How do I construct the Tits-Freudenthal magic square?

Thanks to Mia Hughes and John Huerta for the helpful discussions on this topic.

I am here taking another quick jab at trying to understand the construction of the Tits-Freudenthal Magic square. Let’s see if we can get into Vinberg’s mindset when he wrote down Vinberg’s construction.

Let’s say we knew the following theorem: $$\text{ the derivations of  } \mathcal{J}_3(\mathbb{O}) = f_4$$ We want to write down derivations of other algebras, $\mathbb{O} \otimes_{\mathbb{R}} \mathbb{D}$, where $\mathbb{D}$ is a division algebra.

Let’s see how we might derive the fact that $$\text{der}(\mathcal{J}_3(\mathbb{A})) \simeq a_3(\mathbb{A}) \oplus \text{der}(\mathbb{A})$$

where $a_3$ denotes the 3×3 trace-free antisymmetric matrices, and $$\text{der}(A) = \text{Lie}(\text{Aut}(A))$$

Continue reading How do I construct the Tits-Freudenthal magic square?

What does an algebraic integer have to be?

What does an integer have to be?

  • No matter how you extend $\mathcal{Q}$, the integers which lie in $\mathcal{Q}$ must lie in $\mathcal{Z}$.
  • If $\alpha$ is an integer, then so are its conjugates.
  • The sums and products of integers are also integers.

From this we may describe what an algebraic integer must be.

Start with a root $\alpha$.

Look at all of it’s conjugates. Continue reading What does an algebraic integer have to be?