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 ${\mathcal{A}}_q(n)$ defined in \cite{HU} and its truncated objects ${\mathcal{A}}_q(n,\mathbf{m})$. 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 ${\mathcal{A}}_q^{(s)}(n)$ or ${\mathcal{A}}_q^{(s)}(n,\mathbf{m})$, 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 $({\Omega }_q(n,\mathbf{m}),d^{\bullet })$. 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.

26 pages

References in corpus (1)

• Localization of modules for a semisimple Lie algebra in prime characteristic

Cited by in corpus (2)

• Quantum (dual) Grassmann superalgebra as ${\mathcal{U}}_q(\mathfrak{gl}(m|n))$-module algebra and beyond
• Realization of $U_q({\mathfrak{sp}}_{2n})$ within the Differential Algebra on Quantum Symplectic Space