Rational cohomology and Zariski dense subgroups of solvable linear algebraic groups

Milana Golich, Mark Pengitore
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Abstract

In this article, we establish results concerning the cohomology of Zariski dense subgroups of solvable linear algebraic groups. We show that for an irreducible solvable $\mathbb{Q}$-defined linear algebraic group $\mathbf{G}$, there exists an isomorphism between the cohomology rings with coefficients in a finite dimensional rational $\mathbf{G}$-module $M$ of the associated $\mathbb{Q}$-defined Lie algebra $\mathfrak{g_\mathbb{Q}}$ and Zariski dense subgroups $\Gamma \leq \mathbf{G}(\mathbb{Q})$ that satisfy the condition that they intersect the $\mathbb{Q}$-split maximal torus discretely. We further prove that the restriction map in rational cohomology from $\mathbf{G}$ to a Zariski dense subgroup $\Gamma \leq \mathbf{G}(\mathbb{Q})$ with coefficients in $M$ is an injection. We then derive several results regarding finitely generated solvable groups of finite abelian rank and their representations on cohomology.
可解线性代数群的有理同调与扎里斯基密集子群
在这篇文章中,我们建立了有关可解线性代数群的 Zariskidense 子群同调的结果。我们证明,对于不可还原的 $\mathbb{Q}$ 定义线性代数群 $\mathbf{G}$、的无穷维有理 $\mathbf{G}$ 模块 $M$ 的同调环之间存在同构。定义的李代数 $\mathfrak{g_\mathbb{Q}}$ 和 Zariski 二重群 $\Gamma \leq \mathbf{G}(\mathbb{Q})$ 满足它们与 $\mathbb{Q}$ 分离的最大环离散相交的条件。我们进一步证明,在有理同调中,从 $\mathbf{G}$ 到扎里斯基密集子群 $\Gamma \leq \mathbf{G}(\mathbb{Q})$ 的系数在 $M$ 中的限制映射是一个注入。然后,我们推导出关于有限无性秩的有限生成可解群及其表征同调的几个结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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