论文标题
关于魔术广场和简单谎言代数的“特殊统一”的观点
A perspective on the Magic Square and the 'special unitary' realizations of simple Lie algebras
论文作者
论文摘要
本文包含在IFWGP2012上提供的迷你课程“空间：透视图”的最后部分。在这里，我处理了迷你课程的部分，该部分集中在与简单的真实谎言群体相关的分类问题上。我回顾了魔术广场“ a la freudenthal”的原始介绍，重点是四个规范的分区代数$ {\ mathbb r}，{\ mathbb c}，{\ mathbb h}，{\ mathbb h}，{\ mathbb o} $。然后，我探讨了某些代数$ {\ mathbb k} $或张紧产品$ {\ Mathbb k} _1 \ otimes {\ Mathbb k} _2 $的可能性，我认为只有适当的设置可以使某个构造的规定允许，但我认为不得不适当的设置可以使某个构造的规定，因此我探索了某些代数$ {\ Mathbb k} $上的“特殊统一”的可能性。四季度和八元。这样，我们获得了一个“大魔法广场”，并填写了涵盖该方案中所有简单真实谎言代数的所有真实形式所需的细节。本文以所有对“特殊统一”（或仅在$ n = 2 $时仅统一的''代数的实现的完整列表结束，这是两个张张量的产品，包括两个$*$*$ - 代数$ {\ Mathbb k} _1，{\ Mathb k} _2 $，在所有情况下是$ { c}，{\ mathbb h}，{\ mathbb o} $及其拆分版本作为集合，以$*$ - 共轭赋予它们，通常但并非总是自然是天然复杂，Quaternionic或Octonionionic共轭。
This article contains the last part of the mini-course `Spaces: a perspective view' delivered at the IFWGP2012. Here I deal with the part of the mini-course which centers on the classification questions associated to the simple real Lie groups. I review the original introduction of the Magic Square `a la Freudenthal', putting the emphasis in the role played in this construction by the four normed division algebras ${\mathbb R}, {\mathbb C}, {\mathbb H}, {\mathbb O}$. I then explore the possibility of understanding some simple real Lie algebras as `special unitary' over some algebras ${\mathbb K}$ or tensor products ${\mathbb K}_1\otimes {\mathbb K}_2$, and I argue that the proper setting for this construction is not to confine only to normed division algebras, but to allow the split versions of complex, quaternions and octonions as well. This way we get a `Grand Magic Square' and we fill in the details required to cover all real forms of simple real Lie algebras within this scheme. The paper ends with the complete lists of all realizations of simple real Lie algebras as `special unitary' (or only `unitary' when $n=2$) over some tensor product of two $*$-algebras ${\mathbb K}_1, {\mathbb K}_2$, which in all cases are obtained from ${\mathbb R}, {\mathbb C}, {\mathbb H}, {\mathbb O}$ and their split versions as sets, endowing them with a $*$-conjugation which usually but not always is the natural complex, quaternionic or octonionic conjugation.
