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You don't explicitly say your representation is complex but I think your example shows that this is the case you're interested in. If so, then $V_0$ has a $G$-invariant complement $V_1$ by Maschke's T …

This looks like the kind of thing that one might be able to prove by filtering $U$ by word-length over $g$ and then passing to the associated graded ring. The idea is that this would reduce the proble …

This is only an attempt at answering question 2. If $G$ is a group with a non-trivial finite dimensional representation $V$ then it has a finite dimensional representation on a complex Hilbert space $ …

Yes, this is basically correct. A character $\chi\colon G\to \mathbb{C}^\times$ [note it should take values in the multiplicative group of non-zero complex numbers] induces an algebra homomorphism $\t …

Since the sum of all the trivial subrepresentations of $U(\mathfrak{g})$ with the adjoint action is its centre, if you could do this then the centre must have a monomial basis but it does not. For exa …

The answer is now yes, I think http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.0896v2.pdf Edit: as requested: http://arxiv.org/abs/1011.0896

To fill out my comment with a partial answer to the question. Note that given any (left) $\mathfrak{g}$-module $V$ one can compute $H_i(\mathfrak{g},V)$ as $\mathrm{Tor}_i^{U(\mathfrak{g})}(\mathbb{C …

This is an attempt to prove the (refined) conjecture I made in the comments of my previous answer. Let $\mathfrak{g}$ be a f.d soluble Lie algebra over $\mathbb{C}$. Let $\mathfrak{n}$ be its derived …

Proposition 1.2.9 of http://math.columbia.edu/~scautis/dmodules/hottaetal.pdf explains that if $M$ and $N$ are both left $D$-modules and $M'$ and $N'$ are both right $D$-modules then (a) $M\otimes_{ …

If $A$ happens to be (left) Noetherian then to show $f$ makes $B$ a faithfully flat right $A$-module it is enough to check that $B\otimes_A S\neq 0$ for every simple (left) $A$-module $S$ since, in th …

For motivation I would advise starting with Brauer's original paper. You'll need a JSTOR login though: https://www.jstor.org/stable/1968843?origin=crossref&seq=1#metadata_info_tab_contents