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

The paper explores the indecomposable submodule structures of quantum divided power algebra A_q(n) defined in [23] and its truncated objects A_q(n,m). An "intertwinedly-lifting" method is established to prove the indecomposability of a module when its socle is non-simple. The Loewy filtrations are described for all homogeneous subspaces A_q^(s)(n) or A_q^(s)(n,m), the Loewy layers and dimensions are determined. The rigidity of these indecomposable modules is proved. An interesting combinatorial identity is derived from our realization model for a class of indecomposable u_q(sl_n)-modules. Meanwhile, the quantum Grassmann algebra Ω_q(n) over A_q)(n) is constructed, together with the quantum de Rham complex (Ω_q(n), d^•) via defining the appropriate q-differentials, and its subcomplex (Ω_q(n, m), d^•). For the latter, the corresponding quantum de Rham cohomology modules are decomposed into the direct sum of some sign-trivial u_q(sl_n)-modules.