Higher-order and Weil Grassmannian as a space of subalgebras of a Weil algebra

Abstract

Let $G^r_{k,m}$ be the subgroup of the jet group $G^r_k$ formed by elements projectable to $G^r_m \simeq G^r_m \times \{j^r_0\id_{\er^{k-m}} \}$. We define a foliation $\cv $ on $\reg J^r_0(\er^k ,\er^m )_0$ of $G^r_m$-orbits with respect to the left action defined by the jet composition. Elements of $\reg J^r_0(\er^k ,\er^m )_0$ are considered as vectors from $\er^k$ with the standard inner product over them. The $r$-th order Grassmannian $\G (r,k,m)$ is defined as the basis of the principal bundle $\hat{p}^{\#}:\reg J^r_0(\er^k ,\er^m )_0 \to \cv $ identified with the reduction of the principal bundle $\hat{p}:G^r_k \to G^r_{k,m} \backslash G^r_k$ to the structure group $G^r_m \simeq G^r_m \times \{ j^r_0\id_{\er^{k-m}} \}$. Adding the claim of the first-order orthonormality and modifying $\cv $ to $\cv_{\Ort}$ we obtain the geometrical structure over $\G (r,k,m)$ in the form of the so-called orthonormal Grassmann bundle $\hat{p}^{\# }_{\Ort }$. We construct an atlas on $\G (r,k,m)$ from a finite system of local sections of $\hat{p}^{\# }_{\Ort }$ and define the $r$-th order Grassmannian bundle functor with standard fiber $\G (r,k,m)$ on the category $\cmf_m$ of $m$-dimensional manifolds and local diffeomorphism.

For the jet algebra $\D^r_k$, a Weil algebra $A = \D^r_k /I$ and the projection homomorphism $p_A:\D^r_k \to A$ we define the partition $\cv_A$ on $\reg T^A_0\er^m$ formed by orbits of the left action of the group of all $T^A_0h; h \in \Diff_0\er^m$, which is by \cite{TOM} identified with $G^r_m$. We prove that the local sections of $\hat{p}^{\# }_{\Ort }$ above satisfy some kind of $T^A$-respecting property and determine an atlas on $\cv_A$. We define the Weil Grassmannian $\G (A,m) = \cv_A$ and the bundle functor $\G^A$ defined on $\cmf_m$ with $\G (A,m)$ as its standard fiber. We prove the coincidence of $\G (A,m)$ with the quotient of $\cv$ by the map $[\Vl{p}_{A,\er^m}]:\cv \to \cv_A$ induced by $p_A$.

We define the principle bundle $\hat{p}^{\# }_A :\reg J^r_0(\er^k ,\er^m )_0 \to \G (A,m)$ with the structure group $G^r_m$.
We define a partition $\cv^A$ on $\reg T^A_0\er^m$ coarser then $\cv_A$ and some auxiliary foliations within $\reg T^A_0\er^m$. It is proved that the factorization
$[\hat{p}^{\# }_A]: \reg T^A_0\er^m \to \cv_A$ of $\hat{p}^{\# }_A$ to $\reg T^A_0\er^m$ can be considered as the disjoint union of bundles with standard fiber. identified with an $m$-wide sublagebra of $A$.