I am teaching Calc I, for the first time, and I haven't seriously revisited the subject in quite some time. An interesting pedagogy question came up: How misleading is it to regard $\frac{dy}{dx}$ as a fraction? There is one strong argument against this: We tell students that $dy$ and $dx$ mean ...

For complex semisimple Lie algebras, the maximal dimension of an abelian subalgebra was determined by Mal'cev in 1945. For $E_7$, for example, it is $27$, and is the radical of the $E_6$ parabolic. What about in characteristic $p$ for $p>0$? I suspect the answer is nearly, but not quite, the sam...

This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers. Let $\mathfrak g$ be a Lie algebra over $k$. One can define the universal enveloping algebra $U\mathfrak g...