arXiv subjects

William J. Ugalde

Publications and source records attributed to William J. Ugalde.

7 recordsLinked to original sources

arXiv2026

Symmetry of some noncommutative sphere algebras

Two known $q$-deformed (or `quantum') $7$-spheres, both denoted $\mathbb{S}^7_q$ in the literature, may be distinguished by the presence or absence of symmetry under $\mathrm{SU}_q(2)$. The quaternionic version of $\mathbb{S}^7_q$ has been shown by Brain and Landi to support such a symmetry. Here we show that this is not the case for the older $\mathbb{S}^7_q$ introduced by Vaksman and Soibelman: and as a consequence, these quantum $7$-spheres are not isomorphic.

math.QA↗

arXiv2024

Characterization of principal bundles: the noncommutative algebraic case

We review Hopf-Galois extensions, in particular faithfully flat ones, accepted to be the noncommutative algebraic dual of a principal bundle. We also make a short digression into how quantum groups relate to Hopf-Galois extensions. Several examples are given, in order to provide a satisfactory understanding of each topic.

math.QA↗

arXiv2023

Characterization of principal bundles: the commutative case

A review of the characterization of principal bundles, through the different properties of the action of a group and its related canonical and translation maps, is presented. The work is divided in three stages: a topological group acting on a topological space, a discrete group acting on a smooth manifold, and a Lie group acting on a smooth manifold.

math.GM↗

arXiv2007

Some Conformal Invariants from the Noncommutative Residue for Manifolds with Boundary

We review previous work of Alain Connes, and its extension by the author, on some conformal invariants obtained from the noncommutative residue on even dimensional compact manifolds without boundary. Inspired by recent work of Yong Wang, we also address possible generalizations of these conformal invariants to the setting of compact manifolds with boundary.

math.DG↗

arXiv2005

A construction of critical GJMS operators using Wodzicki's residue

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical {\sf GJMS} operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue of a pseudo-differential operator of order $-2,$ originally defined by A. Connes, acting on middle dimension forms.

math.DG↗

arXiv2003

Differential forms canonically associated to even-dimensional compact conformal manifolds

On a 6-dimensional, conformal, oriented, compact manifold $M$ without boundary, we compute a whole family of differential forms $Ω_6(f,h)$ of order 6, with $f,h \in C^\infty(M).$ Each of these forms will be symmetric on $f,$ and $h,$ conformally invariant, and such that $\int_M f_0 Ω_6(f_1,f_2)$ defines a Hochschild 2-cocycle over the algebra $C^\infty(M).$ In the particular 6-dimensional conformally flat case, we compute the unique one satisfying $\Wres(f_0[F,f][F,h]) = \int_M f_0Ω_6(f,h)$ for $(\cH,F)$ the Fredholm module associated by A. Connes \cite{Con1} to the manifold $M,$ and $\Wres$ the Wodzicki residue.

math.DG↗

arXiv2003

Differential forms and the Wodzicki residue

For a pseudodifferential operator $S$ of order 0 acting on sections of a vector bundle $B$ on a compact manifold $M$ without boundary, we associate a differential form of order dimension of $M$ acting on $C^\infty(M)\times C^\infty(M)$. This differential form $Ω_{n,S}$ is given in terms of the Wodzicki 1-density $\wres([S,f][S,h])$. In the particular case of an even dimensional, compact, conformal manifold without boundary, we study this differential form for the case $(B,S)=(\cH,F)$, that is, the Fredholm module associated by A. Connes to the manifold $M.$ We give its explicit expression in the flat case and then we address the general case.

math.DG↗