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I asked a similar question to Daniel Huybrechts some time ago, in the form of "If I have a derived equivalence between two varieties, what is this telling me about the relation between the two varieti …

For me, the exterior algebra is the free polynomial algebra in anti-commutative variables. Of course, this begs the question, why do anti-commutative variables come up so much? As a homological alge …

Let $\mathbb{H}$ denote the quaternions. Let $G$ be a group, and define a representation of $G$ on $\mathbb{H}^n$ in the natural way; that is, its a map $\rho:G\rightarrow Hom_{\mathbb{H}-}(\mathbb{ …

I enjoyed Pavel Etingof's lecture notes for his representation theory class, which can be found here: http://www-math.mit.edu/~etingof/replect.pdf (there is a link to it on his website) They move fast …

Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$, let $U$ be its enveloping algebra, and let $M$ be a $\mathfrak{g}$-module (not necessarily finite dimensional). Call the invariant dim …

I believe an approach that works is to define the Chevalley-Eilenberg complex as a kind of `Koszul complex over the ring of functions'. The enveloping algebra $U$ is relatively quadratic over the rin …