Cartan matrix

A Cartan matrix is a way of writing down how a group of objects, called root elements, are related to each other. To understand this, let's pretend we have a bunch of toy blocks, each with a different shape and color. We want to know how these blocks are related, so we line them up and figure out which blocks can transform into other blocks by rotating or flipping them.

In the same way, a Cartan matrix helps us understand how root elements can be transformed into each other, but instead of rotating or flipping, we use a mathematical operation called multiplication. We write down the transformations between each root element as a matrix. Each row and column represents a different root element, and the number in the intersection of the row and column tells us how many times we need to multiply one root by another to get a third.

This might seem complicated, but it helps us understand how different root elements are related and how they can be combined to create new elements. Just like how we can use toy blocks to build different structures, we can use root elements to form complex structures in mathematics and physics.