On the Construction of $\mathbb{Z}_2^n-$Supergrassmannians as Homogeneous $\mathbb{Z}_2^n-$Superspaces

TL;DR

This paper constructs $ $-supergrassmannians as homogeneous $ $-superspaces by gluing superdomains, describing the action of the super Lie group $GL( extbf{m})$, and proving transitivity and local chart gluing.

Contribution

It provides an explicit construction of $ $-supergrassmannians as homogeneous superspaces with detailed group action descriptions.

Findings

Constructed $ $-supergrassmannians via superdomain gluing.

Described the $GL( extbf{m})$ action explicitly in functor of points.

Proved the transitivity of the group action and local chart gluing.

Abstract

In this paper, we construct the $\mathbb Z_{2}^{n}-$supergrassmannians by gluing of the $\mathbb Z_{2}^{n}-$superdomains and give an explicit description of the action of the $\mathbb Z_{2}^{n}-$super Lie group $GL(\overrightarrow{\textbf{m}})$ on the $\mathbb Z_{2}^{n}-$supergrassmannian $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ 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 supergrassmannian.

Equations

$$⟨.,.⟩$$

$$⟨a,b⟩$$

$$(0,0,0)<(0,1,1)<(1,0,1)<(1,1,0)<(0,0,1)<(0,1,0)<(1,0,0)<(1,1,1)$$

$${\gamma }_0<{\gamma }_1<\dots <{\gamma }_{\text{q}}.$$

$$a.b=(-1{)}^{⟨\tilde{a},\tilde{b}⟩}b.a$$

$${\xi }_i{\xi }_j=(-1{)}^{⟨{\gamma }_i,{\gamma }_j⟩}{\xi }_j{\xi }_i.$$

$$\mathbb{R}^{\overrightarrow{\textbf{m}}}:=\Big{(}\mathbb{R}^{m_{0}},C^{\infty}_{\mathbb{R}^{m_{0}}}(-)[[\xi_{1}^{1},\ldots,\xi_{1}^{m_{1}},\xi_{2}^{1},\ldots,\xi_{2}^{m_{2}},\ldots,\xi_{3}^{1},\ldots,\xi_{\textbf{q}}^{m_{\textbf{q}}}]]\Big{)}$$

$$C_{{\mathbb{R}}^{m_0}}^{\infty }(U)[[{\xi }_1^1,\dots ,{\xi }_1^{m_1},{\xi }_2^1,\dots ,{\xi }_2^{m_2},\dots ,{\xi }_3^1,\dots ,{\xi }_{\text{q}}^{m_{\text{q}}}]],$$

$${\xi }_i^j{\xi }_k^l=(-1{)}^{⟨{\gamma }_i,{\gamma }_k⟩}{\xi }_k^l{\xi }_i^j.$$

$$E\stackrel{ϵ}{\to }X\stackrel{\alpha }{\underset{\beta }{\rightrightarrows }}Y,$$

$$\begin{array}{c}X(.):\mathcal{C}\to \text{SET}\\ S\mapsto X(S),\\ \\ X(.):Ho{m}_{\mathcal{C}}(S,T)\to Ho{m}_{\text{SET}}(X(T),X(S))\\ \phi \mapsto X(\phi ),\end{array}$$

$$\begin{array}{c}\mathcal{Y}:\mathcal{C}\to [\mathcal{C},\text{SET}]\\ X\mapsto X(.)\\ \psi \mapsto \psi (.).\end{array}$$

$$Ho{m}_{\mathcal{C}}(X,Y)⟶Ho{m}_{[\mathcal{C},\text{SET}]}(X(.),Y(.)),$$

$$\begin{array}{cc}{\overline{j}}_p:\{0\}\to \overline{M},& {j_p}^{\ast }:{\mathcal{O}}_M\to \mathbb{R}\\ & g\mapsto \tilde{g}(p)=:e{v}_p(g).\end{array}$$

$${\hat{p}}_{{}_{{}_T}}:T\to {\mathbb{R}}^{\overrightarrow{\text{0}}}\stackrel{j_p}{\to }M,$$

$$\mu \circ (\mu \times 1_G)$$

$$\mu \circ (1_G\times {\hat{e}}_G)\circ {△}_G$$

$$\mu \circ (1_G\times i)\circ {△}_G$$

$${\mu }_{{}_T}:{\mathbb{R}}^{1|1|1|1}(T)\times {\mathbb{R}}^{1|1|1|1}(T)$$

$$(g_0,g_1,g_2,g_3),(g_0^{\prime },g_1^{\prime },g_2^{\prime },g_3^{\prime })$$

$$C_{{\mathbb{R}}^{m_0^2+\dots +m_{\text{q}}^2}}^{\infty }[[{\xi }_1^1,\dots ,{\xi }_1^{t_1},{\xi }_2^1,\dots ,{\xi }_2^{t_2},\dots ,{\xi }_3^1,\dots ,{\xi }_{\text{q}}^{t_{\text{q}}}]].$$

$$\overline{X}=\prod _{i}GL(V_i)\subset \prod _{i}End(V_i).$$

$$X=\Big{(}\prod_{i}GL(V_{i}),\mathcal{A}|_{\prod_{i}GL(V_{i})}\Big{)}.$$

$$Hom(T,X)=Hom\Big{(}\mathcal{A}(X),\mathcal{O}(T)\Big{)}.$$

$$\left[\begin{array}[]{c|c|c|c}B_{00}&B_{01}&\ldots&B_{0\textbf{q}}\\ \hline\cr\ldots&&&\ldots\\ \hline\cr B_{\textbf{q}0}&B_{\textbf{q}1}&\ldots&B_{\textbf{q}\textbf{q}}\\ \end{array}\right].$$

$$r_x:=\mu \circ (1_G\times {\hat{x}}_G)\circ {\Delta }_G,$$

$$l_x:=\mu \circ ({\hat{x}}_G\times 1_G)\circ {\Delta }_G,$$

$$r_x^{\ast }:=(1_{\mathcal{O}(G)}\otimes e{v}_x)\circ {\mu }^{\ast },$$

$$l_x^{\ast }:=(e{v}_x\otimes 1_{\mathcal{O}(G)})\circ {\mu }^{\ast }.$$

$$M\times G\times G\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\begin{array}{ccc}a\circ (1_M\times \mu )=a\circ (a\times 1_G),& & a\circ (1_M\times {\hat{e}}_M)\circ {\Delta }_M=1_M.\end{array}$$

$$a(.):M(.)\times G(.)\to M(.).$$

$$a_p:G\to M,a^g:M\to M,$$

Full text

Keywords: $\mathbb{Z}^{n}_{2}-$Super Lie group, Homogeneous superspace, $\mathbb{Z}^{n}_{2}-$ Supermanifold, $\mathbb{Z}^{n}_{2}-$ Supergrassmannian

Introduction

There is growing interest in studying generalized supergeometry, that is, geometry of graded manifolds where the grading group is not $\mathbb{Z}_{2}$, but $\mathbb{Z}^{n}_{2}=\mathbb{Z}_{2}\times\ldots\times\mathbb{Z}_{2}$. The foundational aspects of the theory of $\mathbb{Z}^{n}_{2}-$supermanifolds were recently studied in [6], [7], [8] and [9]. This generalization is used in string theory and parastatistics in physics, see [1], [16]. Also in Mathematics, there exist many examples of $\mathbb{Z}^{n}_{2}-$graded $\mathbb{Z}^{n}_{2}-$commutative algebras: quaternions and Clifford algebras, the algebra of Deligne differential superforms, etc. Moreover, there exist interesting examples of $\mathbb{Z}^{n}_{2}-$supermanifolds. In this paper, we study the $\mathbb{Z}^{n}_{2}-$supergrassmannians as $\mathbb{Z}^{n}_{2}-$supermanifolds and their constructions.

In the context of supermanifolds, homogeneous superspaces have been defined and investigated extensively using the functor of points approach in [3], [4] and [5]. In this paper, we show that $\mathbb{Z}^{n}_{2}-$supergrassmannians $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ are homogeneous, c.f. section 2. To this end, we show that the $\mathbb{Z}^{n}_{2}-$super Lie group $GL(\overrightarrow{\textbf{m}})$, c.f. section 1, acts transitively on $\mathbb{Z}^{n}_{2}-$supergrassmannian $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ , c.f. section 2.

In the first section, we recall briefly all necessary basic concepts such as $\mathbb{Z}^{n}_{2}-$ grading spaces, $\mathbb{Z}^{n}_{2}-$supermanifolds, $\mathbb{Z}^{n}_{2}-$super Lie groups and an action of a $\mathbb{Z}^{n}_{2}-$ super Lie group on a $\mathbb{Z}^{n}_{2}-$supermanifold. we use these concepts in the case of $\mathbb{Z}_{2}-$supergeometry in [3] and [5].

In section 2, we study the $\mathbb{Z}^{n}_{2}-$supergrassmannians extensively. The $\mathbb{Z}_{2}-$ supergrassmannians are introduced by Manin in [11], but here by developing an efficient formalism, we fill in the details of the proof of this statement.

In section 3, by a functor of points approach, an action of the super Lie group $GL(\overrightarrow{\textbf{m}})$ on the supergrassmannian $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ is defined by gluing local actions. Finally it is shown that this action is transitive.

1. Preliminaries

Let $\mathbb{Z}^{n}_{2}=\mathbb{Z}_{2}\times\ldots\times\mathbb{Z}_{2}$ be the $n-$fold Cartesian product of $\mathbb{Z}_{2}.$ From now on, we set $\textbf{q}:=2^{n}-1$ and by $\overrightarrow{\textbf{k}}$ and $\overrightarrow{\textbf{m}},$ we mean $(k_{0},k_{1},\ldots,k_{\textbf{q}})$ and $(m_{0},m_{1},\dots,m_{\textbf{q}})$ respectively such that $k_{i},m_{j}\in\mathbb{N}$. Consider the bi-additive map

[TABLE]

The even subgroup $(\mathbb{Z}^{n}_{2})_{0}$ consists of elements $\gamma\in\mathbb{Z}^{n}_{2}$ such that $\langle\gamma,\gamma\rangle=0,$ and the set $(\mathbb{Z}^{n}_{2})_{1}$ consists of odd elements $\gamma\in\mathbb{Z}^{n}_{2}$ such that $\langle\gamma,\gamma\rangle=1.$

One can fix an ordering on $\mathbb{Z}^{n}_{2}$; based on this ordering, each even element is smaller than each odd element. Given two even (odd) elements $(a_{1},a_{2},\ldots,a_{n})$ and $(b_{1},b_{2},\ldots,b_{n})$, the first one is smaller than the second one for the lexicographical order, if $a_{i}<b_{i}$, for the first i where $a_{i}$ and $b_{i}$ differ. For example, the lexicographical ordering on $\mathbb{Z}_{2}^{3}$ is

[TABLE]

Obviously, $\mathbb{Z}^{n}_{2}$ with lexicographical ordering is totally ordered set. Thus it may be diagrammed as an ascending chain as follows

[TABLE]

In the supergeometry, the sign rules between generators of the algebra are completely determined by their parity. One can define a grading by (1) such that $\epsilon(a,b)=(-1)^{\langle a,b\rangle}$ will be a sign rule what will lead to $\mathbb{Z}^{n}_{2}-$supergeometry. Also, it has been shown that any other sign rule for finite number of coordinates is obtained from the above sine rule for sufficiently big $n.$ See [6] for more details.

1.1. $\mathbb{Z}^{n}_{2}-$supergeometry

The $\mathbb{Z}^{n}_{2}-$graded objects like $\mathbb{Z}^{n}_{2}-$superlagebras, $\mathbb{Z}^{n}_{2}-$super ringed spaces, $\mathbb{Z}^{n}_{2}-$ superdomains and $\mathbb{Z}^{n}_{2}-$supermanifolds have been studied in [6], [8] and [9]. In the following, we recall the necessary definitions from these references.

By definition, a $\mathbb{Z}^{n}_{2}-$super vector space is a direct sum $V=\bigoplus_{\gamma\in\mathbb{Z}^{n}_{2}}V_{\gamma}$ of vector spaces $V_{\gamma}$ over a field $\mathbb{K}$ (with characteristic [math]). For each $\gamma\in\mathbb{Z}^{n}_{2},$ the elements of $V_{\gamma}$ is called homogeneous with degree $\gamma.$ If $x\in V_{\gamma}$ be a homogeneous element of $V,$ then the degree of $x$ is represented by $\tilde{x}=\gamma.$

A $\mathbb{Z}^{n}_{2}-$superring $\mathcal{R}=\bigoplus_{\gamma\in\mathbb{Z}^{n}_{2}}\mathcal{R}_{\gamma}$ is a ring such that its multiplication satisfy $\mathcal{R}_{\gamma_{1}}\mathcal{R}_{\gamma_{2}}\subset\mathcal{R}_{\gamma_{1}+\gamma_{2}}.$ A $\mathbb{Z}^{n}_{2}-$superring $\mathcal{R}$ is called $\mathbb{Z}^{n}_{2}-$commutative, if for any homogeneous elements $a,b\in\mathcal{R}$

[TABLE]

Example 1.1**.**

Let $R$ be a ring and $\xi_{1},\ldots,\xi_{\textbf{q}}$ be indeterminates with degree $\gamma_{1},\ldots,\gamma_{\textbf{q}}\in\mathbb{Z}^{n}_{2}$ respectively such that

[TABLE]

Then $R[[\xi_{1},\ldots,\xi_{\textbf{q}}]]$ is the $\mathbb{Z}^{n}_{2}-$commutative associative unital $R-$algebra of formal series in the $\xi_{a}$ with coefficients in $R.$

By a $\mathbb{Z}^{n}_{2}-$super ringed space, we mean a pair $(X,\mathcal{O}_{X})$ where $X$ is a topological space and $\mathcal{O}_{X}$ is a sheaf of $\mathbb{Z}^{n}_{2}-$commutative $\mathbb{Z}^{n}_{2}-$graded rings on $X$. A morphism between two $\mathbb{Z}^{n}_{2}-$super ringed spaces $(X,\mathcal{O}_{X})$ and $(Y,\mathcal{O}_{Y})$ is a pair $\psi:=(\overline{\psi},\psi^{*})$ such that $\overline{\psi}:X\rightarrow Y$ is a continuous map and $\psi^{*}:\mathcal{O}_{Y}\rightarrow\overline{\psi}_{*}\mathcal{O}_{X}$ is a homomorphism of weight zero between the sheaves of $\mathbb{Z}^{n}_{2}-$commutative $\mathbb{Z}^{n}_{2}-$graded rings. Let $\textbf{q}=2^{n}-1,$ the $\mathbb{Z}^{n}_{2}$-ringed space

[TABLE]

is called $\mathbb{Z}^{n}_{2}$-superdomain such that $C^{\infty}_{\mathbb{R}^{m_{0}}}$ is the sheaf of smooth functions on $\mathbb{R}^{m_{0}}$. Also for each open $U\subset\mathbb{R}^{m_{0}}$,

[TABLE]

is the $\mathbb{Z}^{n}_{2}-$commutative associative unital $\mathbb{Z}^{n}_{2}-$superalgebra of formal power series in formal variables $\xi_{i}^{j}$’s of degrees $\gamma_{i}$ which commuting as follows:

[TABLE]

By evaluation of $f=\sum f_{I}\xi^{I}$ at $x\in U$, denoted by $ev_{x}(f)$, we mean $f_{\phi}(x)$.

A $\mathbb{Z}^{n}_{2}$-supermanifold of dimension $\overrightarrow{\textbf{m}}$ is a $\mathbb{Z}^{n}_{2}$-ringed space $(\overline{M},\mathcal{O}_{M})$ that is locally isomorphic to $\mathbb{R}^{\overrightarrow{\textbf{m}}}$. In addition $\overline{M}$ is a second countable and Hausdorff topological space. A morphism between two $\mathbb{Z}^{n}_{2}$-supermanifolds $M=(\overline{M},\mathcal{O}_{M})$ and $N=(\overline{N},\mathcal{O}_{N})$ is a local morphism between two local $\mathbb{Z}^{n}_{2}$-ringed spaces.

Analogous with supergeometry, one can obtain a $\mathbb{Z}^{n}_{2}-$supermanifold by gluing $\mathbb{Z}^{n}_{2}-$superdomains. We will use this method to construct the $\mathbb{Z}^{n}_{2}-$ supergrassmannian as a $\mathbb{Z}^{n}_{2}-$supermanifold in section 2.

1.2. Category theory

By a locally small category, we mean a category such that the collection of all morphisms between any two of its objects is a set. Let $X$, $Y$ are objects in a category and $\alpha,\beta:X\rightarrow Y$ are morphisms between these objects. An universal pair $(E,\epsilon)$ is called equalizer if the following diagram commutes:

[TABLE]

i.e., $\alpha\circ\epsilon=\beta\circ\epsilon$ and also for each object $T$ and any morphism $\tau:T\rightarrow X$ which satisfy $\alpha\circ\tau=\beta\circ\tau$, there exists unique morphism $\sigma:T\rightarrow E$ such that $\epsilon\circ\sigma=\tau$. If equalizer existed then it is unique up to isomorphism. For example, in the category of sets, which is denoted by SET, the equalizer of two morphisms $\alpha,\beta:X\rightarrow Y$ is the set $E=\{x\in X|\alpha(x)=\beta(x)\}$ together with the inclusion map $\epsilon:E\hookrightarrow X.$

Let $\mathcal{C}$ be a locally small category, and $X$ be an object in $\mathcal{C}$. By $T$-points of $X$, we mean $X(T):=Hom_{\mathcal{C}}(T,X)$ for any $T\in Obj(\mathcal{C})$. The functor of points of $X$ is a functor which is denoted by $X(.)$ and is defined as follows:

[TABLE]

where $X(\varphi):f\mapsto f\circ\varphi.$ A functor $F:\mathcal{C}\to\textbf{SET}$ is called representable if there exists an object $X$ in $\mathcal{C}$ such that $F$ and $X(.)$ are isomorphic. Then one may say that $F$ is represented by $X$. The category of functors from $\mathcal{C}$ to SET is denoted by $[\mathcal{C},\textbf{SET}]$. It is shown that the category of all representable functors from $\mathcal{C}$ to SET is a subcategory of $[\mathcal{C},\textbf{SET}]$.

Corresponding to each morphism $\psi:X\rightarrow Y$, there exists a natural transformation $\psi(.)$ from $X(.)$ to $Y(.)$. This transformation corresponds the mapping $\psi(T):X(T)\rightarrow Y(T)$ with $\xi\mapsto\psi\circ\xi$ for each $T\in Obj(\mathcal{C})$. Now set:

[TABLE]

Obviously, $\mathcal{Y}$ is a covariant functor and it is called Yoneda embedding.

Lemma 1.2**.**

The Yoneda embedding is full and faithful functor, i.e. the map

[TABLE]

is a bijection for each $X,Y\in Obj(\mathcal{C}).$

Proof.

see [5]. ∎

Thus according to this lemma, $X,Y\in Obj(\mathcal{C})$ are isomorphic if and only if their functor of points are isomorphic. The Yoneda embedding is an equivalence between $\mathcal{C}$ and a subcategory of representable functors in $[\mathcal{C},\textbf{SET}]$ since not all functors are representable.

1.3. $\mathbb{Z}^{n}_{2}-$super Lie groups

Let ZSM be the category of $\mathbb{Z}^{n}_{2}-$ supermanifolds. This is a category whose objects are $\mathbb{Z}^{n}_{2}-$supermanifolds whose morphisms are morphisms between two $\mathbb{Z}^{n}_{2}-$supermanifolds. Obviously, ZSM is a locally small category and has finite product property. In addition it has a terminal object $\mathbb{R}^{\overrightarrow{\textbf{0}}}$, that is the constant sheaf $\mathbb{R}$ on a singleton $\{0\}$.

Let $M=(\overline{M},\mathcal{O}_{M})$ be a $\mathbb{Z}^{n}_{2}-$supermanifold and $p\in\overline{M}$. There is a map $j_{p}=(\overline{j}_{p},{j_{p}}^{*})$ where:

[TABLE]

So, for each $\mathbb{Z}^{n}_{2}-$supermanifold $T$, one can define the morphism

[TABLE]

as a composition of $j_{p}$ and the unique morphism $T\rightarrow\mathbb{R}^{\overrightarrow{\textbf{0}}}$.

By $\mathbb{Z}^{n}_{2}-$super Lie group, we mean a group-object in the category ZSM. More precisely, it is defined as follows:

Definition 1.3**.**

A $\mathbb{Z}^{n}_{2}-$super Lie group $G$ is a $\mathbb{Z}^{n}_{2}-$supermanifold $G$ together with the morphisms of weight zero $\mu:G\times G\rightarrow G,\quad i:G\rightarrow G,\quad e:\mathbb{R}^{\overrightarrow{\textbf{0}}}\rightarrow G$ called multiplication, inverse and unit morphisms respectively, such that the following equations are satisfied

[TABLE]

where $1_{G}$ is identity on $G$ and $\hat{e}_{G}$ is the composition of $e$ and the unique morphism $G\to R^{\overrightarrow{\textbf{0}}}$. In addition $\Delta_{G}$ is the diagonal map on $G$.

Note that, there is a Lie group associated with each $\mathbb{Z}^{n}_{2}-$super Lie group. Indeed, let $G$ be a $\mathbb{Z}^{n}_{2}-$super Lie group and $G_{0}$ is reduced manifold associated to $G$ and $\mu_{0},i_{0},e_{0}$ are reduced morphisms associated to $\mu,i,e$ respectively. Since $G\rightarrow G_{0}$ is a functor, $(G_{0},\mu_{0},i_{0},e_{0})$ is a group-object of the category of differentiable manifolds.

Remark 1.4*.*

Simply, one can show that any $\mathbb{Z}^{n}_{2}-$super Lie group $G$ induced a group structure over its $T$-points for any arbitrary $\mathbb{Z}^{n}_{2}-$supermanifold $T$. This means that the functor $T\rightarrow G(T)$ takes values in category of groups. Moreover, for any other $\mathbb{Z}^{n}_{2}-$supermanifold $S$ and morphism $T\rightarrow S$, the corresponding map $G(S)\rightarrow G(T)$ is a homomorphism of groups. One can also define a $\mathbb{Z}^{n}_{2}-$super Lie group as a representable functor $T\rightarrow G(T)$ from category ZSM to category of groups. If such functor represented by a $\mathbb{Z}^{n}_{2}-$supermanifold $G$, then the maps $\mu,i,e$ are obtained by Yoneda’s lemma and the maps $\mu_{{}_{T}}:G(T)\times G(T)\rightarrow G(T),\quad i_{{}_{T}}:G(T)\rightarrow G(T)$ and $e_{{}_{T}}:\mathbb{R}^{\overrightarrow{\textbf{0}}}(T)\rightarrow G(T)$.

Remark 1.5*.*

Consider the $\mathbb{Z}^{n}_{2}-$superdomain $\mathbb{R}^{\overrightarrow{\textbf{m}}}$ and an arbitrary $\mathbb{Z}^{n}_{2}-$ supermanifold $T$. Let $f^{j}_{i}\in\mathcal{O}(T)_{\gamma_{i}}$, $0\leq i\leq\textbf{q}$, $1\leq j\leq m_{j}$, be $\gamma_{i}-$degree elements. By Theorem 6.8 in [6] (Fundamental theorem of $\mathbb{Z}^{n}_{2}$-morphisms), One may define a unique morphism $\psi:T\rightarrow\mathbb{R}^{\overrightarrow{\textbf{m}}}$, by setting $\xi_{i}^{j}\mapsto f_{i}^{j}$ where $(\xi_{i}^{j})$ is a global coordinates system on $\mathbb{R}^{\overrightarrow{\textbf{m}}}$. Thus $\psi$ may be represented by $(f_{i}^{j})$.

Example 1.6**.**

Let $(t,\xi,\eta,\nu)$ be a global coordinates system on $\mathbb{Z}_{2}^{2}-$ superdomain $\mathbb{R}^{1|1|1|1}$. Let $T$ be an arbitrary supermanifold, we define:

[TABLE]

where $g_{i},g_{i}^{\prime}\in\mathcal{O}(T)_{\gamma_{i}}$, $i=0,1,2,3.$ It follows that $\mathbb{R}^{1|1|1|1}(T)$ with $\mu_{{}_{T}}$ is a common group. Thus, by Remark 1.4, $\mathbb{R}^{1|1|1|1}$ is a $\mathbb{Z}_{2}^{2}-$super Lie group.

Analogously, one may show that the $\mathbb{Z}^{n}_{2}-$supersuperdomain $\mathbb{R}^{\overrightarrow{\textbf{m}}}$ is a $\mathbb{Z}^{n}_{2}-$super Lie group.

Example 1.7**.**

Let $V$ be a finite dimensional $\mathbb{Z}^{n}_{2}$-super vector space of dimension $m_{0}|m_{1}|\ldots|m_{\textbf{q}}$ and let $\{R_{1},\ldots,R_{m_{0}},R_{m_{0}+1},\ldots,R_{m_{0}+m_{1}},\ldots,R_{m_{0}+\ldots+m_{\textbf{q}}}\}$ be a basis of $V$ for which the elements $R_{m_{0}+...+m_{i-1}+k},$ $1\leq k\leq m_{i},$ are of weight $\gamma_{i}$ for $0\leq i\leq\textbf{q}$. Consider the functor $F$ from the category ZSM to GRP the category of groups which maps each $\mathbb{Z}^{n}_{2}-$seupermanifold $T$ to $Aut_{\mathcal{O}(T)}(\mathcal{O}(T)\otimes V)$ the group of zero weight automorphisms of $\mathcal{O}(T)\otimes V$. Consider the $\mathbb{Z}^{n}_{2}-$supermanifold \textbf{End}(V)=\Big{(}\prod_{i}End(V_{i}),\mathcal{A}\Big{)} where $\mathcal{A}$ is the following sheaf

[TABLE]

where $t_{k}=\sum_{\gamma_{i}+\gamma_{j}=\gamma_{k}}m_{i}m_{j}.$ Let $F_{ij}$ be a linear transformation on $V$ defined by $R_{k}\mapsto\delta_{ik}R_{j}$, then $\{F_{ij}\}$ is a basis for $\textbf{End}(V)$. If $\{f_{ij}\}$ is the corresponding dual basis, then it may be considered as a global coordinates on $\textbf{End}(V)$. Let $X$ be the open subsupermanifold of $\textbf{End}(V)$ corresponding to the open set:

[TABLE]

Thus, we have

[TABLE]

It can be shown that the functor $F$ may be represented by $X$. For this, one may show that $Hom(T,X)\cong Aut_{\mathcal{O}(T)}(\mathcal{O}(T)\otimes V)$. To this end, first, note that

[TABLE]

It is known that each $\psi\in Hom(\mathcal{A}(X),\mathcal{O}(T))$ may be uniquely determined by $\{g_{ij}\}$ where $g_{ij}=\psi(f_{ij})$, see [15]. Now set $\Psi(R_{j}):=\Sigma g_{ij}R_{i}$. One may consider $\Psi$ as an element of $Aut_{\mathcal{O}(T)}(\mathcal{O}(T)\otimes V)$. Obviously $\psi\mapsto\Psi$ is a bijection from $Hom(T,X)$ to $Aut_{\mathcal{O}(T)}(\mathcal{O}(T)\otimes V)$. Thus the $\mathbb{Z}^{n}_{2}-$supermanifold $X$ is a $\mathbb{Z}^{n}_{2}-$super Lie group and denoted it by $GL(V)$ or $GL(\overrightarrow{\textbf{m}})$ if $V=\mathbb{R}^{\overrightarrow{\textbf{m}}}$. Therefore $T$- points of $GL(\overrightarrow{\textbf{m}})$ are the $\overrightarrow{\textbf{m}}\times\overrightarrow{\textbf{m}}$ invertible $\mathbb{Z}^{n}_{2}-$supermatrices of weight zero

[TABLE]

where the elements of the $m_{k}\times m_{u}$ block $B_{ku}$ have degree $\gamma_{k}+\gamma_{u}$ and the multiplication is the matrix product.

Let $x\in\overline{G}$, one can define the left and right translation by $x$ as

[TABLE]

respectively. One can show that pullbacks of above morphisms are as following

[TABLE]

One may also use the language of functor of points to describe two morphisms (1.4) and (1.5).

Definition 1.8**.**

Let $M$ be a $\mathbb{Z}^{n}_{2}-$supermanifold and let $G$ be a $\mathbb{Z}^{n}_{2}-$super Lie group with $\mu,i$ and $e$ as its multiplication, inverse and unit morphisms respectively. A morphism $a:M\times G\rightarrow M$ is called a (right) action of $G$ on $M$, if the following diagrams commute

[TABLE]

where $\hat{e}_{{}_{M}},\Delta_{M}$ are as above. In this case, we say G acts from right on $M$. One can define left action analogously.

According to the above diagrams, one has:

[TABLE]

By Yoneda lemma (Lemma 1.2), one may consider, equivalently, the action of G as a natural transformation:

[TABLE]

Thus for each supermanifold $T$, the morphism $a(T):M(T)\times G(T)\rightarrow M(T)$ is an action of group $G(T)$ on the set $M(T)$. This means:

$(\mathcal{P}.\mathcal{Q}_{1}).\mathcal{Q}_{2}=\mathcal{P}.(\mathcal{Q}_{1}\mathcal{Q}_{2}),\qquad\forall\mathcal{Q}_{1},\mathcal{Q}_{2}\in G(T),\forall\mathcal{P}\in M(T).$ 2. 2.

$\mathcal{P}.\hat{e}_{{}_{T}}=\mathcal{P},\qquad\forall\mathcal{P}\in M(T).$

Let $p\in\overline{M},$ define

[TABLE]

where $\hat{p}_{{}_{G}}$ and $\hat{g}_{{}_{M}}$ are the morphism (1.2) for $p\in\overline{M}$ and $g\in\overline{G}$ respectively. Equivalently, these maps may be defined as

[TABLE]

One may easily show that $a_{p}$ has constant rank(see Proposition 8.1.5 in [5], for more details). Before next definition, we recall that a morphism between $\mathbb{Z}^{n}_{2}-$ supermanifolds, say $\psi:M\rightarrow N$ is a submersion at $x\in\overline{M}$, if $(d\psi)_{x}$ is surjective and $\psi$ is called submersion, if it is surjective at each point. (For more details, refer to [15], [5]). Also $\psi$ is a surjective submersion, if in addition $\psi_{0}$ is surjective.

Definition 1.9**.**

Let $G$ acts on $M$ with action $a:M\times G\rightarrow M$. The action $a$ is called transitive, if there exist $p\in\overline{M}$ such that $a_{p}$ is a surjective submersion.

It is shown that, if $a_{p}$ is a submersion for one $p\in\overline{M}$, then it is a submersion for all point in $\overline{M}$. The following proposition will be required in the last section.

Proposition 1.10**.**

Let $a:M\times G\rightarrow M$ be an action. Then $a$ is transitive if and only if for a $p\in\overline{M},$ $(a_{p})_{\mathbb{R}^{\overrightarrow{\textbf{r}}^{\prime}}}:G(\mathbb{R}^{\overrightarrow{\textbf{r}}^{\prime}})\rightarrow M(\mathbb{R}^{\overrightarrow{\textbf{r}}^{\prime}})$ is surjective, where $\overrightarrow{\textbf{r}}=(r_{0},r_{1},\ldots,r_{\textbf{q}})$ is the dimension of $G$ and $\overrightarrow{\textbf{r}}^{\prime}:=(0,r_{1},\ldots,r_{\textbf{q}}).$

Proof.

The proof is the same as the proof of proposition 9.1.4 in [5] with appropriate modifications. ∎

Definition 1.11**.**

Let $G$ be a $\mathbb{Z}^{n}_{2}-$super Lie group and let $a$ be an action of $G$ on $\mathbb{Z}^{n}_{2}-$supermanifold $M$. By stabilizer of $p\in\overline{M},$ we mean a $\mathbb{Z}^{n}_{2}-$supermanifold $G_{p}$ equalizing the diagram

[TABLE]

Proposition 1.12**.**

Let $a:M\times G\rightarrow M$ be an action, then

The following diagram admits an equalizer $G_{p}$

[TABLE] 2. 2.

$G_{p}$* is a *$\mathbb{Z}^{n}_{2}-$subsuper Lie group of $G$. 3. 3.

The functor $T\rightarrow(G(T))_{\hat{p}_{{}_{T}}}$ is represented by $G_{p}$, where $(G(T))_{\hat{p}_{{}_{T}}}$ is the stabilizer in $\hat{p}_{{}_{T}}$ of the action of $G(T)$ on $M(T)$.

Proof.

The proof is the same as the proof of proposition 8.4.7 in [5] with appropriate modifications. ∎

We end this section with the next proposition which is a straightforward $\mathbb{Z}^{n}_{2}$ generalization of the proposition 6.5 in [3].

Proposition 1.13**.**

Suppose $G$ acts transitively on $M$. There exists a $G$-equivariant isomorphism

[TABLE]

2. $\mathbb{Z}^{n}_{2}$-supergrassmannian

Supergrassmannians $G_{k|l}(m|n)$ are introduced by Manin in [11] and the authors have studied them in more details in [2] and [12]. In this section, we introduce the $\mathbb{Z}_{2}^{n}$-supergrassmannian which is denoted by $G_{k_{0}|k_{1}|\ldots|k_{\textbf{q}}}(m_{0}|m_{1}|\ldots|m_{\textbf{q}})$ or $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ in short. For convenience from now, we set

[TABLE]

and also decompose any $\mathbb{Z}^{n}_{2}-$supermatrix into $2^{n}\times 2^{n}$ blocks

[TABLE]

such that the elements of block $B_{ku}$ have degree $\gamma_{k}+\gamma_{u}$. By a $\mathbb{Z}^{n}_{2}$-supergrassmannian, $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$, we mean a $\mathbb{Z}^{n}_{2}$-supermanifold which is constructed by gluing $\mathbb{Z}^{n}_{2}$- superdomains \mathbb{R}^{\overrightarrow{\beta}}=\Big{(}\mathbb{R}^{\beta_{0}},\,C^{\infty}_{\mathbb{R}^{\beta_{0}}}(-)[[\xi_{1}^{1},\ldots,\xi_{1}^{\beta_{1}},\xi_{2}^{1},\ldots,\xi_{2}^{\beta_{2}},\ldots,\xi_{\textbf{q}}^{1},\ldots,\xi_{\textbf{q}}^{\beta_{\textbf{q}}}]]\Big{)} as follows:

For $i=0,1,\ldots,\textbf{q},$ let $I_{i}\subset\{1,\ldots,m_{i}\}$ be a sorted subset in ascending order with $k_{i}$ elements. The elements of $I_{i}$ are called $\gamma_{i}$-degree indices. The multi-index $\overrightarrow{\textbf{I}}=(I_{1},...,I_{\textbf{q}})$ is called $\overrightarrow{\textbf{k}}$-index. Set $\mathcal{U}_{\overrightarrow{\textbf{I}}}:=(\overline{U}_{\overrightarrow{\textbf{I}}},\mathcal{O}_{\overrightarrow{\textbf{I}}})$, where

[TABLE]

Let each $\mathbb{Z}^{n}_{2}$-superdomain $\mathcal{U}_{\overrightarrow{\textbf{I}}}$ be labeled by a $\mathbb{Z}^{n}_{2}-$supermatrix $\overrightarrow{\textbf{k}}\times\overrightarrow{\textbf{m}}$ of weight zero, say $A_{\overrightarrow{\textbf{I}}}$, with $2^{n}\times 2^{n}$ blocks $B_{ij}$ each of which is a $k_{i}\times m_{j}$ matrix. In addition, except for columns with indices in $I_{0}\cup I_{1}\cup\ldots\cup I_{\textbf{q}}$, which together form a minor denoted by $M_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{I}}}$, the matrix is filled from up to down and left to right by $x_{a}^{\overrightarrow{\textbf{I}}},\xi_{b}^{\overrightarrow{\textbf{I}}}$, the free generators of $\mathcal{O}_{\overrightarrow{\textbf{I}}}(\mathbb{R}^{\beta_{0}})$ each of them sits in a block with same degree. This process impose an ordering on the set of generators. In addition $M_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{I}}}$ is supposed to be the identity matrix.

For example, consider $G_{1|2|1|1|}{2|2|2|2}$. Then let $I_{0}=\{1\},I_{1}=\{1,2\},I_{2}=\{1\},I_{3}=\{2\}$, so $\overrightarrow{\textbf{I}}$ is a $1|2|1|1$-index. In this case the set of generators of $\mathcal{O}_{\overrightarrow{\textbf{I}}}(\mathbb{R}^{\beta_{0}})$ is

[TABLE]

and $A_{\overrightarrow{\textbf{I}}}$ is:

[TABLE]

Note that, in this example,

[TABLE]

is corresponding total ordered set of generators.

By $\widetilde{U}_{{\overrightarrow{\textbf{I}}},{\overrightarrow{\textbf{J}}}},$ we mean the set of all points of $\overline{U}_{\overrightarrow{\textbf{I}}}$, on which $M_{\overrightarrow{\textbf{J}}}A_{\overrightarrow{\textbf{I}}}$ is invertible. Obviously $\widetilde{U}_{{\overrightarrow{\textbf{I}}},{\overrightarrow{\textbf{J}}}}$ is an open set. The transition map between the two $\mathbb{Z}^{n}_{2}$-superdomains $U_{\overrightarrow{\textbf{I}}}$ and $U_{\overrightarrow{\textbf{J}}}$ is denoted by

[TABLE]

Note that $g_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}=(\overline{g}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}},g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}),$ where $g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}$ is an isomorphism between sheaves determined by defining on each entry of $D_{\overrightarrow{\textbf{I}}}(A_{\overrightarrow{\textbf{I}}})$ as a rational expression which appears as the corresponding entry provided by the pasting equation

[TABLE]

where $D_{\overrightarrow{\textbf{I}}}(A_{\overrightarrow{\textbf{I}}})$ is a matrix which is remained after omitting $M_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{I}}}$. Clearly, the left hand side of (2.2) is defined whenever $M_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{J}}}$ is invertible. The morphism $g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}$ induces the continuous map $\overline{g}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}$ (in the case $n=1$, see [13], lemma 3.1).

For example in $G_{1|2|1|1}(2|2|2|2)$ suppose

[TABLE]

so $\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}$ are $1|2|1|1$-indices. We have:

[TABLE]

[TABLE]

The maps $g_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}$ are gluing morphisms. In fact, a straightforward computation shows the following proposition holds.

Proposition 2.1**.**

Let g_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}=\big{(}\overline{g}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}},g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}\big{)} be as above, then

$g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{I}}}}=id.$** 2. 2.

$g^{*}_{{}_{\overrightarrow{\textbf{J}},\overrightarrow{\textbf{I}}}}\circ g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}=id.$** 3. 3.

$g^{*}_{{}_{\overrightarrow{\textbf{S}},\overrightarrow{\textbf{I}}}}\circ g^{*}_{{}_{\overrightarrow{\textbf{J}},\overrightarrow{\textbf{S}}}}\circ g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}=id.$**

Proof.

For first equality, note that the map $g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{I}}}}$ is obtained from the following equality:

[TABLE]

where the matrix $M_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{I}}}$ is identity. So $g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{I}}}}$ is defined by the following equality:

[TABLE]

This shows the first equality. For second equality, let $\overrightarrow{\textbf{J}}$ be an another $\overrightarrow{\textbf{k}}$-index, so $g^{*}_{{}_{\overrightarrow{\textbf{J}},\overrightarrow{\textbf{I}}}}$ is obtained by the following equality:

[TABLE]

One may see that $g^{*}_{{}_{\overrightarrow{\textbf{J}},\overrightarrow{\textbf{I}}}}\circ g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}$ is obtained by following equality:

[TABLE]

For left side, we have

[TABLE]

Accordingly the map $g^{*}_{{}_{\overrightarrow{\textbf{J}},\overrightarrow{\textbf{I}}}}\circ g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}$ is obtained by $D_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{I}}}=D_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{I}}}$ and it shows that this map is identity. For third equality, it is sufficient to show that the map $g^{*}_{{}_{\overrightarrow{\textbf{S}},\overrightarrow{\textbf{I}}}}\circ g^{*}_{{}_{\overrightarrow{\textbf{J}},\overrightarrow{\textbf{S}}}}\circ g^{*}_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}$ is obtained from

[TABLE]

This case obtains from case $2$ analogously. ∎

So the sheaves $(\overline{U}_{\overrightarrow{\textbf{I}}},\mathcal{O}_{\overrightarrow{\textbf{I}}})$ may be glued through the $g_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}$ to construct the $\mathbb{Z}^{n}_{2}-$supergrassmannian $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$. Indeed, according to [15], the conditions of the above proposition are necessary and sufficient for gluing.

3. $\mathbb{Z}^{n}_{2}-$Supergrassmannian as homogeneous $\mathbb{Z}^{n}_{2}-$superspace

Let $G=(\overline{G},\mathcal{O}_{G})$ be a $\mathbb{Z}^{n}_{2}-$super Lie group and $H=(\overline{H},\mathcal{O}_{H})$ be a closed $\mathbb{Z}^{n}_{2}-$sub super Lie group of $G$. One can define a $\mathbb{Z}^{n}_{2}-$supermanifold structure on the topological space $\overline{X}=\overline{G}/\overline{H}$ as follows:

Let $\mathfrak{g}=Lie(G)$ and $\mathfrak{h}=Lie(H)$ be the $\mathbb{Z}^{n}_{2}-$super Lie algebras corresponding with $G$ and $H$. For each $Z\in\mathfrak{g}$, let $D_{Z}$ be the left invariant vector field on $G$ associated with $Z$. For $\mathfrak{h},$ a $\mathbb{Z}^{n}_{2}-$subalgebra of $\mathfrak{g},$ set:

[TABLE]

On the other hand, for any open subset $U\subset\overline{G}$ set:

[TABLE]

where $r_{x_{0}}$ is the right translation by $x_{0}$ in (1.5). If $\overline{H}$ is connected, then $\mathcal{O}_{inv}(U)=\mathcal{O}_{\mathfrak{h}}(U)$. Let $\overline{\pi}:\overline{G}\rightarrow\overline{X}$ be the natural projection. For each open subset $W\subset\overline{X}=\overline{G}/\overline{H}$, the structure sheaf $\mathcal{O}_{X}$ is defined as following

[TABLE]

where $U=\overline{\pi}^{-1}(W).$ One can show that $\mathcal{O}_{X}$ is a sheaf on $\overline{X}$ and the ringed space $X=(\overline{X},\mathcal{O}_{X})$ is a $\mathbb{Z}^{n}_{2}-$superdomain locally (See [3] for more details). So $X$ is a $\mathbb{Z}^{n}_{2}-$supermanifold and is called homogeneous $\mathbb{Z}^{n}_{2}-$superspace. In this section, we want to show that the $\mathbb{Z}^{n}_{2}-$supergrassmannian $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ is a homogeneous $\mathbb{Z}^{n}_{2}-$superspace. According to the section 1, it is enough to find a $\mathbb{Z}^{n}_{2}-$super Lie group which acts on $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ transitively. For this, we need the following remark and the next lemma

Remark 3.1*.*

Let $\mathcal{X}$ be an element of $U_{\overrightarrow{\textbf{I}}}(T)$ where $\overrightarrow{\textbf{I}}$ is an arbitrary index. One can correspond to $\mathcal{X}$ a $\mathbb{Z}^{n}_{2}-$supermatrix $\overrightarrow{\textbf{k}}\times\overrightarrow{\textbf{m}}$ called $[\mathcal{X}]_{\overrightarrow{\textbf{I}}}$ as follows: Except for columns with indices in $I_{0}\cup I_{1}\cup\ldots\cup I_{\textbf{q}}$, the blocks are filled from up to down and left to right by $f_{i},g_{j}$’s where

[TABLE]

according to the ordering (2.1), where $(x_{i};\xi_{j})$ is the global coordinates of the $\mathbb{Z}^{n}_{2}-$superdomain $U_{\overrightarrow{\textbf{I}}}$. The columns with indices in $I_{0}\cup I_{1}\cup\ldots\cup I_{\textbf{q}}$ form an identity matrix.

Lemma 3.2**.**

Let $\psi:T\rightarrow\mathbb{R}^{\overrightarrow{\textbf{r}}}$ be a $T$-point of $\mathbb{R}^{\overrightarrow{\textbf{r}}}$ and $(z_{tu})$ be a global coordinates of $\mathbb{R}^{\overrightarrow{\textbf{r}}}$ with ordering as the one introduced in (2.1). If $B=(\psi^{*}(z_{tu}))$ is the $\mathbb{Z}^{n}_{2}-$supermatrix corresponding to $\psi$, then the $\mathbb{Z}^{n}_{2}-$supermatrix corresponding to \big{(}g_{{}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}}\big{)}_{{}_{T}}(\psi) is as follows:

[TABLE]

where $[B]_{\overrightarrow{\textbf{J}}}$ is introduced in Remark 3.1.

Proof.

Note that $g^{*}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}$ may be represented by a $\mathbb{Z}^{n}_{2}-$supermatrix as follows:

[TABLE]

where $A_{\overrightarrow{\textbf{J}}}$ is the label of $U_{\overrightarrow{\textbf{J}}}.$ Let $M_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{J}}}=(m_{tu})$ and $(M_{\overrightarrow{\textbf{I}}}A_{\overrightarrow{\textbf{J}}})^{-1}=(m^{tu})$. If $z=(z_{ij})$ be a coordinates system on $U_{\overrightarrow{\textbf{I}}}$, then one has

[TABLE]

Then

[TABLE]

For second equality one may note that $\psi^{*}$ is a homomorphism of $\mathbb{Z}^{n}_{2}-$superalgebras and $m^{tk}(z)$ is a rational function of $z$. Obviously, the last expression is the $(t,u)$-entry of the matrix D_{\overrightarrow{\textbf{I}}}\Big{(}(M_{\overrightarrow{\textbf{I}}}[B]_{\overrightarrow{\textbf{J}}})^{-1}[B]_{\overrightarrow{\textbf{J}}}\Big{)}. This completes the proof. ∎

Theorem 3.3**.**

The $\mathbb{Z}^{n}_{2}-$super Lie group $GL(\overrightarrow{\textbf{m}})$ acts on $\mathbb{Z}^{n}_{2}-$ supergrassmannian $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$.

Proof.

First, we have to define a morphism $a:G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})\times GL(\overrightarrow{\textbf{m}})\rightarrow G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$. For this, by Yoneda lemma, it is sufficient to define $a_{{}_{T}}$:

[TABLE]

for each $\mathbb{Z}^{n}_{2}-$supermanifold $T$ or equivalently define

[TABLE]

where $\mathcal{P}$ is a fixed arbitrary element in $GL(\overrightarrow{\textbf{m}})(T)$. For brevity, we denote $(a_{{}_{T}})^{\mathcal{P}}$ by A. One may consider $GL(\overrightarrow{\textbf{m}})(T)$, as the set of $\overrightarrow{\textbf{m}}\times\overrightarrow{\textbf{m}}$ invertible $\mathbb{Z}^{n}_{2}-$supermatrices with entries in $\mathcal{O}(T)$, but there is not such a description for $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})(T)$, because it is not a $\mathbb{Z}^{n}_{2}-$superdomain. We know each $\mathbb{Z}^{n}_{2}-$supergrassmanian is constructed by gluing $\mathbb{Z}^{n}_{2}-$superdomains (c.f. section 2), so one may define the actions of $GL(\overrightarrow{\textbf{m}})$ on $\mathbb{Z}^{n}_{2}-$superdomains $(\overline{U}_{\overrightarrow{\textbf{I}}},\mathcal{O}_{\overrightarrow{\textbf{I}}})$ and then shows that these actions glued to construct $a_{{}_{T}}$.

For defining A, it is needed to refine the covering $\{U_{\overrightarrow{\textbf{I}}}(T)\}_{\overrightarrow{\textbf{I}}}$. Set

[TABLE]

where $[\mathcal{P}]$ is the matrix form of the fixed arbitrary element $\mathcal{P}$ in $GL(\overrightarrow{\textbf{m}})(T)$, see [5] and [15]. One can show that $\{U_{\overrightarrow{\textbf{I}}}^{\overrightarrow{\textbf{J}}}(T)\}_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}$ is a covering for $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})(T)$ and \textbf{A}\Big{(}U_{\overrightarrow{\textbf{I}}}^{\overrightarrow{\textbf{J}}}(T)\Big{)}\subseteq U_{\overrightarrow{\textbf{J}}}(T). Now consider all maps

[TABLE]

where, $[\psi]_{\overrightarrow{\textbf{I}}}$ is as above. We have to show that these maps may be glued to construct a global map on $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})(T)$. For this, it is sufficient to show that the following diagram commutes:

[TABLE]

where \big{(}g_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}\big{)}_{{}_{T}} is the induced map from $g_{\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}}}$ on $T$-points. The following proposition is used to show commutativity of the above diagram. ∎

Proposition 3.4**.**

The last diagram commutes.

Proof.

We have to show that

[TABLE]

for arbitrary $\overrightarrow{\textbf{k}}$-indices $\overrightarrow{\textbf{I}},\overrightarrow{\textbf{J}},\overrightarrow{\textbf{Q}},\overrightarrow{\textbf{L}}.$ Let $\psi\in U_{\overrightarrow{\textbf{I}}}^{\overrightarrow{\textbf{J}}}(T)\cap U_{\overrightarrow{\textbf{Q}}}^{\overrightarrow{\textbf{L}}}(T)$ be an arbitrary element. One has $\psi\in U_{\overrightarrow{\textbf{I}}}^{\overrightarrow{\textbf{J}}}(T)$, so

[TABLE]

From left side of (3.1), we have:

[TABLE]

‎ For right side of equation (3.1), we have

[TABLE]

This shows that the above diagram commutes. ∎

Therefore $GL(\overrightarrow{\textbf{m}})$ acts on $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ with action $a$. Now it is needed to show that this action is transitive.

Theorem 3.5**.**

$GL(\overrightarrow{\textbf{m}})$* acts on $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ transitively.*

Proof.

By proposition 1.10, it is sufficient to show that the map

[TABLE]

is surjective, where $\overrightarrow{\textbf{r}}=(r_{0},r_{1},\ldots,r_{{}_{\textbf{q}}})$ is dimension of $GL(\overrightarrow{\textbf{m}})$ and $\overrightarrow{\textbf{r}}^{\prime}=(0,r_{1},\ldots,r_{{}_{\textbf{q}}}).$ Let

[TABLE]

be an element and $\bar{p}_{0},\bar{p}_{1},\ldots,\bar{p}_{{}_{\textbf{q}}}$ be the matrices corresponding to subspaces $p_{0},p_{1},\ldots,p_{\textbf{q}}$ respectively. As an element of $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})(T)$, one may represent $\hat{p}_{{}_{T}}$, as follows

[TABLE]

where $T$ is an arbitrary $\mathbb{Z}^{n}_{2}-$supermanifold. For surjectivity, let

[TABLE]

be an arbitrary element. According to (1.3), we have to show that there exists an element $V\in GL(\overrightarrow{\textbf{m}})(\mathbb{R}^{\overrightarrow{\textbf{r}}^{\prime}})$ such that $\hat{p}_{{}_{T}}V=W$. Since the Lie group $GL(m_{i})$ acts on manifold $G_{k_{i}}(m_{i})$ transitively, then there exists an invertible matrix $H_{ii}\in GL(m_{i})$ such that $\bar{p}_{i}H_{ii}=W_{ii}$. In addition, the equations $\bar{p}_{i}Z=W_{ij}$ have solutions since $rank(\bar{p}_{i})=k_{i}$. Let $H_{ij}$ be solutions of these equations respectively. Clearly, One can see

[TABLE]

satisfy in the equation $\hat{p}_{{}_{T}}V=W$. So $(a_{p})_{\mathbb{R}^{\overrightarrow{\textbf{r}}^{\prime}}}$ is surjective. By Proposition 1.10, $GL(\overrightarrow{\textbf{m}})$ acts on $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ transitively. ∎

Thus according to Proposition 1.13, $G_{\overrightarrow{\textbf{k}}}(\overrightarrow{\textbf{m}})$ is a homogeneous $\mathbb{Z}^{n}_{2}-$superspace.