It is hard to explain the value of higher math, but it's worth it.
Perhaps a better answer to "What are Lie algebras?" is to respond in terms that mean something to your audience. Avoid words like "vector" and "combinatorics".
Instead use metaphors. Like Rubix cubes. Tell them Lie algebra is a way to solve Rubix cubes faster. And also other similar puzzles that are way harder than Rubix cubes. It's true enough for casual conversation and probably more interesting than a vaguer answer.
I realize that using words like "vector" and "combinatorics" are poor choices. This is part of the problem. It is difficult to come up with a good metaphor that is simultaneously interesting and meaningful. I think the Rubik's cube is a great example. Thanks, I'll be using that in the future.
"There are a lot of hard problems out there, sometimes they are toys like the Rubik's Cube and sometimes its quantum physics. I work on the math that let's people solve them."
Every software developer has had this conversation themselves.
>"Telling them what it can do or what it is used for is almost always the best way to talk about something you do with the totally uninitiated."
Although I'd guess for most researchers in Maths what it can be used for is many years away (if it actually has a "practical use") and what it can do is far too esoteric for the layman.
Perhaps a better answer to "What are Lie algebras?" is to respond in terms that mean something to your audience. Avoid words like "vector" and "combinatorics".
Instead use metaphors. Like Rubix cubes. Tell them Lie algebra is a way to solve Rubix cubes faster. And also other similar puzzles that are way harder than Rubix cubes. It's true enough for casual conversation and probably more interesting than a vaguer answer.