The paper gives a new representation of conformal groups in n dimensions in terms of hyperquaternions defined as tensor products of quaternion algebras (or a subalgebra thereof). Being Clifford algebras, hyperquaternions provide a good representation of pseudo-orthogonal groups such as \(O(p+1,q+1)\) isomorphic to the nD conformal group with \(n=p+q.\) The representation yields simple expressions of the generators, independently of matrices or operators. The canonical decomposition and the invariants are discussed. As application, the 4D relativistic conformal group is detailed together with a worked example. Finally, the formalism is compared to the operator representation. Potential uses include in particular, conformal geometry, computer graphics and conformal field theory.

该论文根据定义为四元数代数（或其子代数）的张量积的超四元数给出了n维共形群的新表示。作为 Clifford 代数，超四元数提供了伪正交群的良好表示，例如\(O(p+1,q+1)\)同构到nD共形群，其中\(n=p+q.\)表示产生生成器的简单表达式，独立于矩阵或运算符。讨论了规范分解和不变量。作为应用，4D相对论共形群与一个工作示例一起详细说明。最后，将形式主义与运算符表示进行比较。潜在用途特别包括共形几何、计算机图形学和共形场论。