To truly understand Lie groups, you need to know a lot of abstract algebra, topology, and differential geometry. Algebraic geometry and a thorough understanding of linear algebra also help.

But since you’re probably not a professional mathematician, I’ll try to explain the essence of a Lie group without using a lot of vernacular.

Algebraists like to study objects that have an algebraic structure. One such object is a group. A group has elements (think of them as numbers), and we can perform operations on these elements such as addition or multiplication, and sometimes even “division”.

Topologists, on the other hand, are interested in objects that have a more geometric structure: these objects are called topological spaces. A topological space is distinguished by its shape. Two topological spaces are essentially the same if one can be turned into the other by reshaping it without creating any holes or gaps.  The classic example is of a doughnut and a coffee mug: to a topologist, these two objects are the same.  It is conceivable that you could reshape the doughnut by pinching and stretching it so that it looks like a mug.

A Lie group is an object that has both a topological structure (a shape) and an algebraic structure (a multiplication-like operation), so that the two structures interact nicely. By ‘nicely’, I mean that we can essentially do calculus on the structure.

Now that you know 1 minute of information more on Lie groups than any of your friends do, you should (a) talk about them at parties, and (b) ask me any questions you might have. I’d love to talk more about Lie theory or help you find resources so that you can learn on your own.