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

TL;DR

This paper investigates the structure and cohomology of quantum divided power algebras, establishing indecomposability, Loewy filtrations, and quantum de Rham cohomology, revealing new algebraic and combinatorial insights.

Contribution

It introduces an intertwinedly-lifting method for module indecomposability, describes Loewy filtrations, and constructs quantum de Rham complexes with cohomology decompositions.

Findings

Proved indecomposability of modules with non-simple socles.

Described Loewy filtrations and determined layers and dimensions.

Decomposed quantum de Rham cohomology into direct sums of trivial modules.

Abstract

The paper explores the indecomposable submodule structures of quantum divided power algebra $\mathcal{A}_q(n)$ defined in \cite{HU} and its truncated objects $\mathcal{A}_q(n, \bold m)$. 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 $\mathcal{A}^{(s)}_q(n)$ or $\mathcal{A}_q^{(s)}(n, \bold m)$, 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 $\mathfrak{u}_q(\mathfrak{sl}_n)$-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 quantum de Rham cohomology modules are decomposed into the direct sum of some sign-trivial $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules.