Quantum Supersymmetries (II): Loewy Filtrations and Quantum de Rham Cohomology over Quantum Grassmann Superalgebra
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| Format: | Preprint |
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2024
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| _version_ | 1866910679976378368 |
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| author | Feng, Ge Hu, Naihong Rosso, Marc |
| author_facet | Feng, Ge Hu, Naihong Rosso, Marc |
| contents | We explore the indecomposable submodule structure of quantum Grassmann super-algebra $Ω_q(m|n)$ and its truncated objects $Ω_q(m|n,\textbf{r})$ 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 $Ω_q^{(s)}(m|n,\textbf{r})$ and $Ω_q^{(s)}(m|n)$ 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 $\mathcal D_q(m|n)=Ω_q(m|n)\otimes \sqcap_q(m|n)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04901 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Supersymmetries (II): Loewy Filtrations and Quantum de Rham Cohomology over Quantum Grassmann Superalgebra Feng, Ge Hu, Naihong Rosso, Marc Quantum Algebra We explore the indecomposable submodule structure of quantum Grassmann super-algebra $Ω_q(m|n)$ and its truncated objects $Ω_q(m|n,\textbf{r})$ 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 $Ω_q^{(s)}(m|n,\textbf{r})$ and $Ω_q^{(s)}(m|n)$ 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 $\mathcal D_q(m|n)=Ω_q(m|n)\otimes \sqcap_q(m|n)$ 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. |
| title | Quantum Supersymmetries (II): Loewy Filtrations and Quantum de Rham Cohomology over Quantum Grassmann Superalgebra |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2410.04901 |