arXiv subjects

Stephen Kwok

Publications and source records attributed to Stephen Kwok.

5 recordsLinked to original sources

arXiv2020

Local Forms of Morphisms of Colored Supermanifolds

In \cite{Covolo:2016}, \cite{Covolo:2012} and \cite{Poncin:2016}, we introduced the category of colored supermanifolds ($\mathbb{Z}_2^n$-super\-ma\-ni\-folds or just $\mathbb{Z}_2^n$-manifolds ($\mathbb{Z}_2^n=\mathbb{Z}_2\times\ldots\times\mathbb{Z}_2$ ($n$ times))), explicitly described the corresponding $\mathbb{Z}_2^n$-Berezinian and gave first insights into $\mathbb{Z}_2^n$-integration theory. The present paper contains a detailed account of parts of the $\mathbb{Z}_2^n$-differential calculus and of the $\mathbb{Z}_2^n$-variants of the trilogy of local theorems, which consists of the inverse function theorem, the implicit function theorem and the constant rank theorem.

math.DG↗

arXiv2016

Differential calculus on $\mathbb{Z}^n_2$-supermanifolds

The concept of $\Zn$-supermanifold has been recently proposed as a natural generalization of classical ($\Zs$-graded) supergeometry, allowing for more complicated commutativity constraints. Here we continue the study of $\Zn$-supergeometry by developing the foundations of differential calculus on $\Zn$-supermanifolds.

math.DG↗

arXiv2016

The Frobenius theorem for $\mathbb{Z}^n_2$-supermanifolds

We continue the development of $\mathbb{Z}^n_2$-supergeometry, a natural generalization of classical ($\mathbb{Z}_2$-graded) supergeometry, by proving the Frobenius theorem for integrable distributions on differentiable $\mathbb{Z}^n_2$-supermanifolds. Both the local and global versions of the theorem are addressed.

math.DG↗

arXiv2014

The geometry of $Π$-invertible sheaves

Using the fact that $Π$-invertible sheaves can be interpreted as locally free sheaves of modules for the super skew field $\mathbb{D}$, we give a new construction of the $Π$-projective superspace $\mathbb{P}^n_{Π, B}$ over affine $k$ superschemes $B$, $k$ an algebraically closed field. We characterize morphisms into $\mathbb{P}^n_{Π, B}$ and give a new interpretation of the composition of $Π$-invertible sheaves in terms of the algebra of $\mathbb{D}$.

math.AG↗

arXiv2013

Super Morita Theory

We develop the basics of Morita theory for super rings. As an application, we produce a more explicit super Morita equivalence in the case of super Azumaya algebras.

math.RA↗