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The paper explores the indecomposable submodule structures of quantum dividedpower algebra $\mathcal{A}_q(n)$ defined in \cite{HU} and its truncated objects$\mathcal{A}_q(n, \bold m)$. An "intertwinedly-lifting" method is establishedto prove the indecomposability of a module when its socle is non-simple. TheLoewy filtrations are described for all homogeneous subspaces$\mathcal{A}^{(s)}_q(n)$ or $\mathcal{A}_q^{(s)}(n, \bold m)$, the Loewy layersand dimensions are determined. The rigidity of these indecomposable modules isproved. An interesting combinatorial identity is derived from our realizationmodel for a class of indecomposable $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules.Meanwhile, the quantum Grassmann algebra $\Omega_q(n)$ over $\mathcal{A}_q(n)$is constructed, together with the quantum de Rham complex $(\Omega_q(n),d^\bullet)$ via defining the appropriate $q$-differentials, and its subcomplex$(\Omega_q(n,\bold m), d^\bullet)$. For the latter, the corresponding quantumde Rham cohomology modules are decomposed into the direct sum of somesign-trivial $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules.

Loewy filtration and quantum de Rham cohomology over quantum divided power algebra