论文标题
数字量子组
Digital quantum groups
论文作者
论文摘要
我们在字段$ \ bbb f_2 = \ {0,1 \} $上找到并分类所有Bialgebras和Hopf代数或尺寸$ \ le 4 $的“量子组”。我们将结果总结为颤抖的结果，其中顶点是不相等的代数，并且每个不相等的bialgebra或Hopf代数都有一个由代数构建的箭头源的代数和箭头目标的代数双重箭头。有314个截然不同的双gebras，其中25个霍普夫代数最多从一个顶点到另一个顶点。我们发现了一个独特的最小非公务和非协商性，它是自偶联的，类似于$ u_q（sl_2）$的数字版本。我们还在一个anyonic变量$ x^4 = 0 $中找到了独特的自dual Hopf代数。对于我们所有的HOPF代数，我们确定了积分和相关的傅立叶变换操作员，被视为颤抖的表示。我们还发现了我们的Hopf代数上的所有Quasitriangular或“通用R-Matrix”结构。这些在任何表示形式中都诱导了杨巴克斯特或辫子关系的解决方案。
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.
