We find and classify all bialgebras and Hopf algebras or `quantum groups' of dimension $\le 4$ over the field $\Bbb F_2=\{0,1\}$. We summarise our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras, and among them 25 Hopf algebras with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover self-dual and resembles a digital version of $u_q(sl_2)$. We also find a unique self-dual Hopf algebra in one anyonic variable $x^4=0$. For all our Hopf algebras we determine the integral and associated Fourier transform operator, viewed as a representation of the quiver. We also find all quasitriangular or `universal R-matrix' structures on our Hopf algebras. These induce solutions of the Yang-Baxter or braid relations in any representation.

中文翻译：

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数字量子群

我们在 $\Bbb F_2=\{0,1\}$ 域上找到并分类所有维度为 $\le 4$ 的双代数和 Hopf 代数或“量子群”。我们将我们的结果总结为一个箭袋，其中顶点是不等价代数，每个不等价双代数或 Hopf 代数都有一个箭头，这些代数是从箭头源处的代数和箭头目标处的代数对偶构建的。有 314 个不同的双代数，其中有 25 个 Hopf 代数，从一个顶点到另一个顶点最多只有一个。我们找到了一个独特的最小非对易和非可交换的，而且是自对偶的，类似于 $u_q(sl_2)$ 的数字版本。我们还在一个任意变量 $x^4=0$ 中找到了一个独特的自对偶 Hopf 代数。对于我们所有的 Hopf 代数，我们确定积分和相关的傅立叶变换算子，被视为箭袋的代表。我们还在 Hopf 代数上找到了所有拟三角或“通用 R 矩阵”结构。在任何表示中，这些都会导致 Yang-Baxter 或辫子关系的解。