论文标题
使用Quaternion的$ W \ left（f_4 \ right）$和$ w \ left（e_8 \右）$的特殊Weyl Group $ w \的构造，使用Quaternion和16维欧几里得空间中的晶格
A construction of Exceptional Weyl Group $W\left(F_4\right)$ and $W\left(E_8\right)$ using Quaternion, and the lattice in 16-dimensional Euclidean space
论文作者
论文摘要
据提到，交替组的$ 2 \ cdot a_4 $是次级$ 2 \ cdot a_4 $作为Quaternion的亚代词，而整数和半智能者称为Hurwitz Quaternionic Integers $ \ MATHSCR {H MATHSCR {h} n by J.H.Conway和Neil neil neil J.A. A. A. A. A. A. Sloane。在本文中，我遵循了这本书，并扩展了整数和半智商的四元组，以具有双重性，并证明了它的亚代词与杰出的Weyl Group $ w（F_4）$同构。我还发现了一种构建$ 16 $维晶格$λ_{16} $的方法，这似乎与称为barnes-wall lattice $λ_ {\ text {\ barnes-wall}} $的晶格是同构的，目前认为这是非常密集的（尽管这仍然是非常密集的（尽管仍然是用来讨论的）。最后，我简要提及如何使用八元和双季节来构建一个杰出的Weyl oft $ w（e_8）$。
It is mentioned that there is a subalgebra isomorphic to the alternating group $2 \cdot A_4$ as a subalgebra of the Quaternion over integers and half-integers called Hurwitz quaternionic integers $\mathscr{H}$ in the book by J.H.Conway and Neil J. A. Sloane. In this paper, I have followed this book and extended Quaternion over integers and half-integers to have duality, and proved that a subalgebra in it isomorphic to Exceptional Weyl group $W(F_4)$. I have also found a method of constructing the $16$-dimensional lattice $Λ_{16}$ which seems to be isomorphic to the lattice called the Barnes-Wall lattice $Λ_{\text {Barnes-Wall }}$, which is currently considered to be very dense (although this remains to be discussed) using the Dual Quaternion. Lastly, I briefly mention how to construct an exceptional Weyl group $W(E_8)$ using an Octonion and Dual Quaternion.
