On the construction of $ \mathbb Z^n_2- $grassmannians as homogeneous $ \mathbb Z^n_2- $spaces

<abstract><p>In this paper, we construct the $ \mathbb Z^n_2- $grassmannians by gluing of the $ \mathbb Z^n_2- $domains and give an explicit description of the action of the $ \mathbb Z^n_2- $Lie group $ GL(\overrightarrow{\textbf{m}}) $ on the $ \mathbb Z^n_2- $grassmannian $ G_{ \overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}}) $ in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the $ \mathbb Z^n_2- $grassmannian.</p></abstract>

On the construction of supergrassmannians as homogeneous superspaces

In this paper, we give an explicit description of the action of the super Lie group [Formula: see text] on supergrassmannian [Formula: see text] in the functor of points language. In particular, we give a concrete proof of the transitively of this action, and the gluing of the local charts of the supergrassmannian.

Klein and conformal superspaces, split algebras and spinor orbits

We discuss [Formula: see text] Klein and Klein-conformal superspaces in [Formula: see text] space-time dimensions, realizing them in terms of their functor of points over the split composition algebra [Formula: see text]. We exploit the observation that certain split forms of orthogonal groups can be realized in terms of matrix groups over split composition algebras. This leads to a natural interpretation of the sections of the spinor bundle in the critical split dimensions [Formula: see text] and [Formula: see text] as [Formula: see text], [Formula: see text] and [Formula: see text], respectively. Within this approach, we also analyze the non-trivial spinor orbit stratification that is relevant in our construction since it affects the Klein-conformal superspace structure.

A COMPARISON OF THE FUNCTORS OF POINTS OF SUPERMANIFOLDS

We study the functor of points and different local functors of points for smooth and holomorphic supermanifolds, providing characterization theorems and fully discussing the representability issues. In the end we examine applications to differential calculus including the transitivity theorems.

ON ALGEBRAIC SUPERGROUPS AND QUANTUM DEFORMATIONS

We give the definitions of affine algebraic supervariety and affine algebraic supergroup through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroups GL(m|n) and SL(m|n) and give explicitly the Hopf algebra structure of the algebra representing the functors of points. In the end we also give the quantization of GL(m|n) together with its coaction on suitable quantum spaces according to Manin's philosophy.

On the notion of compactness in supergeometry

We discuss the problem of finding an analogue of the concept of topological space in supergeometry, motivated by the search for a procedure to compactify supermanifolds along odd coordinates. In particular, we examine the topologies arising naturally on the sets of points of locally ringed superspaces, and show that in the presence of a nontrivial odd sector such topologies are never compact. The main outcome of our discussion is the following new observation: not only the usual framework of supergeometry (the theory of locally ringed spaces), but the more general approach of the functor of points, need to be further enlarged.