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Perhaps part of the confusion is that you didn't actually discuss triangular decompositions at all (that certainly confused me at first). A triangular decomposition for a Cherednik algebra comes from …

A very important example is the quotient of $U(\mathfrak{g})$ (where $\mathfrak{g}$ is a simple complex Lie algebra) by the central elements killing a finite dimensional representation. This has a un …

I would argue that this notion doesn't have one natural generalization. One obvious one is what I would call an $R$-bimodule algebra, that is, an algebra which is an R-bimodule in such a way that lef …

This is extremely false: for generic $\mathbf{c}$, the algebra $H_{0,\mathbf{c}}(W)$ is Morita equivalent to the functions on Calogero-Moser space, a finite dimensional smooth affine variety, and cate …

This is essentially Nakayama's lemma (not literally, but the same proof). More generally, the result is that a graded module map $\phi\colon N\to M$ is surjective if and only if the induced map $N\to …

Yes, this is true; it's essentially just a restatement of Artin-Wedderburn. All you need to do is note that by Artin-Wedderburn, a finite dimensional algebra with trivial Jacobson radical is a sum of …

I assume you're allowing A to be non-commutative. In this case, things can go wrong in all kinds of ways. For example, all Verma modules have endomorphisms given by $\mathbb{C}$, but loads of those …