What does the notation $U\mathfrak{sl}_2$ mean, and why is the $U$ written in a different typeface to the $\mathfrak{sl}$? A representation theory homework problem asks me to determine the finite dimensional irreducible representations and the finite dimensional indecomposable representations of $U\mathfrak{sl}_2$. I suppose this is standard notation, but I'm not sure what is meant by it.

Question. What does the notation $U\mathfrak{sl}_2$ mean, and why is the $U$ written in a different typeface to the $\mathfrak{sl}$?
What's the general pattern here, and how do you puzzle these things out yourself?


This probably means the universal enveloping algebra, though it is unusual to see it written without brackets (I would usually expect it to be written as $U(\mathfrak{sl}_2)$).
