I'm reading a paper by Mitchell Rothstein (Sheaves with connections on abelian varieties) and he defines $$ \mathfrak{g} = H^1(X, \mathcal{O}_X) $$ where $X$ is some ableian variety over $k$ (algebraically closed) and then states $$ End(\mathfrak{g}^*) \simeq Ext(\mathcal{O}_X,\mathfrak{g}^* \otimes \mathcal{O}_X). $$ I believe $\mathfrak{g}^* = Hom_{\Gamma(x)}(\mathfrak{g},\Gamma(X))$, but Im not certain.
Either way, if anyone could show me the isomorphism above I would be very grateful, I think its probably simple so even just a hint would be great.
I mean these are certainly isomorphic but you want a canonical one? Is this supposed to be respecting some structure I'm not seeing?
There are canonical isomorphisms
$$\text{End}(\mathfrak{g}^\ast)\cong \mathfrak{g}\otimes\mathfrak{g}^\ast$$
and
$$\text{Ext}(\mathcal{O}_X,\mathfrak{g}^\ast\otimes\mathcal{O}_X)=\text{Ext}(\mathcal{O}_X,\mathcal{O}_X)\otimes\mathfrak{g}^\ast=H^1(X,\mathcal{O}_X)\otimes\mathfrak{g}^\ast=\mathfrak{g}\otimes\mathfrak{g}^\ast$$
I don't know what the purpose of all of this is though.
Hopefully that helps.