# Tagged Questions

56 views

### how can I show $H^1(g , Hom_C(g,M))=0$?

For a simple Lie algebra $g$ and a finite dimensional vector space $M$ with a trivial $g-$action, how can I show $H^1(g , Hom_C(g,M))=0$?
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### Corestriction map in lie algebra cohomology

Given a lie algebra $\mathfrak{g}$ over a field $k$, we can define the cohomology groups of $\mathfrak{g}$ as follows: $$H^n(\mathfrak{g},k):=\mathrm{Ext}_{U(\mathfrak{g})}^n(k,k)$$ where ...
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### Weight space for a finite-dimensional $\mathfrak{g}-$module $M$.

Let $\mathfrak{g}$ a semisimple Lie algebra, $M$ finite-dimensional $\mathfrak{g}-$module, $\mu\in\mathfrak{h}^*_{\mathbb{Z}}$ and $s_i$ simple reflection such that ...
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### Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
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### Motivation?: Lie algebra and algebraic group Cohomology

This is just an a-priori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
### How to prove $H^2(\mathfrak{g}, J(\mathfrak{g}))\neq0$, where $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$?
$\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over a field $k$ with $\mathrm{char}k=0$. $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$. That is, the kernel of ...