arXiv subjects

Kwalombota Ilwale

Publications and source records attributed to Kwalombota Ilwale.

4 recordsLinked to original sources

arXiv2023

Symmetric $(σ,τ)$-algebras and $(σ,τ)$-Hochschild cohomology

On an associative algebra, we introduce the concept of symmetric $(σ,τ)$-derivations together with a regularity condition and prove that strongly regular symmetric $(σ,τ)$-derivations are inner. Symmetric $(σ,τ)$-derivations are $(σ,τ)$-derivations that are simultaneously $(σ,τ)$-derivations as well as $(τ,σ)$-derivations, generalizing a property of commutative algebras. Motivated by this notion, we explore the geometry of symmetric $(σ,τ)$-algebras and prove that there exist a unique strongly regular symmetric $(σ,τ)$-connection. Furthermore, we introduce $(σ,τ)$-Hochschild cohomology and show that, in first degree, it describes the outer $(σ,τ)$-derivations on an associative algebra. Along the way, examples are provided to illustrate the novel concepts.

math.QA↗

arXiv2022

On the geometry of $(σ,τ)$-algebras

We introduce $(σ,τ)$-algebras as a framework for twisted differential calculi over noncommutative, as well as commutative, algebras with motivations from the theory of $σ$-derivations and quantum groups. A $(σ,τ)$-algebra consists of an associative algebra together with a set of $(σ,τ)$-derivations, and corresponding notions of $(σ,τ)$-modules and connections are introduced. We prove that $(σ,τ)$-connections exist on projective modules, and introduce notions of both torsion and curvature, as well as compatibility with a hermitian form, leading to the definition of a Levi-Civita $(σ,τ)$-connection. To illustrate the novel concepts, we consider $(σ,τ)$-algebras and connections over matrix algebras in detail.

math.QA↗

arXiv2022

Levi-Civita connections on quantum spheres

We introduce $q$-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with $q$-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics. We also give metric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.

math.QA↗

arXiv2020

On q-deformed Levi-Civita connections

We explore the possibility of introducing q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. For the module of 1-forms on the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics satisfying a certain reality condition. Finally, we construct metric connections on a class of projective modules over the quantum 2-sphere.

math.QA↗