give sufficient conditions that guarantee there is a morphism of abelian groups $K_0 \to \mathbb{Z}[t]/(t^2)$ compatible with the tensor product of bimodules. Applications include the standard Podleś sphere $S^2_q$ and a quantum $4$-sphere $S^4_q$ coming from quantum symplectic groups. For the latter, the K-theory is generated by the Euler class of the instanton bundle. We give explicit formulas for the projections of vector bundles on $S^4_q$ associated to the principal $SU_q(2)$-bundle $S^7_q \to S^4_q$ via irreducible corepresentations of $SU_q(2)$, and compute their characteristic classes.

math.QA↗

arXiv2024

Atiyah sequences of braided Lie algebras and their splittings

Associated with an equivariant noncommutative principal bundle we give an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. For the $SU(2)$-principal bundle over the sphere $S^{4}_θ$ an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra $so_θ(5,1)$ yields a five parameter family of splittings. On the principal $SO_θ(2n,\mathbb{R})$-bundle of orthonormal frames over the sphere $S^{2n}_θ$, the splitting of the sequence leads to the Levi-Civita connection for the `round' metric on the $S^{2n}_θ$. The corresponding Riemannian geometry of $S^{2n}_θ$ is worked out.

math.QA↗

arXiv2024

Reduction of Quantum Principal Bundles over non affine bases

In this paper we develop the theory of reduction of quantum principal bundles over projective bases. We show how the sheaf theoretic approach can be effectively applied to certain relevant examples as the Klein model for the projective spaces; in particular we study in the algebraic setting the reduction of the principal bundle $\mathrm{GL}(n) \to \mathrm{GL}(n)/P= \mathbf{P}^{n-1}(\mathbb{C})$ to the Levi subgroup $G_0$ inside the maximal parabolic subgroup $P$ of $\mathrm{GL}(n)$. We characterize reductions in the sheaf theoretic setting.

math.QA↗

arXiv2024

On the geometry of quantum spheres and hyperboloids

We study two classes of quantum spheres and hyperboloids which are $*$-quantum spaces for the quantum orthogonal group $\mathcal{O}(SO_q(3))$. We construct line bundles over the quantum homogeneous space of invariant elements for the quantum subgroup $SO(2)$ of $SO_q(3)$. These are associated to the quantum principal bundle via corepresentations of $SO(2)$ and are given by finitely-generated projective modules $\mathcal{E}_n$ of rank $1$ and even degree $-2n$. The corresponding idempotents, representing classes in K-theory, are explicitly worked out. For $q$ real, we diagonalise the Casimir operator of the Hopf algebra ${\mathcal{U}_{q^{1/2}}(sl_2)}$ dual to $\mathcal{O}(SO_q(3))$.

math.QA↗

arXiv2022

Braided Hopf algebras and gauge transformations II: $*$-structures and examples

We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible $*$-structures, encompassing the quasitriangular case.

math.QA↗

arXiv2022

Braided Hopf algebras and gauge transformations

We study infinitesimal gauge transformations of an equivariant noncommutative principal bundle as a braided Lie algebra of derivations. For this, we analyse general $K$-braided Hopf and Lie algebras, for $K$ a (quasi)triangular Hopf algebra of symmetries, and study their representations as braided derivations. We then study Drinfeld twist deformations of braided Hopf algebras and of Lie algebras of infinitesimal gauge transformations. We give examples coming from deformations of abelian and Jordanian type. In particular we explicitly describe the braided Lie algebra of gauge transformations of the instanton bundle and of the orthogonal bundle on the quantum sphere $S^4_θ$.

math.QA↗

arXiv2020

The Gauge Group of a Noncommutative Principal Bundle and Twist Deformations

We study noncommutative principal bundles (Hopf-Galois extensions) in the context of coquasitriangular Hopf algebras and their monoidal category of comodule algebras. When the total space is quasi-commutative, and thus the base space subalgebra is central, we define the gauge group as the group of vertical automorphisms or equivalently as the group of equivariant algebra maps. We study Drinfeld twist (2-cocycle) deformations of Hopf-Galois extensions and show that the gauge group of the twisted extension is isomorphic to the gauge group of the initial extension. In particular, noncommutative principal bundles arising via twist deformation of commutative principal bundles have classical gauge group. We illustrate the theory with a few examples.

math.QA↗

arXiv2017

A class of differential quadratic algebras and their symmetries

We study a multi-parametric family of quadratic algebras in four generators, which includes coordinate algebras of noncommutative four-planes and, as quotient algebras, noncommutative three spheres. Particular subfamilies comprise Sklyanin algebras and Connes--Dubois-Violette planes. We determine quantum groups of symmetries for the general algebras and construct finite-dimensional covariant differential calculi.

math.QA↗

arXiv2016

Noncommutative principal bundles through twist deformation

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.

math.QA↗

arXiv2015

A 4-sphere with non central radius and its instanton sheaf

We build an SU(2)-Hopf bundle over a quantum toric four-sphere whose radius is non central. The construction is carried out using local methods in terms of sheaves of Hopf-Galois extensions. The associated instanton bundle is presented and endowed with a connection with anti-selfdual curvature.

math.QA↗

arXiv2012

Quantized Matrix Algebras and Quantum seeds

We determine explicit quantum seeds for classes of quantized matrix algebras. Furthermore, we obtain results on centers and block diagonal forms {of these algebras.} In the case where $q$ is {an arbitrary} root of unity, this further determines the degrees.

math.QA↗

arXiv2011

Deformation of tensor product (co)algebras via non-(co)normal twists

We study new coalgebra structures on the tensor product of two coalgebras $C$ and $D$ by twisting the tensor product coalgebra via a twist map $Ψ: C \otimes D \rightarrow D \otimes C$. We deal with the general case in which the counit of the tensor product coalgebra is deformed as well. Some classes of such deformations are analyzed and a notion of equivalence of twists is discussed. We also present the dual deformation of tensor product algebras and provide examples.

math.RA↗

arXiv2010

The quantum Cartan algebra associated to a bicovariant differential calculus

We associate to any (suitable) bicovariant differential calculus on a quantum group a Cartan Hopf algebra which has a left, respectively right, representation in terms of left, respectively right, Cartan calculus operators. The example of the Hopf algebra associated to the $4D_+$ differential calculus on $SU_q(2)$ is described.

math.QA↗

arXiv2008

Noncommutative families of instantons

We construct $θ$-deformations of the classical groups SL(2,H) and Sp(2). Coacting on the basic instanton on a noncommutative four-sphere $S^4_θ$, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the quantum quotient of $SL_θ(2,H)$ by $Sp_θ(2)$.

math.QA↗

arXiv2005

A Hopf bundle over a quantum four-sphere from the symplectic group

We construct a quantum version of the SU(2) Hopf bundle $S^7 \to S^4$. The quantum sphere $S^7_q$ arises from the symplectic group $Sp_q(2)$ and a quantum 4-sphere $S^4_q$ is obtained via a suitable self-adjoint idempotent $p$ whose entries generate the algebra $A(S^4_q)$ of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere $S^4$. We compute the fundamental $K$-homology class of $S^4_q$ and pair it with the class of $p$ in the $K$-theory getting the value -1 for the topological charge. There is a right coaction of $SU_q(2)$ on $S^7_q$ such that the algebra $A(S^7_q)$ is a non trivial quantum principal bundle over $A(S^4_q)$ with structure quantum group $A(SU_q(2))$.

math.QA↗

arXiv2001

Finite group discretization of Yang-Mills and Einstein actions

Discrete versions of the Yang-Mills and Einstein actions are proposed for any finite group. These actions are invariant respectively under local gauge transformations, and under the analogues of Lorentz and general coordinate transformations. The case Z_n \times Z_n \times...\times Z_n is treated in some detail, recovering the Wilson action for Yang-Mills theories, and a new discretized action for gravity.

hep-th↗