I would not be surprised by downvotes since this question is at the same time very naïve, very vague and might be asking about things well known for decades. Specifically vagueness comes from the word “special” in the title. As for that third weakness, I have a weak defence against it by choosing the reference request tag.

The starting point is the trivial observation that for a finite-dimensional semisimple Lie algebra $mathfrak g$, the algebra $mathfrak g(t,t^{-1})$ of Laurent polynomials with coefficients in $mathfrak g$ can be viewed as the algebra of polynomial global sections of the trivial vector bundle with fibre $mathfrak g$ over the punctured plane $mathbb C^times$.

My first question is whether it is possible to somehow modify this bundle in such a way that the global sections will turn out to be the affine Lie algebra $hat{mathfrak g}$ (the universal central extension of $mathfrak g(t,t^{-1})$); similarly for twisted versions of $hat{mathfrak g}$.

Second question – is it known what does one obtain with nontrivial vector bundles, and with some projective curve in place of $mathbb C^times$? I have no idea whether one indeed obtains a Lie algebra at all or something different. Does the algebraic group structure of $mathbb C^times$ have any significance in this context? Accordingly, is it important whether I take an elliptic curve or curves of all genera give more or less similar results?

Finally, I remember that although sections of the tangent bundle form a Lie algebra, it is not a Lie algebra bundle in the sense that fibres do not have any Lie algebra structure. Still, if I am not mistaken, the Virasoro algebra is the universal central extension of the algebra of global sections of a tangent bundle, right? Again, is there some modified form of the tangent bundle such that the Virasoro algebra would be global sections of this modified bundle? And again, is the group structure significant here? What are algebras of sections of tangent bundles over projective curves, in particular, over elliptic curves? Do they have nontrivial central extensions?