Quantum Supersymmetries (II): Loewy Filtrations and Quantum de Rham Cohomology over Quantum Grassmann Superalgebra

TL;DR

This paper investigates the indecomposable module structure and Loewy filtrations of quantum Grassmann superalgebras at roots of unity, and constructs a quantum super de Rham cohomology theory with nontrivial cohomology groups.

Contribution

It introduces a net-like weave-lifting method to analyze indecomposability and describes Loewy filtrations, also constructing a quantum super de Rham cohomology with nontrivial results.

Findings

All homogeneous super subspaces are indecomposable modules.

Loewy layers and dimensions are explicitly determined.

Quantum super de Rham cohomology is nontrivial for truncated complexes.

Abstract

We explore the indecomposable submodule structure of quantum Grassmann super-algebra $\Omega_q(m|n)$ and its truncated objects $\Omega_q(m|n,\textbf{r})$ in the case when $q=\varepsilon$ is an $\ell$-th root of unity. A net-like weave-lifting method is developed to show the indecomposability of all the homogeneous super subspaces $\Omega_q^{(s)}(m|n,\textbf{r})$ and $\Omega_q^{(s)}(m|n)$ as $\mathcal U_q(\mathfrak{gl}(m|n))$-modules by defining "energy grade" to depict their "$\ell$-adic" phenomenon. Their Loewy filtrations are described, the Loewy layers and dimensions are determined by combinatorial identities. The quantum super de Rham cochain short complex $(\mathcal D_q(m|n)^{(\bullet)},d^\bullet)$ is constructed and proved to be acyclic (Poincar\'e Lemma), where $\mathcal D_q(m|n)=\Omega_q(m|n)\otimes \sqcap_q(m|n)$ and $\sqcap_q(m|n)$ is the quantum exterior super-algebra, over which we define the $q$-differentials. %such that the product structure of $\sqcap_q(m|n)$, the quantum exterior super-algebra, is well-matched everywhere. However, the truncated quantum de Rham cochain subcomplexes $(\mathcal D_q(m|n,\textbf{r})^{(\bullet)},d^\bullet)$ we mainly consider are no longer acyclic and the resulting quantum super de Rham cohomologies $H^s_{DR}(\mathcal D_q(m|n, \mathbf r)^{(\bullet)})$ are highly nontrivial.