Then $U^{*}\otimes U$ is embedded into $C^{0}(G)$ as a $G\otimes G$-module. A theorem of Peter-Weyl type states that ${\cal{J}}(G,{\C})$ is a dense subspace of $C^{0}(G)$ and that ${\cal{J}}(G,{\C})$ is a direct sum of $U^{*}\otimes U$, where $U$ runs through a complete set of representatives of equivalence classes of finite-dimensional irreducible $G$-modules.

To start vertex operator algebra theory, let us recall from \[FLM\] and \[FHL\] (see also \[B\]) the definition of vertex operator algebra, which will be the official definition for this paper. [*A vertex operator algebra*]{} is a ${\Z}$-graded vector space $V=\coprod_{n\in {\Z}}V_{(n)}$ such that $\dim V_{(n)}<\infty$ for all $n\in {\Z}$ and $V_{(n)}=0$ for $n$ sufficiently small, equipped with a bilinear “vertex multiplication” $$\begin{aligned}
Y(\cdot,x)\cdot: & &V\otimes V\rightarrow V((x))\nonumber\\ 
& &u\otimes v\mapsto Y(u,x)v=\sum_{n\in {\Z}}u_{n}vx^{-n-1}\end{aligned}$$ such that for $m, n, k\in {\Z},\;u\in V_{(k)}$, $$\begin{aligned}
\label{egrad}
u_{n}V_{(m)}\subset V_{(m+k-n-1)},\end{aligned}$$ and such that the [*Jacobi identity*]{} holds for $u,v\in V$: $$\begin{aligned}
\label{e0jacobi}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
Y(u,x_{1})Y(v,x_{2})
-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
Y(v,x_{2})Y(u,x_{1})\nonumber\\
& &=x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
Y(Y(u,x_{0})v,x_{2}).\end{aligned}$$ It is equipped with a vector ${\bf 1}$, called the [*vacuum vector*]{}, such that for $v\in V$, $$\begin{aligned}
& &Y({\bf 1},x)v=v,\\
& &Y(v,x){\bf 1}\in V[[x]]\;\;\;\mbox{ and  }
\;\lim_{x\rightarrow 0}Y(v,x){\bf 1}=v.\end{aligned}$$ It is also equipped with a vector $\omega$, called the [*Virasoro element*]{}, such that 