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

Abstract

We explore the indecomposable submodule structure of quantum Grassmann super-algebra ${\Omega }_q(m|n)$ and its truncated objects ${\Omega }_q(m|n,\text{r})$ in the case when $q=\epsilon$ 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 ${\Omega }_q^{(s)}(m|n,\text{r})$ 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 (Poincaré 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 $({\mathcal{D}}_q(m|n,\text{r}{)}^{(\bullet )},d^{\bullet })$ we mainly consider are no longer acyclic and the resulting quantum super de Rham cohomologies $H_{DR}^s({\mathcal{D}}_q(m|n,\mathbf{r}{)}^{(\bullet )})$ are highly nontrivial.

29 pages