量子除幂代数上的Loewy滤子与量子de Rham上同调
Loewy filtration and quantum de Rham cohomology over quantum divided power algebra
摘要：
研究了量子除幂代数$\mathcal{A}_q(n)$在\cite{HU}中的不可分解子模结构及其截断对象$\mathcal{A}_q(n, \bold m)$ .建立了一种“交织提升”方法，证明了柱脚为非单的模的不可分解性。对所有的齐次子空间$\mathcal{A}^{(s)}_q(n)$或$\mathcal{A}_q^{(s)}(n, \bold m)$描述了Loewy滤子，确定了Loewy层和维数.证明了这些不可分解模的刚性。对于一类不可分解的$\mathfrak{u}_q(\mathfrak{sl}_n)$ -模，从我们的实现模型中导出了一个有趣的组合恒等式.同时，通过定义适当的$q$ -微分，构造了$\mathcal{A}_q(n)$上的量子Grassmann代数$Ω_q(n)$和量子de Rham复形$(Ω_q(n), d^\bullet)$，以及它的子复形$(Ω_q(n,\bold m), d^\bullet)$ .对于后者，相应的量子de Rham上同调模被分解为若干符号平凡$\mathfrak{u}_q(\mathfrak{sl}_n)$ -模的直和.
英文摘要：
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 $Ω_q(n)$ over $\mathcal{A}_q(n)$ is constructed, together with the quantum de Rham complex $(Ω_q(n), d^\bullet)$ via defining the appropriate $q$-differentials, and its subcomplex $(Ω_q(n,\bold m), d^\bullet)$. 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.
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