A new representation of the Poincaré groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincaré generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators. An extended multivector calculus is introduced (allowing a possible sign change of the metric or of the exterior product). The Poincaré groups are formulated as a dual extension of hyperquaternion pseudo-orthogonal groups. The canonical decomposition and the invariants are discussed. As concrete example, the 4D Poincaré group is examined together with a numerical application. Finally, the hyperquaternion representation is compared to the quantum mechanical one. Potential applications include in particular, moving reference frames and computer graphics.

通过对偶超四元数，在n维上发展了庞加莱族的新表示形式，超四元数被定义为四元数代数（或其子代数）的张量积。这种形式主义产生了唯一定义的乘积和庞加莱生成器的简单表达式，具有直接的物理意义，揭示了与矩阵或运算符无关的代数结构。引入了扩展的多矢量演算（允许度量或外部乘积的可能符号更改）。庞加莱基团被定义为超四元数拟正交基团的双重扩展。讨论了规范分解和不变量。作为具体示例，4 D庞加莱群与数值应用一起被检验。最后，将超四元数表示与量子力学表示进行了比较。潜在的应用尤其包括移动参考系和计算机图形。