We study the first and second cohomology groups of the [Formula: see text]-algebras of the universal unitary and orthogonal quantum groups [Formula: see text] and [Formula: see text]. This provides valuable information for constructing and classifying Lévy processes on these quantum groups, as pointed out by Schürmann. In the case when all eigenvalues of [Formula: see text] are distinct, we show that these [Formula: see text]-algebras have the properties (GC), (NC) and (LK) introduced by Schürmann and studied recently by Franz, Gerhold and Thom. In the degenerate case [Formula: see text], we show that they do not have any of these properties. We also compute the second cohomology group of [Formula: see text] with trivial coefficients — [Formula: see text] — and construct an explicit basis for the corresponding second cohomology group for [Formula: see text] (whose dimension was known earlier, thanks to the work of Collins, Härtel and Thom).

中文翻译：

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用于在与通用紧致量子群相关的代数上生成泛函的 Lévy-Khintchine 分解

我们研究了[公式：见文本]-通用酉和正交量子群[公式：见文本]和[公式：见文本]的代数的第一和第二上同调群。正如 Schürmann 所指出的，这为在这些量子群上构建和分类 Lévy 过程提供了有价值的信息。在 [Formula: see text] 的所有特征值都不同的情况下，我们证明这些 [Formula: see text]-代数具有 Schürmann 引入并由 Franz 最近研究的性质 (GC)、(NC) 和 (LK) ，格霍尔德和汤姆。在退化的情况下[公式：见正文]，我们证明它们不具有任何这些属性。我们还计算了 [Formula: see text] 的第二个具有平凡系数的上同调群 - [Formula: see text] - 并为 [Formula: