Difference between two definitions of affine Lie algebras

Question

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, we have the notion of affinization of $\mathfrak{g}$, which is the central extension of the corresponding loop algebra.

However, we can see two slightly different definitions of loop algebras in literature. For a commutative $\mathbb{C}$-algebra $R$, we can make $\mathfrak{g}\otimes R$ a Lie algebra by taking $$[x\otimes f,y\otimes g]=[x,y]\otimes fg.$$ The first definition of loop algebra considers $R=\mathbb{C}[t,t^{-1}]$ while the second considers $R=\mathbb{C}((t))$.

I wonder if there is a significant difference between these two definitions. What are they good for in specific situations?

Answer 1

In my experience, the Laurent polynomial construction is more suited to the "algebraic" aspects of the theory--in particular, if the power of $t$ corresponds to the coefficient of $\delta$ in the root space decomposition, the construction over $\mathbb{C}[t, t^{-1}]$ forces all elements to be a finite sum of root vectors. That of course will get lost if you work over $\mathbb{C}((t))$, since you now have infinite formal sums. But, the representation theory you build for them (highest weight integrable modules, etc) all works out essentially the same way.

But, if you are interested in geometric aspects like affine flag varieties, affine Grassmannians, conformal blocks on curves and the Verlinde formula, etc., it is better to work over the "positive completion" by using $\mathbb{C}((t))$. From this perspective it is common to treat $\mathbb{C}((t))$ as functions locally around a punctured disk/curve with local coordinate $t$.