On the construction of \(\mathbb{Z}^n_2\)-Grassmannians as homogeneous \(\mathbb{Z}^n_2\)-spaces (Q2127484)

In this paper, a generalization of super-Grassmannian is defined in the category of \(\mathbb{Z}^{2n}\)-manifolds. This space is called \(\mathbb{Z}^{2n}\)-Grassmannian. The approach to constructing this space is gluing \(\mathbb{Z}^{2n}\)-domains, which are defined as follows \[ \mathbb{R}^{\vec{m}}=\left(\mathbb{R}^{m_0},\mathcal{O}_{\mathbb{R}^{\vec{m}}}[\![\xi_1^1,\ldots,\xi_1^{m_1},\xi_2^1,\ldots,\xi_2^{m_2},\ldots,\xi_q^{m_q}]\!]\right), \] where \(\mathcal{O}_{\mathbb{R}^{\vec{m}}}\) is the sheaf of smooth functions on \(\mathbb{R}^{\vec{m}}\) and \(q=2n-1\). Furthermore, by gluing the local actions, the action of the \(\mathbb{Z}^{2n}\)-Lie group \(GL(\vec{m})\) is defined and shown to be transitive.