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

Abstract

The paper explores the indecomposable submodule structures of quantum divided power algebra Aq(n) defined in [23] and its truncated objects Aq(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 Aq(s)(n) or Aq(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 uq(sln)-modules. Meanwhile, the quantum Grassmann algebra Ωq(n) over Aq(n) is constructed, together with the quantum de Rham complex (Ωq(n),d•) via defining the appropriate q-differentials, and its subcomplex (Ωq(n,m),d•). For the latter, the corresponding quantum de Rham cohomology modules are decomposed into the direct sum of some sign-trivial uq(sln)-modules.