Borel subgroup

A Borel subgroup is a specific kind of subgroup within a larger mathematical structure called a Lie group or algebraic group. Think of it as a "building block" that captures certain symmetrical properties of the whole group. It is maximal among connected, solvable subgroups—meaning it is as large as possible without losing a property called solvability, which relates to how the subgroup's structure can be broken down into simple parts. Borel subgroups play a key role in understanding the overall symmetry and structure of more complex mathematical objects, especially in algebraic geometry and group theory.