Borel subgroup

Imagine you have a big group of toys to play with, but you want to make smaller groups to play different games. A Borel subgroup is like one of those smaller groups, but it only has the toys that can be lined up in a straight line - like blocks, cars, or figures. The other toys that can't line up in a straight line - like balls, stuffed animals, or oddly shaped figures - aren't in the Borel subgroup.

Similarly, in math, a Borel subgroup is a smaller group of matrices (which are like mathematical toys) that can be written as a combination of matrices that only move points in a straight line. These matrices are called upper triangular matrices, and they're like the toys that can be lined up in a straight line. The other matrices that move points around in more complicated ways aren't in the Borel subgroup.

So just like you could make different groups of toys to play different games, mathematicians can study different subgroups of matrices to solve different problems in algebra and geometry!