Link between two definitions of Loop algebra

Question

Good morning,

I’m currently reading 「Conformal Field Theory and Topology」, Toshitake Kohno.

Here $\mathfrak{g}$ is a Lie algebra associated to a Lie group $G$

The loop group $LG$ is defined as the set of smooth maps from $\mathbb{S}^{1} = \{z \in \mathbb{C}, |z| =1 \}$ to $G$, the product being defined by $\gamma_{1}・\gamma_{2}(z) = \gamma_{1}(z)\gamma_{2}(z) $

The loop algebra $L\mathfrak{g} $ is considered to be the complexification of the Lie algebra associated to $LG$

In the book, professor Kohno sets $L\mathfrak{g} = \mathfrak{g}\otimes \mathbb{C}((t))$

And I struggle to make the link with the geometrical definition $L \mathfrak{g} = \mathfrak{g} \otimes C^{\infty}(\mathbb{S}^{1}) $

Could you please enlighten me about the link between these two definitons ?

Thanks for your help,

Dearly

Answer 1

I guess it's just Fourier transform on $C^{\infty}(S^1)$. For $f\in C^{\infty}(S^1)$, $f=\sum_i a_i \exp(inz)$, Then $t=\exp(iz)$.

I'm not an expert in analysis, not sure whether there're equivilent or other subtlety.